An Invitation to Coarse Groups

Arielle Leitner and Federico Vigolo

July 9, 2021

Abstract In this monograph we lay the fundations for a theory of coarse groups and coarse actions. Coarse groups are group objects in the category of coarse spaces, and can be thought of as sets with operations that satisfy the group axioms “up to uniformly bounded error”. In the ﬁrst part of this work, we develop the theory of coarse homomorphisms, quotients, and subgroups, and prove that coarse versions of the Isomorphism Theorems hold true. We also initiate the study of coarse actions and show how they relate to the fundamental observation of Geometric Group Theory. In the second part we explore a selection of more specialized topics, such as the study of coarse structures on groups, and groups of coarse automorphisms. The main aim is to show how the theory of coarse groups connects with classical subjects. These include: natural connections with number theory; the study of bi-invariant metrics; quasimorphisms and stable commutator length; Out(Fn) and topological group actions. We see this text as an invitation and a stepping stone for further research.

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1 Contents

1 Introduction 5 1.1 Background and motivation...... 5 1.2 On this manuscript...... 8 1.3 List of ﬁndings I: basic theory...... 9 1.4 List of ﬁndings II: selected topics...... 14 1.5 Acknowledgements...... 19

I Basic Theory 20

2 Introduction to the coarse category 20 2.1 Some notation for subsets and products...... 20 2.2 Coarse structures...... 21 2.3 Controlled maps...... 24 2.4 The category of coarse spaces...... 26 2.5 Binary products...... 28 2.6 Equicontrolled maps...... 29

3 Properties of the category of coarse spaces 30 3.1 Monomorphisms and epimorphisms...... 30 3.2 Pull-back and push-forward...... 30 3.3 Controlled thickenings, asymptoticity and indexed partial covering...... 31 3.4 Coarse subspaces, images and quotients...... 32 3.5 Containments and intersections of coarse subspaces...... 34

4 Coarse Groups 37 4.1 The diagrammatic approach...... 37 4.2 The point-set approach...... 39 4.3 Making sets into coarse groups...... 42 4.4 Adapted and cancellative operations...... 45 4.5 Coarse groups are determined locally...... 48 4.6 Sets of neighbourhoods of the identity...... 50

5 Coarse homomorphisms, subgroups and quotients 52 5.1 Coarse homomorphisms...... 52 5.2 Coarse subgroups...... 58 5.3 Coarse quotients...... 61

6 Coarse actions 63 6.1 Basic deﬁnitions...... 63 6.2 Coarse action by conjugation...... 66 6.3 Coarse orbits...... 68 6.4 The Fundamental Observation of Geometric Group Theory...... 71 6.5 Quotient coarse actions...... 73 6.6 Coarse cosets spaces...... 74

2 7 Coarse kernels 79 7.1 Coarse preimages and kernels...... 79 7.2 The Isomorphism Theorems...... 81 7.3 A criterion for the existence of coarse kernels...... 84 7.4 Some comments and questions...... 87

II Selected topics 88

8 Coarse structures on set-groups 88 8.1 Bi-invariant metrics...... 88 8.2 Metric vs arbitrary coarsiﬁcations...... 90 8.3 Intrinsically bounded set-groups...... 90

9 Coarse structures on Z 92 9.1 Coarse Structures Generated by Topologies...... 92 9.2 Proﬁnite coarse structures...... 94 9.3 Coarse structures via Cayley graphs and other questions...... 96

10 On cancellation metrics 97 10.1 Computing the cancellation metric on free groups...... 98 10.2 Cancellation metrics on ﬁnitely generated set-groups and their subgroups...... 100

11 A quest for coarse groups that are not coarsiﬁed set-groups 102 11.1 General observations...... 102 11.2 A conjecture on coarse groups that are not coarsiﬁed set groups...... 104 11.3 A few more questions...... 106

12 On coarse homomorphisms and coarse automorphisms 106 12.1 Elementary constructions of coarse automorphisms...... 106 12.2 Coarse homomorphisms into Banach spaces...... 108 12.3 Hartnick-Schweitzer quasimorphisms...... 109 12.4 Coarse homomorphisms of groups with cancellation metrics...... 113

13 Proper coarse actions 116 13.1 Properness via bornologies...... 117 13.2 Proper actions are determined locally...... 117 13.3 Topological coarse actions...... 119

14 Spaces of controlled maps 121 14.1 Fragmentary coarse structures...... 121 14.2 The fragmentary coarse space of controlled maps...... 123 14.3 Coarse actions revisited...... 127 14.4 Frag-coarse groups of controlled transformations...... 129

III Appendices 132

3 A Categorical aspects of Coarse 132 A.1 Limits and colimits...... 132 A.2 The category of fragmentary coarse spaces...... 134 A.3 Enriched coarse categories...... 136 A.4 The pre-coarse categories...... 137

B Metric Groups 137 B.1 Groups in the 1-LipMet category...... 137 B.2 Groups in the LipMet category...... 138

Index 144

List of Symbols 146

4 1 Introduction

1.1 Background and motivation Coarse geometry. Coarse geometry is the study of the large scale geometric features of a space or, more precisely, of those properties that are invariant up to “uniformly bounded error”. This language allows us to formalize the idea that two spaces such as Zn and Rn look alike when seen “from very far away”. In other words, we imagine looking at spaces through a lens that blurs the picture so that all the ﬁne details become invisible and we only recognise the “coarse” shape of it. There are many reasons why one may wish to do this. For example, to use topological/analytic techniques on a discrete space like Z by identifying it with a continuous one like R or, vice versa, to use discrete/algebraic methods on continuous spaces. The language of coarse geometry allows us to study any space where there is a notion of “uniform boundedness”, topological or otherwise. It is convenient to start exploring this idea in the context of metric spaces because the metric provides us with an intuitive meaning of uniform boundedness. Namely, a family (Bi)i∈I of subsets of a metric space (X, dX) is uniformly bounded if there is an uniform upper bound on their diameters: diam(Bi) ≤ C for every i ∈ I. When this is the case, any two sequences (xi)i∈I and (yi)i∈I with xi, yi ∈ Bi coincide up to uniformly bounded error and are hence “coarsely indistinguishable”. Analogously, we say that two functions f1, f2 : Z → X coincide up to uniformly bounded error (a.k.a. they are close) if there exists C ∈ R so that dX( f1(z), f2(z)) ≤ C for all z ∈ Z. Again, we think of close functions as coarsely indistinguishable from one another. Given two metric spaces (X, dX), (Y, dY ) we only wish to consider the functions f : X → Y that preserve uniform boundedness, i.e. which send uniformly bounded subsets of X to uniformly bounded subsets of Y. Explicitly, this happens when there exists some increasing control function ρ: [0, ∞) → [0, ∞) 0 0 0 such that dY ( f (x), f (x )) ≤ ρ dX(x, x ) for every x, x ∈ X. When this is the case, we say that f : X → Y is a controlled function. Two metric spaces are coarsely equivalent if there exist controlled functions f : X → Y and g: Y → X whose compositions f ◦ g and g ◦ f are close to idY and idX respectively. For instance, we now see that Zn and Rn are coarsely equivalent via the natural inclusion and the integer ﬂoor function.

To provide some context and motivation, we note that the coarse geometry of metric spaces is the backbone of Geometric Group Theory. Any finitely generated group can be seen as a discrete geodesic metric space by equipping it with a word metric (equivalently, identifying the group elements with the vertices of a Cayley graph considered with the graph metric). These metrics depend on the choice of a finite generating set. However, they are unique up to coarse equivalence. As a consequence, it is possible to characterize algebraic properties of groups in terms of coarse geometric invariants of their Cayley graphs. Two outstanding such results are Gromov’s theorem showing that a finitely generated group is virtually nilpotent if and only if it has polynomial growth, and Stallings’ result that a group has a non trivial splitting over a finite subgroup if and only if its Cayley graph has more than one end. The Milnor–Švarc Lemma illustrates another fundamental feature of coarse equivalences. This result states that a finitely generated group G acting properly cocompactly on a metric space X is actually coarsely equivalent to X. For example, if G = π1(M) is the fundamental group of a compact Riemannian manifold, then the action by deck transformations induces a coarse equivalence between G and the universal cover Me. This important observation provides a seamless dictionary between algebraic properties of groups, their actions, and geometry of metric spaces. It is fair to say that this neat correspondence is one of the main raisons d’être of Geometric Group Theory: the ability to conflate groups and spaces on which they act proves to be a very powerful tool. To mention one exceptional achievement: Agol and Wise exploited this philosophy to prove the virtual Haken Conjecture, a long standing open problem in low-dimensional topology. Other important applications have a more analytic flavor. For instance, Roe proved a version of the

5 Atiyah–Singer index theorem for open manifolds and formulated a coarse version of the Baum–Connes conjecture. G. Yu proved that a huge class of manifolds satisfies this coarse Baum–Connes Conjecture and, as a consequence, the Novikov Conjecture. Remark 1.1.1. In Geometric Group Theory one generally speaks about quasi-isometries rather than coarse equivalences. However, it is an easy exercise to show that a controlled function between geodesic metric spaces is in fact coarsely Lipschitz (i.e. the control function ρ can be assumed to be affine ρ(t) = Lt + A). Since Cayley graphs are geodesic metric spaces, a coarse equivalence between finitely generated groups is indeed a quasi-isometry. As already mentioned, the coarse geometric perspective is useful beyond the realm of metric spaces. For instance, to identify a topological group with any of its cocompact lattices. In this general setting however, it is more complicated to understand what the meaning of “uniformly bounded” should be. Fortunately, Roe [Roe03] developed an elegant axiomatization of the notion of uniform boundedness in terms of what he calls coarse structures. To better understand Roe’s approach, it is useful to revisit the metric example. Given a metric space 0 0 0 0 (X, d), equip the product X × X with the `1-metric d((x1, x2), (x1, x2)) B d(x1, x1) + d(x2, x2). With this choice of metric, the distance between two points x, y ∈ X is equal to the distance of the pair (x, y) from the diagonal ∆X ⊂ X × X. It follows that a family (Bi)i∈I of subsets of X is uniformly bounded if and only S if the union i∈I Bi × Bi is contained in a bounded neighborhood of ∆X. This suggests that in order to have a notion of uniform boundedness on a set X it is sufficient to describe the “bounded neighborhoods” of the diagonal ∆X ⊂ X × X. Taking this point of view, a coarse structure on a set X is a family E of subsets of X × X satisfying certain axioms (Definition 2.2.1). These sets ought to be thought of as the bounded neighborhoods of ∆X. S A family (Bi)i∈I of subsets X is uniformly bounded (a.k.a. controlled) if the union i∈I Bi × Bi is contained in some E ∈ E. The axioms on E are chosen so that subsets and uniformly bounded neighbourhoods of uniformly bounded sets remain uniformly bounded. A coarse space is a set X equipped with a coarse structure E. In other words, the slogan is that a coarse space is a space where it makes mathematical sense to say that some property holds “up to uniformly bounded error”. The functions between coarse spaces that we wish to consider are those functions that preserve uniform boundedness. Explicitly, if (X, E), (Y, F ) are coarse spaces, we are interested in those functions f : X → Y such that f × f : X × X → Y × Y sends elements in E to elements in F . Such functions are called controlled. The reader may have realised that the definition for a coarse space is a direct analog of the classical notion of uniform space [Bou66]. This is no surprise: both languages seek to axiomatize some notion of “uniformity” whose meaning is clear for metric spaces but elusive in general. Uniform spaces extend the idea of “uniformly small” and uniform continuity; coarse spaces extend the idea of “uniformly bounded”. As we did for metric spaces, we now wish to identify “close functions”. Two functions between coarse spaces f, f 0 : (X, E) → (Y, F ) are close if f (x) and f 0(x) are uniformly close as x varies, i.e. the sets { f (x), f 0(x)} are uniformly bounded. A coarse map between coarse spaces (X, E), (Y, F ) is an equivalence class [ f ] of controlled functions X → Y, where functions are equivalent if they are close. A coarse equivalence is an coarse map [ f ]: (X, E) → (Y, F ) with a coarse inverse, i.e. a coarse map [g]: (Y, F ) → (X, E) with [ f ] ◦ [g] = [idY ] and [g] ◦ [ f ] = [idX]. Two coarse spaces are coarsely equivalent if there is a coarse equivalence between them. Coarse geometry may be thought of as the study of properties of coarse spaces that are defined up to coarse equivalence. To close the circle of ideas and conclude this brief introduction to coarse geometry, we remark that every metric space has a natural metric coarse structure Ed (consisting of all subsets of bounded neighborhoods of the diagonal). In this case, it is easy to verify that the notions of uniformly bounded, controlled, close and coarse equivalence for the coarse space (X, EdX ) coincide with the corresponding notions for the metric space (X, dX).

6 Coarse groups. The category of coarse spaces Coarse is the category with coarse spaces as objects and coarse maps as morphisms. From now on, we will use bold italic symbols to denote anything related to Coarse, so that X = (X, E) denotes a coarse space and f : X → Y denotes a coarse map (i.e. an equivalence class of controlled maps f : (X, E) → (Y, F )). In one sentence, a coarse group is a group object in the category of coarse spaces. More concretely, a coarse group is a coarse space G = (G, E) with a unit e ∈ G and coarse maps ∗: G × G → G and [-]−1 : G → G that obey the usual group laws “up to uniformly bounded error”. It is an empirical observation that the group objects in an interesting category turn out to be very interesting in their own right. For example: algebraic groups, Lie groups and topological groups are all deﬁned as group objects in their relevant categories. Given this premise, it is natural to be interested in coarse groups! One bewildering feature we came to appreciate while developing this work is that even basic questions regarding coarse groups end up being connected with the most disparate topics. To mention a few, we see that the study of coarse structures on Z quickly takes us in the realm of number theory; a similar study on more general groups connects with questions regarding the existence of bi-invariant metrics. Investigating non-abelian free groups leads us towards the theory of (stable) commutator length and Out(Fn). Banach spaces give rise to all sort of interesting examples, and, to conclude, the study of coarse homomorphisms between coarse groups is a natural generalization of the theory of quasimorphisms. We will provide a more detailed explanation of these aspects in the following subsections. To avoid confusion, from now on we will always add an adjective to the word “group” to specify the category we are working in. We will thus have coarse groups, topological groups, algebraic groups, and—rather unconventionally—we will use the term set-group to denote groups in the classical sense, which are group objects in the category of sets.

Coarse actions. Since their conception by Galois, set-group actions have been just as important as the set-groups themselves. It is then natural to try to understand what actions should look like in the category of coarse spaces. Of course, there is a natural definition of coarse action of a coarse group G on a coarse space Y. Namely, it is a coarse map α: G×Y → Y that satisfies the usual axioms for actions of set-groups up to uniformly bounded error. This definition directly generalizes other classical notions, such as that of quasi-action. Apart from its naturality, one interesting aspect of the theory of coarse actions is that it allows us to look at classical results from a different perspective. In a sense that we will make precise later on, there is a strong connection between coarse actions and left-invariant coarse structures. Using this shift of perspective, we can then realize that the Milnor–Švarc Lemma is more or less equivalent to the observation that left-invariant coarse structures on a set-group G are determined by the bounded neighborhoods of the unit e ∈ G.

Purpose and motivation. The goal of this monograph is to lay the foundations for a theory of coarse groups. In a large part, we wrote this text as a service to the community: rather than pushing towards a specific application, we made an effort to develop a self-contained exposition with emphasis on examples. We also point at some of the many connections between this topic and other subjects, and we leave a number of open questions for future research. On a first read, it may be helpful to imagine all coarse spaces as metric spaces: the case of metric coarse spaces already contains many of the conceptual difficulties of the general theory of coarse spaces. However, we found that Roe’s formalism of coarse spaces was of great help in identifying the crucial points of the theory. Another reason for embracing Roe’s language is that it generalizes effortlessly to various other settings and constructs. This gives rise to a richer, more flexible theory. For example, any abelian topological group admits a natural coarse structure where the controlled sets are contained in uniformly compact

7 neighborhoods of the diagonal. These topological coarse structures are such that any abelian topological group is coarsely equivalent to any of its cocompact lattices. One significant case is that of the group of rational numbers: Q is a cocompact lattice in the ring of adeles AQ, hence Q and AQ are isomorphic coarse groups (the former is seen as a discrete topological group, the latter has a natural locally compact topology). This fact is analogous to the obvious remark that Z and R are coarsely equivalent and, using the formalism of coarse spaces, it is just as simple to prove. Our interest in this subject was sparked by Uri Bader’s realization that this observation allows for a unified proof of the fact that the spaces of “quasi-endomorphisms” of Z and Q are naturally identified with R and AQ respectively [Bad21]. The general language of coarse spaces is also well suited for dealing with large (i.e. non finitely generated groups). This is the same reason why Rosendal adopted it in his recent monograph on the coarse geometry of topological groups [Ros17]. Finally, we should also point out that the theory of coarse spaces is commonly used in the operator algebras community in relation with C∗-algebras of geometric origin. We shall not pursue these topics in this monograph, but we leave them for future work.

1.2 On this manuscript This text is organised in two separate parts. The first part is mostly concerned with elementary properties and constructions of coarse groups and coarse actions. Here we prove a number of fairly basic “utility results” and verify that the various pieces of the theory fit well together and work as expected. This part is for the most part self contained and the proofs are rather detailed. We also made an effort to develop this using both Roe’s language of coarse structures [Roe03] and the approach in terms of “controlled coverings” à la Dydak–Hoffland [DH08]. In this regard, it is a recommended exercise to the reader to translate statements and proofs of various results from one language to the other. Of course, not all the material presented in this part is novel. Almost all the general facts about the category of coarse spaces can be found scattered in the literature. Also various ideas that are important in the development of the theory of coarse groups and coarse actions exist in a more or less implicit form in previous works. Besides Roe, other names of mathematicians who played a role in developing the theory of coarse geometry that we use in this text include Bader, Brodskiy, Dikranjan, Dydak, Hoffland, Mitra, Nicas, Protasov, Protasova, Rosendal, Rosenthal, and Zava. More detailed citations are included throughout the text. While writing this manuscript, we decided to recollect and introduce all the necessary facts and conventions to provide a unified framework with consistent nomenclature and notation.

The second part of the text has a rather different flavour. The goal here is to point at possible research directions and connections with more classical subjects. To do so, we selected and developed some specific topics according to our own taste. By design, this part is much less self-contained, as it is largely meant to connect with pre-existing literature. For the most part, these topics can be read independently from one another. Here the proofs are generally briefer than in the first part. We also spend more time in introducing and discussing a number of open questions. This work only scratches the surface of a theory of coarse groups and their coarse actions. All the topics we considered led us towards all sorts of interesting mathematics: more than we could hope to do ourselves. Rather than as a complete treatment, we see this part of the manuscript as an invitation to further research.

We should point out that, despite being inspired by the categorical perspective, we decided to avoid making heavy use of the categorical language in the main body of this text. Instead, we collected various categorical remarks in the appendix.

Warning. Albeit almost definitive, this manuscript is still a draft! The final version of this text will certainly differ on a number of details. This also means that there will be more typos than usual, for

8 which we apologize. Moreover, at this stage we are particularly interested in comments and feedback. We have tried our best to correctly cite the existing literature, but we surely missed some. Any suggestion in this regard would be very welcome.

1.3 List of findings I: basic theory Coarse groups. By definition, if G = (G, E) is a coarse group and ∗: G × G → G is the coarse multiplication map, any of its representatives ∗: G × G → G must be controlled. The first step in our study of coarse groups, is to understand when a multiplication function on a coarse space is controlled. To do so, it is fundamental to observe that the multiplication function ∗ is controlled if and only if both the left multiplication (g ∗ -) and the right multiplication (- ∗ g) are equicontrolled as g varies in G, see Definition 2.6.1. The intuition is that a family of maps ( fi)i∈I between metric spaces is equicontrolled if and only if the maps are all controlled with the same control function ρ: [0, ∞) → [0, ∞). This elementary observation allows us to produce our first non trivial examples of coarse groups. In fact, if G is a set-group equipped with a bi-invariant metric d, then the left and right multiplication are −1 equicontrolled with respect to Ed. It follows that the group operation ∗ is controlled. The inversion (-) is an isometry, hence it is controlled as well. It then follows that (G, Ed) is a coarse group with coarse operations ∗ and [-]−1. Remark 1.3.1. An alternative, more categorical, point of view to the above remark is that groups with bi-invariant metrics are group objects in a category of metric spaces (see AppendixB). Considering metric coarse structures defines a functor from this category to Coarse, and it is easy to observe that this functor sends group objects to group objects. We can now provide concrete examples and begin to acquire a taste for a theory of coarse groups. The easiest examples are given by choosing an abelian group and giving it a left-invariant metric. This of course includes Zn and Rn with their Euclidean metric. Somewhat more exotic examples—which will be useful later—include the additive group of any infinite dimensional Banach space, e.g. the separable Hilbert space `2(N). On the non-abelian side, a fundamental example is given by the free group FS on S generators, equipped with the cancellation metric d×. Formally, d× can be defined as the word metric associated with the set of all conjugates of elements of S (note that this is an infinite set). An alternative definition is obtained by declaring that for every word w ∈ FS the distance d×(e, w) is the minimal number of letters of w that it is necessary to cancel to make it equivalent to the trivial word (the letters can appear in any −1 position in w). The metric is then determined by the identity d×(w1, w2) = d×(e, w1 w2). As a coarse group, (FS , Ed× ) is considerably more complicated than the previous examples because it is not even coarsely abelian (i.e. the distance d×(w1w2, w2w1) is not uniformly bounded as w1, w2 range in FS ).

Remark 1.3.2. Every set-group G can be made into a trivially coarse group (G, Emin) by declaring that the only bounded sets are singletons (i.e. G is given the trivial coarse structure Emin consisting only of diagonal subsets). In other words, the coarse group (G, Emin) is not coarse at all, because no non-trivial error is a bounded error. The importance of this degenerate example is that it shows how to see the theory of coarse groups as an extension of the classical theory of set-groups: every “coarse” statement or deﬁnition reduces to the relevant classical notion when specialised to trivially coarse groups. It will soon become apparent that Geometric Group Theory itself can be thought of as the study of coarse actions of trivially coarse groups.

The above examples of coarse groups are all obtained by equipping some set-group G with a coarse structure E so that G = (G, E) is a coarse group (i.e. G is a coarsiﬁed set-group). In other words, these are coarse groups whose coarse operations admit representatives that satisfy the group axioms “on the nose”. However, this need not be the case in general. As a matter of fact, it turns out that the various group

9 axioms behave rather diﬀerently. It is not hard to show that it is always possible to ﬁnd well-adapted representatives so that the identities g ∗ e = e ∗ g = g and g ∗ g−1 = g−1 ∗ g = e hold for every g ∈ G (Lemma 4.4.3). On the other hand, one can come up with examples where every representative for ∗ fails to be associative. In view of this, one may also say that coarse groups are coarse spaces equipped with a coarsely associative binary operation that has a unit and inverses. It is worth pointing out that the lack of associativity implies that even well-adapted representatives that satisfy the unit and inverse axiom may be rather nasty. In particular, (-)−1 need not be an involution and the left multiplication (g−1 ∗ -) need not be an inverse of the function (g ∗ -) (in fact, these functions might not even be bijective)! On the positive side, we can prove that these issues can be ignored by moving to some isomorphic coarse group:

Theorem 1.3.3 (Theorem 4.4.6). For every coarse group G there exists an isomorphic coarse group Gˇ whose coarse operations admit well-adapted representatives so that (-)−1 is an involution and (g−1 ∗ -) is the inverse of (g ∗ -) for every g ∈ G.ˇ

We would like to mention two more properties of coarse groups. The ﬁrst one is a useful fact that makes it easier to check whether a coarse space endowed with multiplication and inverse operations is a coarse group. Namely, if one already knows that the multiplication function is controlled then the inverse is automatically controlled:

Theorem 1.3.4 (Theorem 4.2.6). Let G = (G, E) be a coarse space, e ∈ G and ∗, (-)−1 be operations that coarsely satisfy the axioms of associativity, unit and inversion. If ∗ is controlled then (-)−1 is controlled as well and G endowed with the coarse operations ∗ and [-]−1 is a coarse group.

Remark 1.3.5. This should be contrasted with the topological setting: it is not true that if a set-group G is given a topology τ so that the multiplication function is continuous then (G, τ) is a topological group. In fact, there are examples where the inversion is not continuous. It is an important property of coarse groups that their coarse structure is determined “locally”. Informally, this means that a family of subsets Ui ⊆ G of a coarse group G is uniformly bounded if and only if there is some bounded set B such that each Ui is contained in some translate of B. A more formal statement is analogous to the fact that the topology of a topological group is completely determined by the set of open neighborhoods of the identity:

Proposition 1.3.6 (Proposition 4.5.2). Let E, E0 be two coarse structures on a set G, and assume that ∗, e, (-)−1 are ﬁxed so that both (G, E) and (G, E0) are coarse groups. Then E ⊆ E0 if and only if every E-bounded set containing e is E0-bounded.

Remark 1.3.7. Once a set G and ∗, e, (-)−1 are ﬁxed, it is possible to explicitly characterize whether a certain family (Ui)i∈I of subsets of G is the family of all E-bounded subsets of G for some coarse structure E so that (G, E) is a coarse group. Informally, this is the case if and only if (Ui)i∈I is closed under ﬁnite products and arbitrary unions of conjugates (the formal statement is given in Proposition 4.6.1). Since products of compact sets are compact, it follows that every abelian topological group (G, τ) can be given a unique coarse structure Eτ so that the Eτ-bounded sets are the relatively compact subsets of G and (G, Eτ) is a coarse group.

Coarse homomorphisms, subgroups and quotients. A coarse homomorphism between coarse groups is a coarse map f : G → H that is compatible with the coarse group operations. Equivalently, if f : G → H is a representative for f then the image f (g1 ∗G g2) must coincide with the product f (g1) ∗H f (g2) up to uniformly bounded error. Quasimorphisms are a good example of coarse homomorphisms. Recall that a R-quasimorphism of a set-group G is a function φ: G → R such that |φ(g1g2) − φ(g1) − φ(g2)| is uniformly bounded. Let

10 E|-| be the metric coarse structure on R associated with the Euclidean metric. If E is a coarse structure on G so that G = (G, E) is a coarse group, then φ: G → (R, E|-|) is a coarse homomorphism as soon as the R-quasimorphism φ is controlled. An examination of the rich literature on R-quasimorphisms and bounded cohomology shows that coarse homomorphisms can diﬀer greatly from usual homomorphisms of set-groups. Remark 1.3.8. The fact that coarse structures on coarse groups are determined locally has some useful consequences for their coarse homomorphisms. Namely:

• Let G = (G, E), H = (H, F ) be coarse groups and let f : G → H be a function that so that f (g1 ∗G g2) and f (g1) ∗H f (g2) (resp. f (eG) and eH) coincide up to uniformly bounded error. If f sends bounded sets to bounded sets then it is automatically controlled, hence f is a coarse homomorphism (Proposition 5.1.14).

• If f : G → H is a proper coarse homomorphism (i.e. so that preimages of bounded sets are bounded) then it is a coarse embedding (Corollary 5.1.9).

Naturally, two coarse groups are said to be isomorphic if there exist coarse homomorphisms f : G → H and g: H → G that are coarse inverses of one another.

At this point the theory becomes more subtle. We wish to declare that a coarse subgroup of a coarse group G is the image of a coarse homomorphism f : H → G. However, coarse homomorphisms are only defined up to some equivalence relation, hence so are their images. Explicitly, we define the equivalence relation of asymptoticity (Definition 3.3.2) on the set of subsets of a coarse space and we define the coarse image of a coarse map f : X → Y to be the equivalence class of f (X) ⊆ Y up to asymptoticity. For intuition, the reader should keep in mind that if X = (X, Ed) is a metric coarse space then two subsets of X are asymptotic if and only if they are a finite Hausdorff distance from one another. A coarse subgroup of G (denoted H < G) is the coarse image of a coarse homomorphism to G. Alternatively, we can characterize the coarse subgroups of G = (G, E) as those equivalence classes [H] −1 where H ⊆ G is a subset so that [H] = [H ∗G H] and [H] = [H ] (Proposition 5.2.2). Strictly speaking, a coarse subgroup H < G is not a coarse group because the underlying set is only defined up to equivalence. However, whenever a representative is fixed H is indeed a coarse group. Moreover, different choices of representatives yield canonically isomorphic coarse groups. We are hence entitled to forget about this subtlety and treat coarse subgroups as coarse groups. Coarse subgroups can be used to appreciate some interesting behaviour in the theory of coarse groups. d d For example, let V ⊂ R √be a k-dimensional affine subspace and let H ⊂ Z be the subset of lattice points that are within distance k from V. Since V + V is at finite Hausdorff distance from V, it follows that [H] d is a coarse subgroup of (Z , Ek-k). A peculiar feature of [H] is that one can choose V with an irrational slope so that H is not asymptotic to any subgroup of Zd. It is also worthwhile noting that quasicrystals in Rd give rise to coarse subgroups as well. We should point out that, although mildly puzzling, these coarse k subgroups are not very interesting as coarse group, as they are always isomorphic to (R , Ek-k). More exotic examples can be obtained considering infinite approximate subgroups. Recall that an approximate subgroup is a subset A of a set-group G so that A = A−1 and the set of products AA is contained into finitely many translates of A (i.e. AA ⊆ XA for some X ⊂ G finite). Now, if E is a coarse structure on G so that (G, E) is a coarse group and every finite subset of G is E-bounded, it follows that for any approximate subgroup A ⊂ G the asymptotic class [A] is a coarse subgroup of (G, E). We will later return to these ideas to construct coarse subgroups of (F2, Ed× ).

Quotients of coarse groups are obtained by enlarging the coarse structure. Morally, the point of coarse geometry is that points in a coarse space can only be distinguished up to uniformly bounded error. If we

11 enlarge the coarse structure, fewer points can be distinguished, therefore this “coarser” coarse space is for all intents and purposes a quotient of the original space. We then say that a coarse quotient of a coarse group G = (G, E) is a coarse group (G, F ) so that E ⊆ F . This deﬁnition behaves as expected: a coarse group Q is isomorphic to a coarse quotient of G if and only if there is a coarsely surjective coarse homomorphism q: G → Q. Remark 1.3.9. Notice that unlike the case of set-groups, a coarse quotient of a coarse group need not be “quotient by a coarse subgroup”. For example, the coarse group (Z, E|-|) is a coarse quotient of the trivially coarse group (Z, Emin), however there is no coarse subgroup [H] < (Z, Emin) so that (Z, E|-|) is obtained by “modding out H” (recall that the Emin is the coarse structure so that points coarsely coincide only if they they are the same, Remark 1.3.2). It is still true that if N E G is a coarsely normal coarse subgroup of G, there is a well-deﬁned coarse quotient G/N (see below for more on this).

Coarse kernels and Isomorphism Theorems. Recall the coarse image of a coarse map is only defined up to asymptoticity. Similarly, we expect that the preimage of some coarse subspace should only be defined up to equivalence. However, the situation is even more delicate, as coarse preimages may not exist at all! The reason for this is already present in Remark 1.3.9. If we consider the coarse map idZ : (Z, Emin) → (Z, E|-|), we see that any E|-|-bounded set can be seen as a reasonable preimage of 0. However, these sets are not asymptotic according to the trivial coarse structure. We are hence unable to make a canonical choice of a coarse subspace of (Z, Emin) to play the role of the coarse preimage of [0] ⊂ (Z, E|-|). Fortunately, for many important examples of coarse maps f : X → Y we can define canonical coarse preimages. Namely it is often the case that all the reasonable candidate preimages of some coarse subspace Z ⊆ Y are asymptotic to one another. When this happens, we say that this equivalence class is the coarse preimage of Z and we denote it by f −1(Z) (Definition 3.5.7). We say that a coarse homomorphism of coarse groups f : G → H has a coarse kernel if [eH] ⊆ H has −1 a coarse preimage. In this case we denote it by Ker( f) B f (eH). Given the premise, it is unsurprising that not every coarse homomorphism has a coarse kernel. However, when it does exist it is well behaved: we prove in Subsection 7.2 that appropriate analogs of the isomorphism theorems relating subgroups, kernels and quotients hold true in the coarse setting as well (Theorems 7.2.1, 7.2.2 and 7.2.3).

Coarse actions. As announced, a coarse action α: G y Y of a coarse group G = (G, E) on a coarse space Y = (Y, F ) is a coarse map α: G × Y → Y that is coarsely compatible with the coarse group operations. That is, any representative α for α is so that α(g1, α(g2, y)) is uniformly close to α(g1 ∗ g2, y) and α(e, y) is uniformly close to y for every g1, g2 ∈ G, y ∈ Y. Once again, in order for α to be controlled it is necessary that the functions α(g, - ) and α(- , y) be equicontrolled.

Remark 1.3.10. If G is a set-group and α: G y (Y, dY ) is an action by isometries then α: (G, Emin) y

(Y, EdY ) is a coarse action of the trivially coarse group. However, it is generally not true that if dG is a bi-invariant metric on G then α: (G, EdG ) y (Y, EdY ) is a coarse action: in order to make sure that α is controlled it is necessary that dY (α(g1, y), α(g2, y)) is uniformly bounded in terms of dG(g1, g2). Prototypical examples of coarse group actions of a coarse group G include the actions of G on itself by left multiplication (more on this later) and by conjugation. A useful feature of the latter is that it allows us to deﬁne coarse abelianity, normality etc. in terms of (coarse) ﬁxed point properties.

Given two coarse actions α: G y Y, α0 : G y Y0 of the same coarse group G, we say that a coarse map f : Y → Y0 is coarsely equivariant if it intertwines the G-actions up to uniformly bounded error (Deﬁnition 6.1.8). As we did for coarse groups, we say that a quotient coarse action of α: (G, E) y (Y, F ) is a coarse action α: (G, E) y (Y, F 0) where F 0 ⊇ F is some coarser coarse structure on Y so that the

12 functions α(g, - ) are still equicontrolled (this is necessary to ensure that α is still controlled). Once again, this is a consistent deﬁnition because a coarsely equivariant map f : (Y, F ) → Z is coarsely surjective if and only if it factors through a coarsely equivariant coarse equivalence (Y, F 0) Z. Given a coarse subgroup [H] = H < G = (G, E), we can deﬁne a coarse action of G on the coarse coset space G/H by considering an appropriate quotient of the coarse action by left multiplication G y G. Namely, G/H is the set G equipped with the smallest coarse structure containing E and such that H is bounded and G y G/H is a coarse action (it is interesting to note that, as in Remark 1.3.9, we may also consider coset spaces G/A where A ⊂ G is any coarse subspace, not necessarily a coarse subgroup). The following result is an ingredient in the proof of the First Isomorphism Theorem.

Theorem 1.3.11 (Theorem 6.6.12). Let H ≤ G be a coarse subgroup. The coarse space G/H can be made into a coarse group with coarse operations that are compatible with the coarse action G y G/H if and only if H is coarsely normal in G.

Something interesting happens when looking at orbit maps and pull-backs of coarse structures. Let α: (G, E) y (Y, F ) be a coarse action and fix a point y ∈ Y. We may then consider the orbit map αy : G → Y given by αy(g) B α(g, y). We can use this function to define a pull-back coarse structure ∗ αy(F ) on G. It follows from the definition of pull-back coarse structure that the orbit map αy defines ∗ a coarse equivalence between (G, αy(F )) and the coarse image αy(G) ⊆ Y. For a set-group G acting by isometries on a metric space (Y, dY ), this is equivalent to defining a (pseudo) metric on G by letting dG(g1, g2) B dY (α(g1, y), α(g2, y)) and giving G the resulting metric coarse structure EdG . ∗ It follows from the construction that E ⊆ αy(F ) and that the left multiplication defines a coarse ∗ ∗ action (G, E) y (G, αy(F )). The orbit map αy : (G, αy(F )) → Y is coarsely equivariant. In turn, if αy is coarsely surjective—in which case we say that the action is cobounded—it follows that (G, E) y (Y, F ) is a coarse quotient of the coarse action by left multiplication of (G, E) on itself. It is also simple to verify that if y0 is a point close to y then the two orbit maps give rise to the same pull-back coarse structure. More generally one can show with a bit of work that if G y Y is a cobounded coarse action then the pull-back coarse structure is uniquely defined up to conjugation. Putting these observations together, one obtains the following characterization of cobounded coarse actions:

Proposition 1.3.12 (Proposition 6.3.8). Let G = (G, E) be a coarse group. There is a natural correspondence: {cobounded coarse actions of G}/coarsely equivariant coarse equivalence and

{F coarse structure | E ⊆ F , left multiplication is a coarse action (G, E) y (G, F )}/conjugation.

Recall that group coarse structures are determined locally (Proposition 1.3.6). The same is true for coarse structures associated with coarse actions. Namely, let (G, E) be a coarse group and let F , F 0 be two coarse structures on G so that the left multiplication deﬁnes coarse actions (G, E) y (G, F ) and (G, E) y (G, F 0) (such coarse structures are said to be equi-left-invariant). Then F ⊆ F 0 if and only if they every F -bounded subset of G is also F 0-bounded. In particular, to check whether the two coarse structures coincide it is enough to verify that they have the same bounded sets. We claim that this simple observation can be seen as the heart of the Milnor–Švarc Lemma. More precisely the Milnor–Švarc Lemma states that if a set-group G has a proper cocompact action on a proper geodesic metric space (Y, dY ) then it is ﬁnitely generated and quasi-isometric to (Y, dY ). To prove this theorem, we start by remarking that cocompactness implies that the coarse action (G, Emin) y (Y, EdY ) ∗ is cobounded, therefore the orbit map is a coarse equivalence between (G, αy(EdY )) and (Y, dY ). Since

13 ∗ the action is proper, the bounded subsets of (G, αy(EdY )) are the finite ones. By the coarse geodesicity of ∗ αy(EdY ), it follows that G is finitely generated. Let dG be a word metric associated with a generating set, then (G, Emin) y (G, EdG ) is also a coarse action. Since the bounded subsets of (G, EdG ) are the finite ∗ ones, it follows that EdG = αy(EdY ) and hence the orbit map is a coarse equivalence between (G, EdG ) and (Y, EdY ). Finally, we already remarked that coarse equivalences between geodesic metric spaces are always quasi-isometries (see Corollary 6.4.3 for a detailed proof). One may say that the above proof of the Milnor–Švarc Lemma bypasses most of the difficulties of the more standard proofs by avoiding working in the metric space Y. Instead, it is based on the observation that the relevant “coarse properties” are well behaved under taking pull-back and this allows us to reduce everything to the study of equi-left-invariant coarse structures on set-groups. We can also appreciate how the argument splits into many independent observations, showing precisely which hypothesis is necessary at any single step. By extension, we anticipate that a coarse geometric perspective will bring a fair amount of clarity into various complicated geometric constructions.

1.4 List of findings II: selected topics Coarsified set groups. The most natural connection between the theory of coarse groups and classical set-group theory is certainly to be found in the study of coarsified set-groups. Namely, those coarse group (G, E) where G is a set-group. Every set-group G has two extreme coarsifications: the trivial one (G, Emin) and the bounded on (G, Emax) (the latter is the coarse structure where the whole of G is bounded). From the coarse point of view, neither of these is an interesting coarsification: the former reduces to classical set-group theory, the latter is always isomorphic to the coarse group having only one point ({e}, Emax). We are therefore interested in those coarsifications that lie in between. To avoid trivial coarse structures, we will be interested in coarsely connected coarsifications. These are coarsifications (G, E) so that every element g ∈ G is E-close to the unit e ∈ G. Typical examples are given by metric coarsifications: if d is some metric on G then for every g ∈ G the distance d(g, e) is some finite number and hence g is Ed-close to e. The reason why coarsely connected coarsifications are particularly interesting for us is that for any coarsification (G, E) the set of points g ∈ G that are E-close to e is a normal subgroup G0 < G and the coarse group (G, E) factors through the quotient (G, E) → (G/G0, Emin). The general question we wish to address is: Question 1.4.1. Which set-groups G admit unbounded coarsely connected coarsifications (G, E)?

Not every infinite set-group has an unbounded coarsely connected coarsification. Specifically, we say that a set-group G is intrinsically bounded if (G, Emax) is the only coarsely connected coarsification of G. Examples of infinite intrinsically bounded set-groups include the dihedral group D∞ and SL(n, Z) with n ≥ 3 (this follows from some bounded generation properties). More generally, it follows from the work of Duszenko and McCammond-Petersen that a Coxeter group is intrinsically bounded if and only if it is of spherical or affine type [Dus12, MP11]. Generalizing the case of SL(n, Z), Gal–Kedra also prove that “many” Chevalley groups are intrinsically bounded [GK11]. Question 1.4.1 is very much related to the existence of unbounded bi-invariant metrics. If d is an unbounded bi-invariant metric on G then (G, Ed) is an unbounded coarsely connected coarsification. For “small” groups this actually characterizes intrinsic boundedness:

Proposition 1.4.2 (Proposition 8.3.4). If a set-group G is normally ﬁnitely generated or countable, then it is intrinsically bounded if and only if it does not admit any unbounded bi-invariant metric.

We should remark that “most” infinite groups are not intrinsically bounded. For example, it follows by the work of Epstein–Fujiwara [EF97] that every infinite Gromov hyperbolic group has an unbounded bi-invariant metric. It follows that a random (in the sense of Gromov) infinite finitely generated group is not intrinsically bounded.

14 Coarsifications of Z. The Euclidean metric shows that Z does have unbounded coarsely connected coarsifications. As it turns out, this is but one of its many coarsifications. It is perhaps unsurprising that any attempt to classify such coarsifications leads to interesting problems in number theory. One source of coarsifications is obtained by picking word metrics dS associated with (infinite) generating sets S ⊂ Z. It is often a delicate problem to understand whether two different generating 0 sets S , S give rise to the same coarse structure EdS = EdS 0 . For example, if S = P is the set of all prime numbers, the equality EdP = Emax is equivalent to the highly non-trivial statement that every natural number is equal to the sum of a bounded number of primes. One may even say that the Goldbach conjecture is a very strong effective version of this result. In general, it appears that the only coarse structures EdS that are relatively well understood are those where S is a union of geometric sequences. A different way of defining coarsifications of Z is by observing that every abelian topological group (G, τ) admits a natural coarse structure Eτ whose bounded sets are the relatively compact subsets of G. This creates a link with the study of group topologies on Z: any such topology τ defines a coarsification 0 0 Eτ. Once again, given different topologies τ , τ it is generally challenging to distinguish Eτ from Eτ. One viable strategy is to find a sequence (an)n∈N that τ-converges to some point a∞ ∈ Z, but so that the set 0 {an | n ∈ N} is not relatively compact according to τ . This technique proves very effective to differentiate between coarsifications arising from profinite completions of Z. More precisely, given any set Q of prime numbers we define a group topology τQ on Z via the diagonal embedding in the product of the pro-p Q completions Z ,→ p∈Q Zp. We can then prove:

0 0 Proposition 1.4.3 (Proposition 9.2.4). Let Q, Q be sets of primes. Then EτQ = EτQ0 if and only if Q = Q . Corollary 1.4.4. There exists a continuum of distinct coarsely connected coarsiﬁcations of Z.

Cancellation metrics. An important means of defining bi-invariant metrics on set-groups is via cancellation metrics (i.e. word metrics associated with conjugation invariant sets). Of course, the word metrics on Z discussed above are special kinds of cancellation metrics. We also mentioned cancellation metrics on non-abelian free groups, as they provided examples of coarse groups that are not coarsely abelian. One reason why these metrics are important is that they are relatively easy to define and they tend to yield “large metrics”. Namely, it is often the case that a set-group G is intrinsically bounded if and only if all its cancellation metrics are bounded. It turns out that it is always possible to recognise whether a coarsification of a set-group arises from a cancellation metric. Coarse structures have a notion of a “generating set of relations” (paragraph before Lemma 2.2.9), and we say that a coarsely connected coarse structure is coarsely geodesic if it is generated by a single relation (Definition 6.4.2). Informally, this definition is an analog of the fact that if (X, d) is a geodesic metric space we can estimate the distance d(x, y) between any two points as the length of the shortest sequence x0,..., xn so that x = x0, y = xn and d(xi, xi+1) ≤ 1 for all 1 ≤ i ≤ n. The following holds:

Proposition 1.4.5 (Proposition 8.2.2). Let G be a set-group. A coarsiﬁcation (G, E) is coarsely geodesic if and only if there exists a cancellation metric d× so that E = Ed× .

S If S is a generating set for a set-group G, we denote by d× the cancellation metric associated with the 0 normal closure of S . If G is finitely generated and S, S are two finite generating sets, then E S = E S 0 . d× d× This means that every finitely generated set-group has a canonical coarsification. Given the success of Geometric Group Theory, it seems all but natural to study it! It is simple to verify that this coarse structure is the finest coarsely connected coarsification. Namely, if G is finitely generated, any coarsely connected coarsification (G, E) is a coarse quotient of (G, E S ) (i.e. d× S E ⊆ E S ). In particular, such a G is intrinsically bounded if and only if d has finite diameter. d× ×

15 At this point we should issue a warning to the geometric group theorists: the inclusion Z ⊂ D∞ shows that cancellation metrics are very poorly behaved when taking ﬁnite index subgroups! This shows that bi-invariant metrics depend rather subtly on the algebraic properties of set-groups and opens the way to all sorts of questions. For example, we do not know whether the canonical coarsiﬁcations of non-abelian free groups Fn and Fm can ever be isomorphic as coarse group if n , m.

Coarse groups that are not coarsified set-groups. The vast majority of the examples of coarse groups that we gave so far are coarsified set groups. This need not be the case: one may find a coarse group G = (G, E) so that no choice of representatives for the coarse operations ∗ and [-]−1 makes G into a set-group. However, we also showed that for every coarse group one can find an isomorphic coarse group whose coarse operations have representatives that fail only to be set-group operations because ∗ is coarsely associative (Theorem 1.3.3). It is reasonable to ask whether every coarse group is in fact isomorphic to some coarsified set group. We believe that this is not the case. That is, we conjecture that there exists a coarse group G that is not isomorphic (as a coarse group) to any coarsified set group. We also describe a suitable candidate.

Let (F2, Ed× ) be the coarsification associated with the canonical cancellation metric of the free group of rank-2. Let also f : F2 → R be a homomorphism sending one generator to 1 and the other to some irrational number α, and let K B f −1([0, 1]). The asymptotic equivalence class [K] is a coarse subgroup of (F2, Ed× ) (this is the coarse kernel of the coarse homomorphism f : (F2, Ed× ) → (R, Ed)), hence it uniquely determines a coarse group up to isomorphism. We conjecture that this kernel is not isomorphic to a coarsified set-group. We prove a criterion to recognise when a coarse group is isomorphic to a coarsified set group (Proposition 11.1.1). Using this criterion, we can reduce our conjecture to a very elementary—albeit complicated—combinatorial problem.

Coarse homomorphisms and coarse automorphisms. For given coarse groups G, H, it is often interesting to study the set HomCrs(G, H) of coarse homomorphisms f : G → H. If G = (G, E), H = (H, F ) are coarsified set groups, it is very natural to compare HomCrs(G, H) with the set of set-group homomorphisms Hom(G, H). One reason for doing so, is that statements relating set-group homomorphisms with coarse homomorphisms can often be seen as rigidity phenomena of the set-group (or lack thereof). Perhaps the simplest non-trivial example is given by the metric coarsification (Z, E|-|) associated with the Euclidean metric on Z. It is not hard to show that a function φ: Z → Z determines a coarse homomorphism if and only if it is an R-quasimorphism. We hence obtain a complete characterization of HomCrs((Z, E|-|), (Z, E|-|)) because it is well-known that the set of R-quasimorphisms of Z up to bounded distance can be identified with R. It is suggestive to note that when we compare coarse homomorphisms Z → Z with R-quasimorphisms we are implicitly using the isomorphism of coarse groups (Z, E|-|) (R, E|-|) to identify HomCrs((Z, E|-|), (Z, E|-|)) with HomCrs((Z, E|-|), (R, E|-|)). In turn, if we focus on the set-group of coarse ∗ automorphisms (i.e. invertible coarse homomorphisms) we see that AutCrs(Z, E|-|) AutCrs(R, E|-|) R . In other words, every coarse automorphism of R is close to a linear transformation. This fact is not special about R: with some care it can be extended to other normed vector spaces:

Proposition 1.4.6 (Proposition 12.2.1). If V1, V2 are real Banach spaces then the set of coarse homomorphisms can be identiﬁed with the space of bounded linear operators

Hom ((V , Ek k ), (V , Ek k )) = B(V , V ). Crs 1 - V1 2 - V2 1 2

n Corollary 1.4.7. For every n ∈ N we have AutCrs(Z , Ek-k) GL(n, R).

16 Recall that an arbitrary coarse homomorphism need not have a well-deﬁned coarse kernel. It is interesting to note that in the linear setting we know precisely when this is the case:

Proposition 1.4.8 (Proposition 12.2.4). Let T : (V1, Ek-k1 ) → (V2, Ek-k2 ) be a coarse homomorphism between Banach spaces and let T be its linear representative. Then T has a coarse kernel if and only if T(V1) is a closed subspace of V2. When this is the case, Ker(T) = [ker(T)]. Another case of interest is where G = (G, EG ) and H = (H, EH ) are the coarsiﬁcations of ﬁnitely d× d× generated set-groups associated with the natural cancellation metrics. In this case there is a natural map Φ: Hom(G, H) → Hom (G, H). It follows from the bi-invariance of EH that for every h ∈ H any Crs d× −1 homomorphism f : G → H is close to its conjugate ch ◦ f (x) B h f (x)h . We thus obtain a quotient map

Φ: Hom(G, H)/conj. → HomCrs(G, H).

In general, Φ need not be injective nor surjective. However, it can be studied on a case-by-case basis to give an idea of the level of compatibility of the algebraic/coarse geometric properties of G and H.

In the case G = H, this Φ also deﬁnes a homomorphism Φ: Out(G) → AutCrs(G, Ed× ). Note that n for G = Z the coarse structure Ed× coincides with the natural metric coarse structure Ek-k and hence n AutCrs(Z , Ed× ) GL(n, R). We then see that Φ is simply given by the inclusion GL(n, Z) ⊂ GL(n, R). This shows that in this case Φ is injective and it is very far from being surjective. One may also say that there are many “exotic” coarse automorphisms that do not arise from automorphisms of Zn.

Another case of interest is that of the non-abelian free set-group Fn. The set-group AutCrs(F2, Ed× ) is rather mysterious. However, we can use the work of Hartnick–Schweitzer [HS16] to prove the following.

Proposition 1.4.9 (Subsection 12.4). The map Φ: Out(Fn) → AutCrs(Fn, Ed× ) is injective. Furthermore,

AutCrs(Fn, Ed× ) is uncountable and contains torsion of arbitrary order. Remark 1.4.10. Hartnick–Schweitzer deﬁne a notion of quasimorphism for non-abelian set-groups. We can show that every set-group G has a canonical coarsiﬁcation (G, EG→R) such that the space of quasimorphisms from G to H á la Hartnick–Schweitzer coincides with the space of coarse homomorphisms

HomCrs((G, EG→R), (H, EH→R)) (Proposition 12.3.2).

Proper coarse actions. Recall the Milnor–Švarc Lemma is tightly related with the fact that equi-left-invariant coarse structures are determined by their bounded sets. More precisely, given an action α: G y (Y, d), we gave an alternative proof for this result based on the study of the pull-back coarse structure ∗ ∗ αy(Ed) and its bounded sets. By definition, a subset B ⊆ G is αy(Ed)-bounded if and only if its image ∗ under the orbit map αy(B) is bounded in Y. One may thus say that the αy(Ed)-bounded sets measure the “lack of properness” of the orbit map. This observation motivates us to develop a language to describe “properness” for coarse actions. Just as coarse structures are used to describe the notion of “uniformly bounded sets”, the correct gadget to describe “bounded sets” is given by bornologies (Definition 13.1.1). A function between bornological spaces f : (X1, B1) → (X2, B2) is bornologous if it sends bounded sets to bounded sets; it is proper if the preimage of bounded sets is bounded. Of course, the family of E-bounded sets for a given coarse structure E is a bornology. A map between coarse spaces is proper if it is proper with respect to the associated bornologies. Let G = (G, E) be a coarse group and let B be a bornology on G containing all the E-bounded subsets of G. A coarse action α:(G, E) × (Y, F ) → (Y, F ) is B-proper (Definition 13.1.2) if the function

(α, πY ):(G, B) × (Y, F ) → (Y, F ) × (Y, F ) is proper, where πY is the projection to Y. We prove the following:

17 Proposition 1.4.11 (Proposition 13.2.1). A coarse action α: (G, E) × (Y, F ) → (Y, F ) is B-proper if and ∗ only if for every y ∈ Y the bornology B coincides with the set of αy(F )-bounded sets. This has a few implications. Firstly, we see that for a given bornology B on (G, E) there exists a B-proper coarse action if and only if there exists some coarse structure E0 on G such that E ⊆ E0, the left multiplication deﬁnes a coarse action (G, E) y (G, E0) and B is the set of E0-bounded subsets of G (this implies that B must be left-invariant). Secondly, the fact that equi-left-invariant coarse structures are determined by their bounded sets shows that all the orbits of B-proper actions are equivariantly coarsely equivalent to one another. This can be seen as a sort of “coarse orbit–stabilizer theorem”: the B-properness condition implies that B is the “coarse stabilizer bornology” of every point y ∈ Y and hence the orbit αy(G) ⊆ (Y, F ) can be equivariantly identiﬁed with an appropriate coarse quotient of the coarse action by left multiplication of G on itself.

It is interesting to test this language in the context of continuous actions of topological groups. Let α: G y Y be such an action, where we also assume that Y is Hausdorﬀ. We may deﬁne a topological left coarse structure Fcpt on Y by declaring that for every compact set K ⊆ Y the family of its left translates g · K, g ∈ G are uniformly bounded sets. If we see G as a trivially coarse group (Remark 1.3.2), we may left then see that α: (G, Emin) y (Y, Fcpt ) is a coarse action. The family K of relatively compact subsets of G is a bornology and it turns out that the continuous action α is topologically proper if and only if the coarse action α is K-proper (Proposition 13.3.1). This fact provides us with a bridge from the topological world to that of coarse geometry. Using this bridge, it is simple to prove the following generalization of the Milnor-Švarc Lemma:

Proposition 1.4.12 (Proposition 13.3.5). Let G be a Hausdorﬀ topological group and (Y, Ed) a coarsely geodesic proper metric space. If there exists a continuous proper and cocompact action α: G y Y by uniform quasi-isometries then G is compactly generated, and for any compact generating set K and y ∈ Y the orbit map αy :(G, dK) → (Y, d) is a quasi-isometry (where dK is the word metric associated with K).

Fragmentary coarse spaces The last topic we address is the study of the space of controlled functions between coarse spaces. Ideally, one would like to define a natural coarse structure on the set of controlled functions f : (X, E) → (Y, F ). Unfortunately, it is not possible to define such a coarse structure that is well-behaved under taking compositions. To give the idea of what goes wrong, consider the following example. Let X be a metric space 0 and ( fi)i∈I, ( fi )i∈I be families of Lipschitz functions from X to itself. One may then say that these two 0 families are uniformly close if there is some constant D so that d( fi(x), fi (x)) ≤ D for every i ∈ I, x ∈ X. When this happens, we think of these families to be coarsely indistinguishable. However, if we are 0 now given two more families of uniformly close Lipschitz functions (gi)i∈I, (gi )i∈I there is no reason 0 0 why the compositions (gi ◦ fi)i∈I and (gi ◦ fi )i∈I should again be uniformly close. We are then in the problematic situation where the composition of two indistinguishable families of functions are no longer indistinguishable. One way to solve this issue is to only consider families of functions with uniformly bounded Lipschitz constants (albeit the bound may be arbitrarily large). As it turns out, this solution can be implemented in the full generality of coarse spaces via means of “fragmentary coarse structures”. A fragmentary coarse structure E on a set X is a family of subsets of X × X that satisfies some axioms which are only marginally weaker than those of a coarse structure (specifically, we do not require that E contains the diagonal ∆X, see Definition 14.1.1). In particular, every coarse structure is also a fragmentary coarse structure. In analogy with coarse spaces, we may then define a category of frag-coarse spaces. Crucially, this category is much better suited for studying spaces of controlled functions. If X = (X, E) and Y = (Y, F ) are (frag-)coarse spaces, we define a frag-coarse structure F E on the set of all controlled functions (X, E) → (Y, F ) (Definition 14.2.3). We denote by YX the resulting frag-coarse space. This definition is natural in the following sense:

18 Theorem 1.4.13 (Theorem 14.2.9). Let X, Y, Z be (frag-)coarse spaces. Then:

1. the evaluation map YX × X → Y sending ( f, x) to f (x) is controlled;

2. the composition map ZY × YX → ZX is controlled;

3. there is a natural bijection

{(frag-)coarse maps Z × X → Y} ←→ {frag-coarse maps Z → YX}.

Remark 1.4.14. In categorical terms, Theorem 1.4.13 means that the category FragCrse of frag-coarse spaces is Cartesian closed and that the category of coarse spaces can be enriched over FragCrse. Theorem 1.4.13 allows us to take a different point of view on coarse actions. For a given coarse space Y, the frag-coarse space YY is a frag-coarse monoid. One may then describe coarse actions G y Y as homomorphisms of frag-coarse monoids G → YY (Proposition 14.3.2). Let X, Y be two coarsely equivalent coarse spaces. Using some care, we may also define a frag-coarse space of coarse equivalences CE(X, Y) so that CE(X, X) is a frag-coarse group (this is somewhat more subtle than just taking the subspace of YX of functions that have a coarse inverse, see Definition 14.4.1). Considering the coarse action by left translation of a coarse group G on itself, we may then prove the following:

Theorem 1.4.15. Every coarse group is isomorphic to a coarse subgroup of a frag-coarsiﬁed set-group.

1.5 Acknowledgements We thank Uri Bader for suggesting this research project, and for many insightful conversations. We also thank Nicolaus Heuer for his helpful comments and insights. Most of the work was carried out while the authors were at the Weizmann Institute of Science, which we thank for the gracious hospitality. Both authors were partially supported by ISF grant 704/08. The second author was also Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044 –390685587, Mathematics Münster: Dynamics–Geometry–Structure.

19 Part I Basic Theory

2 Introduction to the coarse category

In this chapter we introduce the category of coarse spaces, and deﬁne basic notions. We also give examples of coarse structures that are frequently useful. Basic notions include: controlled coverings, controlled maps, closeness, coarse equivalence, binary products, and equicontrolled maps. Most of the material in this section is standard, and appears in several sources, including [Roe03]. For this reason we will not prove everything in detail. The informed reader can skim this section. There are two main approaches to the theory of coarse spaces: via families of controlled coverings and coarse structures. We develop the correspondence between them in the ﬁrst few sections. Each approach has advantages and disadvantages, and in this monograph we will use both approaches.

2.1 Some notation for subsets and products We begin by setting some notation. Given a set X, we denote families of subsets of X with lowercase fraktur letters (a, b,... ) and families of subsets of X × X with uppercase calligraphic letters (E, F ,... ). Subsets of X × X are called relations on X. A partial covering a ⊂ X is any family of subsets of X. It is a covering if X = S{A | A ∈ a}. Let a and a0 be two partial coverings of X, we say a is a refinement of a0 if for every A ∈ a there is an A0 ∈ a0 0 such that A ⊆ A . We call iX B {{x} | x ∈ X} the minimal covering of X (it is a refinement of every other covering). For any subset A ⊆ X and partial covering b of X, the star of A with respect to b is the set St(A, b) B S {B ∈ b | B ∩ A , ∅}. Note that A ⊆ St(A, b) if and only if b covers A—if b is a covering this is always the case. If a is another partial covering of X, the star of a with respect to b is the family St(a, b) B {St(A, b) | A ∈ a}. This is a special instance of a convention we will use often: if F : P(X) → P(Y) is a function sending subsets of X to subsets of Y and a is a partial covering of X, we denote by F(a) the family of images a B {F(A) | A ∈ a}. Now we will describe some operations on relations. Let X, Y be sets, denote by π1 : X × Y → X and π2 : X × Y → Y projections to the first and second coordinate. Given a relation E ⊆ X × Y we let Eop B {(y, x) | (x, y) ∈ E} ⊂ Y × X. For any subset Z ⊆ Y we define E(Z) ⊂ X as

−1 E(Z) B {x ∈ X | ∃z ∈ Z s.t. (x, z) ∈ E} = π1(π2 (Z) ∩ E).

Given points x ∈ X and y ∈ Y we deﬁne sections

xE B {z ∈ X | (x, z) ∈ E} ⊆ Y and Ey B {z ∈ X | (z, y) ∈ E} ⊆ X.

op That is, Ey = E({y}) and xE = E ({x}). Later we will also abuse notation and write E(y). Given 1 E1, E2 ⊆ X × X we denote their composition by:

0 0 0 E1 ◦ E2 B {(x1, x2) | ∃x ∈ X s.t (x1, x ) ∈ E1, (x , x2) ∈ E2} ⊆ X × X.

1We follow Roe’s convention on the order of composition of relations [Roe03]. This is coherent with the pair groupoid structure on X × X and it is so that E1 ◦ E2(Z) = E1(E2(Z)). One can see functions as relations by identify a function f : X → Y with its graphs graph ⊆ X × Y. However, our choice of the order of composition is not coherent with the usual composition of −1 functions. In particular, our conventions are so that f (x) = x(graph( f )) = (graph( f ) )({x}).

20 We denote the diagonal by ∆X ⊂ X × X. Given A ⊆ X, we let ∆A B ∆X ∩ (A × A). In the sequel it will be convenient to move back and forth between relations and partial coverings. Note that the following procedure works particularly well in dealing with relations that contain the diagonal. A relation E ⊆ X ×X naturally deﬁnes a partial covering E(iX) = {Ex | x ∈ X}. Given a partial covering a of X, we deﬁne the “block diagonal” by [ diag(a) B {A × A | A ∈ a} ⊂ X × X.

Note that a is a reﬁnement of (diag(a))(iX). Vice versa, if E ⊆ X × X contains the diagonal ∆X, then E ⊆ diag(E(iX)). We leave it to the reader to prove the following facts about the relationship between the star and composition operations. Given E, F ⊆ X×X with ∆X ⊆ E then E◦F(iX) is a reﬁnement of St(F(iX), E(iF)). Conversely, one can show that diag(St(a, b)) = diag(b)◦diag(a)◦diag(b) for any choice of partial coverings a, b.

Note that for every function f : X → Y, and partial coverings a, b of X the image f (St(a, b)) is a reﬁnement of St( f (a), f (b)). Similarly, if we consider the product function f × f : X × X → Y × Y then for any two relations E1, E2 of X × X we have

f × f (E1 ◦ E2) ⊆ ( f × f (E1)) ◦ ( f × f (E2)).

2.2 Coarse structures We follow [Roe03] to define coarse structures and coarse spaces. We will also explain how to describe coarse structures in terms of controlled partial covering. This alternative description is essentially due to Dydak–Hoffland [DH08]. Definition 2.2.1 (See [Roe03]). A coarse structure on a set X is a collection E of relations on X (subsets of X × X) that is closed under taking subsets and finite unions and such that

(i) ∆X ∈ E (contains the diagonal); (ii) if E ∈ E then Eop ∈ E (closed under symmetry);

(iii) if E1, E2 ∈ E then E1 ◦ E2 ∈ E (closed under composition). The relations E ∈ E are called controlled sets or entourages for the coarse structure E. A set equipped with a coarse structure is called a coarse space. Example 2.2.2. The two extreme examples of coarse structures on a set X are the minimal coarse structure Emin = {∆Z | Z ⊆ X} and the maximal coarse structure Emax = P(X × X) (the maximal coarse structure is × E { | } the only coarse structure containing X X). A slightly less trivial example is fin B E E r ∆X is finite . This is the finite off-diagonal coarse structure (called discrete coarse structure in [Roe03]). Definition 2.2.3. A coarse structure E on X is connected if every pair (x, x0) ∈ X × X is contained in some controlled set. If E is connected, we say that the coarse space (X, E) is coarsely connected. Every coarse space (X, E) can be partitioned into a disjoint union of coarsely connected components. The “finite off-diagonal coarse structure” described above is the minimal connected coarse structure on X. Example 2.2.4. The prototypical example of coarse space is obtained by considering a metric space (X, d) and equipping it with the coarse structure Ed consisting of all the relations E ⊆ X × X for which there is 0 0 an r ∈ R+ so that d(x, x ) ≤ r for every (x, x ) ∈ E.

Warning. For every metric space (X, d) the induced coarse structure Ed is connected.

21 In view of the above, it can be convenient to consider metrics d that take values in the extended positive reals [0, +∞]. To avoid confusion, we will say that such a d is an ∞-metric. We may then define the coarse structure Ed associated with any ∞-metric just as we did for normal metrics. In this case, the coarsely connected components are the subsets of points at finite distance from one another. Remark 2.2.5. One may also define hypermetrics by considering positive valued functions with hyperreal ∗ values d : X × X → R≥0 that satisfy the triangle inequality. Such a hypermetric defines a coarse structure Ed just as a normal metrics does. Two points x, y ∈ X belong to the same coarsely connected component if and only if d(x, y) ∈ R+. This point of view can be useful when dealing with ultralimits of metric spaces. For example, the asymptotic cone Cone(X, e, λ) is the coarsely connected component of the ultralimit ω- limi(X, λid) containing the point e = ω- limi ei [DK18, Chapter 10]. Definition 2.2.6. Let (X, E) be a coarse space. A subset A ⊆ X in bounded if A × A ∈ E. If we want to specify that a set is bounded with respect to a specific coarse structure E, we say that it is E-bounded. Note that the whole space X is E-bounded if and only if E is the maximal coarse structure. Example 2.2.7. Let X be a topological space2. Then the family of sets

Ecpt B {E ⊆ X × X | ∃K ⊆ X compact, E ⊆ (K × K) ∪ ∆X} is the the compact coarse structure. If X is equipped with the discrete topology then Ecpt coincides with E the finite off-diagonal coarse structure fin (hence why the latter is also called discrete coarse structure). Another natural coarse structure on X is the compact fibre coarse structure:

ﬁb Ecpt B {E ⊆ X × X | xE and Ey are relatively compact ∀x, y ∈ X}

fib −1 Here is a sketch of a proof that Ecpt is closed under composition: since (E◦F)y = E(Fy) = π1(E∩π2 (Fy)), −1 it is enough to show that E ∩ π2 (Fy) is relatively compact. By assumption, Fy is contained in a compact −1 3 K is compact and we conclude because E ∩ π2 (K) is compact. fib In general, the coarse structures Ecpt and Ecpt can be seen as extremal examples among the family of coarse structures on X whose bounded sets are precisely the relatively compact subsets of X. If (X, d) is a proper metric space (closed balls are compact), then the controlled entourages of the metric coarse fib structure Ed have compact fibers. It follows that Ecpt ⊆ Ed ⊆ Ecpt. If X has infinite diameter one can also check that the containments are strict. We leave it as an exercise to the reader to check these assertions. Convention 2.2.8. Example 2.2.7 illustrates a notational convention that we will often use to denote mod special coarse structures: we will denote by Eprop the coarse structure whose controlled/bounded sets are described by the property “prop”, possibly with the modifier “mod”. T For any family (Ei)i∈I of coarse structures on X, the intersection i∈ I Ei is a coarse structure on X. One can thus define the coarse structure hEi | i ∈ Ii generated by the relations Ei ⊆ X × X as the minimal coarse structure containing Ei for every i ∈ I.

Lemma 2.2.9. A relation F is contained in hEi | i ∈ Ii if and only if it is contained in a finite composition op F1 ◦ · · · ◦ Fn where each F j is equal to Ei ∪ ∆X or Ei ∪ ∆X for some i ∈ I. 2We are not requiring X to be Hausdorff. For us a set is compact if every open cover has a finite subcover (in literature these are also called quasi-compact). A set is relatively compact if it is contained in a compact set. 3 −1 −1 This can be proved as follows. Let C B E ∩ π2 (K) and note that Cz B π2 (z) ∩ C is compact for every z ∈ K. Let Ui be an open cover of C, we may assume that Ui = Ai × Bi, as the products of open sets form a basis for the topology of X × K. For each z ∈ K, there exists a finite set of indices i1,... in such that Cz ⊆ U1 ∪ · · · ∪ Un. It follows that the finite intersection ∩ · · · ∩ −1 ∪ ∪ Vz B Bi1 Bin is an open neighbourhood of z such that π2 (Vz) is contained in the finite union Ui1 ... Uin . The family {Vz | z ∈ K} is an open cover of K and hence admits a finite subcover. We obtain a finite subcover of C by considering the relevant sets Ui.

22 Sketch of proof. Let F be the collection of relations contained in some E ⊆ F1 ◦ · · · ◦ Fn where the F j op are ﬁnite unions of Ei, Ei and ∆X. All these relations must belong to hEi | i ∈ Ii, it hence suﬃces to show that F is a coarse structure. It is obvious that F contains ∆X and is closed under taking compositions and subsets. Composition and union are distributive operations, i.e. for any E1, E2, F1, F2 ⊆ X × X we have

(E1 ∪ E2) ◦ (F1 ∪ F2) = (E1 ◦ F1) ∪ (E1 ◦ F2) ∪ (E2 ◦ F1) ∪ (E2 ◦ F2).

It follows that F is also closed under finite unions. op To finish the proof it is hence enough to show that any finite union of Ei, Ei and ∆X is contained in op finite compositions of Ei ∪ ∆X and Ei ∪ ∆X. This follows immediately from observing that

E ∪ F ⊆ (E ∪ ∆X) ◦ (F ∪ ∆X). for any pair of relations E, F on X.

Example 2.2.10. Given a family of coarse spaces (Xi, Ei) indexed over a set I, we can define their F disconnected union as the coarse space obtained by giving the disjoint union X B i∈I Xi the coarse E hE | ∈ i ∈ E structure disc B i i I . Concretely, we see that E disc if and only if there exist finitely many Sn indices i1,..., in and controlled sets Ei ∈ Ei such that E ⊆ ∆X ∪ i=1 Ei. As the name suggests, each Xi is coarsely disconnected from any other X j. We are using the term “disconnected union” as opposed to “coarsely disjoint union” because the latter is already used in the literature to denote the minimal coarsely connected coarse structure containing E disc. Remark 2.2.11. Given a coarse structure E on X, it is not hard to show that there exists a metric d on X such that E = Ed if and only if E is generated by a countable family of controlled sets En ∈ E (see E [Roe03, Theorem 2.55] for a full argument). It follows that the finite off-diagonal coarse structure fin on an uncountable set X is not equal to Ed for any choice of metric d on X. The ability to pass from relations to partial coverings and back enables us to describe coarse structures in terms of systems of controlled coverings of X:

Deﬁnition 2.2.12. Given a coarse structure E on a set X, a controlled (partial) covering of X is a (partial) covering a such that diag(a) ∈ E. We will denote by C(E) the family of controlled partial coverings associated with E. If we want to specify that a (partial) covering is controlled with respect to a certain coarse structure E, we say that it is E-controlled.

Remark 2.2.13. One could refer to controlled coverings as “coverings by uniformly bounded sets”. We prefer the term ‘controlled’ because the adjective ‘uniformly’ is somewhat overdetermined.

Example 2.2.14. If (X, d) is a metric space and Ed is the associated bounded coarse structure, then C(Ed) is the family of partial coverings that have a uniform bound on the diameters of their elements. That is, a ∈ C(Ed) if and only if there is a r ≥ 0 such that diam(A) ≤ r for every A ∈ a. We describe the correspondence between controlled coverings and relations more fully: for every op relation E ∈ E and x ∈ X the relation Ex × Ex is contained in E ◦ E . It follows that if E is controlled then the family E(iX) = {Ex | x ∈ X} is a controlled (partial) covering of X. The converse is not true in general, but it does hold for relations E that contain the diagonal. In fact, we already remarked that E ⊆ diag(E(iX)) whenever ∆X ⊆ E. Since a relation E is controlled if and only if E ∪ ∆X is controlled, it follows that the coarse structure E is determined by C(E). Note that the family of controlled partial coverings C(E) associated to the family of relations E satisﬁes:

(C0) if a ∈ C(E) then a ∪ iX ∈ C(E);

23 (C1) if a ∈ C(E) and a0 is a reﬁnement of a, then a0 ∈ C(E);

(C2) given a, b ∈ C(E), then St(a, b) ∈ C(E). Conditions (C1) and (C2) follow from the fact that E is closed under taking subsets and compositions respectively. Since a ∪ b is a refinement of St(a, b ∪ iX), we deduce that C(E) is also closed under finite unions (this can also be deduced from the fact that E is closed under finite unions). Given a family of partial coverings C satisfying (C0)–(C2), it is not hard to see that the family of relations that are contained in diag(a) for some a ∈ C is a coarse structure E such that C(E) = C. We collect these observations in the following. Proposition 2.2.15. For any fixed set X, there is a one-to-one correspondence

coarse structures on X ←→ families of partial coverings of X satisfying (C0)–(C2) .

Remark 2.2.16. In addition to using controlled sets and partial coverings, one can define coarse structures in terms of families of pseudo-metrics di : X × X → [0, ∞] as i ranges over some index set I. Given any such family one can associate the generated coarse structure hEdi | i ∈ Ii. Conversly, given any relation E ∈ E one can define a pseudo metric dE as the largest pseudo-metric such that dE(x, y) ≤ 1 whenever 0 (x, y) ∈ E. If {di | i ∈ I} and {d j | j ∈ J} are families of pseudo-metrics on X and Y that are closed under 0 taking finite minima (i.e. for every i, i ∈ I the the meet di ∧ di0 is again in the family) one can also see that a function f : X → Y is controlled (see below) if and only if for every i ∈ I there is a j ∈ J so that f :(X, di) → (Y, d j) is Lipschitz. We will not make use of this point of view.

2.3 Controlled maps In this section we describe how functions f : X → Y interact with coarse structures E and F on X and Y. We will use the notation f : (X, E) → (Y, F ) or f : X → (Y, F ) if we want to specify that some property of the function f : X → Y depends on E and/or F . Deﬁnition 2.3.1. A function f : (X, E) → (Y, F ) is a controlled map 4 if f × f : X × X → Y × Y sends controlled sets to controlled sets, i.e. ( f × f )(E) ∈ F for every E ∈ E. Equivalently, f is a controlled map if it preserves controlled partial coverings: i.e., f (a) ∈ C(F ) for every a ∈ C(E). Convention 2.3.2. For the sake of clarity, we will use the word ‘function’ to denote any (possibly non controlled) function f : X → Y. We will reserve the word ‘map’ for controlled maps.

Example 2.3.3. Let (X, dX) and (Y, dY ) be metric spaces. Then a function f : (X, EdX ) → (Y, EdY ) is controlled if and only if there is a control function, i.e. an increasing function ρ+ : [0, ∞) → [0, ∞) such that 0 0 dY ( f (x), f (x )) ≤ ρ+(dX(x, x )) for every x, x0 ∈ X. Note that f is a Lipschitz function if and only if it is controlled by a linear control function. We define f to be coarsely Lipschitz if it is controlled by an affine function ρ+(t) B Lt + A for some L, A ≥ 0. Of course, coarsely Lipschitz functions are controlled. Definition 2.3.4. Two functions f, g: X → (Y, F ) are close (denoted f ≈ g) if f × g: X × X → Y × Y sends the diagonal ∆X into a controlled set F ∈ F . Equivalently, f and g are close if the family ( f ∪ g)(iX) B {{ f (x), g(x)} | x ∈ X} is a controlled partial covering of Y.

4In [Roe03] a map as in Definition 2.3.1 is called bornologous. A connected coarse structure E is a special kind of bornology on X × X ([Bourbaki TopVS Ch3]). Definition 2.3.1 can be rephrased as ‘ f × f is a bornologous map’. Furthermore, the family of E-bounded sets in X is again a bornology, and a controlled map f is indeed bornologous with respect to these bornologies. On the other hand, it is not true that every bornologous map f : X → Y is controlled. Definition 2.3.1 captures the notion of ‘uniformly bornologous’.

24 Closeness is an equivalence relation on the set of functions from X to Y. If f1 : (X, E) → (Y, F ) is controlled and f2 : X → Y is close to f1, then also f2 is controlled. If f1, f2 : X → (Y, F ) are close and 0 0 0 g: Z → X is any function, then f1 ◦ g and f2 ◦ g are close. If g : (Y, F ) → (Z , D ) is controlled, then 0 0 also g ◦ f1 and g ◦ f2 are close. Deﬁnition 2.3.5. Let f : (X, E) → (Y, F ) and g: (Y, F ) → (X, E) be controlled maps. We say that g is a right coarse inverse of f if the composition f ◦ g is close to idX; and g is a left coarse inverse of f if g ◦ f is close to idY . The function g is a coarse inverse of f if it is both a left and right coarse inverse (in which case f is also a coarse inverse of g). A coarse equivalence is a controlled function with a coarse inverse. If there is a coarse equivalence between X and Y we call X and Y coarsely equivalent. Deﬁnition 2.3.6. A function between coarse spaces f : (X, E) → (Y, F ) is proper if the preimage of every bounded subset of Y is bounded in X. It is a controlledly proper if ( f × f )−1(F) ∈ E for every F ∈ F . Equivalently, f is controlledly proper if the preimage of every controlled partial covering of Y is a controlled partial covering of X (i.e. f is ‘uniformly proper’). A controlled, controlledly proper map is a coarse embedding .

If Z is a subset of a coarse space (X, E), it inherits a subspace coarse structure E|Z B {E∩Z×Z | E ∈ E}. It is not hard to check that f : (X, E) → (Y, F ) is a coarse embedding if and only if f : (X, E) → (Im( f ), F |Im( f )) is a coarse equivalence. If f : (X, E) → (Y, F ) is a controlled function and f ≈ g, then if f is proper (resp. controlledly proper) then so is g (resp. controlledly proper).

We say that a subset Z ⊂ X is coarsely dense if there is a controlled set E ∈ E such that E(Z) = X. In terms of controlled coverings, Z is coarsely dense if there exists an a ∈ C(E) such that St(Z, a) = X or, equivalently, iX is a reﬁnement of St(iZ, a). Deﬁnition 2.3.7. A function f : X → (Y, F ) is coarsely surjective if f (X) is coarsely dense in Y. Example 2.3.8. It is helpful to translate the above properties for coarse structures associated with metric spaces. Let (X, dX) and (Y, dY ) be metric spaces. Two functions f1, f2 : (X, EdX ) → (Y, EdY ) are close if there exists some R ≥ 0 such that dY ( f1(x), f2(x)) ≤ R for every x ∈ X. A function f : (X, EdX ) → (Y, EdY ) is proper if for every r ≥ 0 there is a R ≥ 0 such that diam( f −1(B(y; r))) ≤ R for every r-ball centred at x ∈ X. The function f is coarsely surjective if there is an R ≥ 0 so that its image is an R-dense net of Y, i.e. the balls B( f (x); R) cover Y. Also note that f is a controlledly proper if and only if there exists an increasing and unbounded control function ρ− : [0, ∞) → [0, ∞) such that

0 0 ρ−(dX(x, x )) ≤ dY ( f (x), f (x )).

Together with Example 2.3.3, this shows that f is a coarse embedding if and only if

0 0 0 ρ−(dX(x, x )) ≤ dY ( f (x), f (x )) ≤ ρ+(dX(x, x )) for some appropriately chosen control functions. We recover the standard deﬁnition of coarse embedding for metric spaces known by geometric group theorists (sometimes also referred to as “uniform embedding”

[Gro93, Section 7.E]). Similarly, we also see that (X, EdX ) and (Y, EdY ) are coarsely equivalent as coarse spaces if and only if (X, dX) and (Y, dY ) are coarsely equivalent as metric spaces [NY12, Section 1.4]. If f is a bi-Lipschitz homeomorphism then it is a coarse equivalence. If f is a coarse embedding that t admits aﬃne control functions ρ−(t) B L − A and ρ+(t) B Lt + A, then f is a quasi-isometric embedding (here we are allowing ρ− to be negative for small values of t). A coarse equivalence where both f and its coarse inverse are quasi-isometric embeddings is a quasi-isometry. Quasi-isometries and quasi-isometric embeddings will not play an important role in this monograph: they will be used sometimes as sources of examples. However, they are the cornerstone of geometric

25 group theory and much of what follows is inspired by them. It is also worth remarking that if X and Y are geodesic spaces (i.e. the distance between any two points is the length of the shortest path connecting them) then X and Y are coarsely equivalent if and only if they are quasi-isometric. The following will not come as a surprise to a geometric group theorist:

Lemma 2.3.9. A controlled map f : (X, E) → (Y, F ) is a coarse equivalence if and only if it is a coarsely surjective coarse embedding.

Sketch of proof. If f has a right coarse inverse, g, then f must be controlledly proper (it is enough to −1 check that f (b) ⊆ St g(b), (idX ∪(g ◦ f ))(iX) for every partial covering b of Y). It is also easy to check that if f has a left coarse inverse then it is coarsely surjective. This proves one implication. For the other one, let b0 be a covering of Y such that St( f (X), b0) = Y. For every y ∈ Y define g(y) by choosing a point x ∈ X such that both f (x) and y belong to the same B0 ∈ b0. The function g is a controlled −1 map because for every controlled covering b of Y, the image g(b) is a refinement of f (St(b, b0)). The map g is a coarse inverse of f since (idY ∪( f ◦ g))(iY ) is a refinement of b0 and (idX ∪(g ◦ f ))(iX) is a −1 refinement of f (St(b0, b0)).

Remark 2.3.10. The controlled surjection idR : (R, Emin) → (R, Ed), where Ed is the coarse structure induced by the Euclidean metric, shows that coarse surjectivity does not imply that there exists a (controlled) left coarse inverse. Similarly, coarse embeddings need not admit right coarse inverses. In fact, one can show that f : X → Y has a right coarse inverse if and only if f is controlledly proper and Y admits a coarse retraction to f (X), i.e. there is a controlled map p: (Y, F ) → ( f (X), F | f (X)) that is a right coarse n inverse for the inclusion f (X) ,→ Y. Note that (R, Ed) does not coarsely retract to {2 | n ∈ N} ⊂ N.

2.4 The category of coarse spaces In this section we formalize the category of coarse spaces, which has objects that are sets with coarse structures, and morphisms that are equivalence classes of controlled maps.

Deﬁnition 2.4.1. A coarse map between two coarse spaces (X, E) and (Y, F ) is an equivalence class of controlled maps (X, E) → (Y, F ), where f, g: X → Y are equivalent if they are close (denoted f ≈ g).

Remark 2.4.2. Here our conventions diverge from Roe’s. In [Roe03] a coarse map is deﬁned as a controlled map (not an equivalence class) that is also proper. The composition of coarse maps is a well-deﬁned coarse map because the closeness relation is well behaved under composition with controlled maps.

Deﬁnition 2.4.3. The category of coarse spaces is the category Coarse whose objects are coarse spaces X = (X, E) and whose morphisms are coarse maps: MorCrs(X, Y) B f : X → Y | f coarse map = f :(X, E) → (Y, F ) | f controlled map / ≈ .

Convention 2.4.4. We use bold symbols when we want to stress that we are working in the Coarse category. So that X will be a coarse space and f : X → Y a coarse map. On the other hand, we keep the notation (X, E) when we want to highlight that a coarse space is a speciﬁc set X with a coarse structure. If [ f ] is the equivalence class of a controlled map f : (X, E) → (Y, F ), then f = [ f ]: X → Y is a coarse map. We use the notation [ f ] to emphasize that f is a representative for the coarse map.

The identity morphism of X is idX B [idX]. A controlled map g is the right coarse inverse of f : (X, E) → (Y, F ) if and only if [ f ] ◦ [g] = [idX]. In particular, f is a coarse equivalence if and only if [ f ] is an isomorphism, in which case [g] is the inverse of [ f ].

26 Note that the notions of coarse left/right inverse, coarse embedding, coarse surjection and properness are all invariant under taking close functions. In particular, they deﬁne properties for coarse maps. In general, we use the adjective ‘coarse’ to denote properties of functions/coarse maps that only make sense up to closeness. If f ≈ g and f is a coarse embedding (resp. left/right inverse, coarse surjection, or proper), then so is g. The set consisting of a single point {pt} is a terminal object in Coarse. Note that any non-empty set equipped with the maximal coarse structure is a terminal object in Coarse (all the non-empty bounded coarse spaces are isomorphic in Coarse). For convenience, in the sequel we will continue using {pt} as a preferred terminal object in Coarse.

Convention 2.4.5. Given a function x: {pt} → X, we identify it with its image x(pt) ∈ X and write x ∈ X. In particular, if x: {pt} → (X, E) is a coarse map, we can identify any choice of a representative x = [x] as a point x ∈ X. We will often use this convention later on, e.g. to treat the unit of a coarse group (G, E) as if it was a point e ∈ G.

Formally, the empty set (∅, P(∅)) is the initial object in Coarse because the empty function is the unique function from ∅ to any other set and it is controlled—we will not need to use this initial object in this paper. Remark 2.4.6. The fact that morphisms in Coarse are defined as equivalence classes of functions creates a certain degree of imbalance. In fact, properties that can be quantified on the codomain of a function have a clear ‘coarse’ analogue (e.g. coarse surjectivity is a well-defined notion), while properties that need to be quantified on the domain (e.g. being a well-defined function) do not have immediate coarse analogues. This can be a nuisance. For example, when constructing a coarse-inverse for a coarsely surjective coarse embedding (Lemma 2.3.9) there is an obvious ‘coarsely well-defined’ candidate but it is then necessary to make some extra unnatural choices to obtain an actual function. One way to resolve this conundrum would be as follows. A function f : X → Y is identified with a subset of X × Y whose sections over every point x ∈ X consist of exactly one point. It is this extra requirement of uniqueness that introduces asymmetry in the definitions. One could have a more ‘symmetric’ notion of mapping by removing it altogether and replacing functions with binary relations E ⊆ X × Y. A “coarsely well-defined function” could then be defined as a relation E ⊆ X × Y so that the sections xE are a controlled partial covering of Y, while the union of the sections Ey is coarsely dense in X. We will not pursue this approach in this monograph.

Every set I can be seen as a coarse space equipped with the minimal coarse structure Emin. Every function f : (I, Emin) → (X, E) is controlled. Further, two functions f, g: (X, E) = (I, Emin) are close if and only if f = g. This shows that Coarse contains Set in a trivial manner.

Deﬁnition 2.4.7. A set equipped with the minimal coarse structure is said to be a trivially coarse space.

Throughout the monograph, we reserve the adjective ‘trivial’ for the minimal coarse structure Emin. The maximal coarse structure Emax will never be referred to as trivial. Instead, it might be called bounded.

Notice every coarse object X is uniquely determined by the coarse maps from trivially coarse spaces into X. Namely, any coarse map f : X → Y induces a map

fI : MorCrs((I, Emin), X) → MorCrs((I, Emin), Y)

by composition. We have:

Lemma 2.4.8. Let f, g: X → Y be coarse maps, then f = g if and only if fI = gI for every trivially coarse set I. Furthermore, f : X → Y is a coarse embedding (resp. coarsely surjective) if and only if fI is injective (resp. surjective) for every trivially coarse set I.

27 Proof. Let X = (X, E), Y = (Y, F ) and f = [ f ], g = [g] for some representatives f, g: X → Y. Since the deﬁnition of morphisms in the coarse category is well-posed, it is automatic that fI = gI whenever f = g. The converse implication is just as simple: let I = X, then fI([idX]) = [ f ]: (X, Emin) → (Y, F ) and gI([idX]) = [g]: (X, Emin) → (Y, F ). That is, if fI([idX]) = gI([idX]) then [ f ] = [g] as controlled maps (X, Emin) → (Y, F ). Since the notion of closeness does not depend on the choice of coarse structure on the domain, it follows that f = g. The ‘furthermore’ statement is only marginally more complicated. It is immediate to check that if f is a coarse embedding then fI is injective. For the converse implication, assume for contradiction that f is not controlledly proper and let I = E equipped with the minimal coarse structure. By assumption, there is a controlled set F ∈ F such that U B ( f × f )−1(F) is not controlled in X. For every E ∈ I = E there is a pair (x1, x2) ∈ U r E and we let g1(E) B x1 and g2(E) B x2. We thus obtained two functions g1, g2 : I → X such that [ f ◦ g1] = [ f ◦ g2] but [g1] , [g2]. The argument for surjectivity is similar. It is again obvious that when f is coarsely surjective then fI is surjective for every I. Vice versa, assume that Y is not contained in St( f (X), c) for any c ∈ C(F ). Let I = C(F ) and choose for every c ∈ C(F ) a yc ∈ Y r St( f (X), c). Then the function h: I → Y sending c to yc is not close to f ◦ g for any choice of g: I → X and hence [h] < Im( fI).

Remark 2.4.9. The fact that composition with f induces an injection (resp. surjection) MorCrs( - , X) → MorCrs( - , Y) whenever f is a coarse embedding (resp. coarsely surjective) is also true when considering non-trivially coarse spaces.

2.5 Binary products Given two coarse spaces (X, E) and (Y, F ) we deﬁne the product coarse structure on X × Y as E ⊗ F B D ⊆ (X × Y) × (X × Y) | π13(D) ∈ E, π24(D) ∈ F , where π13 : (X × Y) × (X × Y) → X × X and π24 : (X × Y) × (X × Y) → Y × Y are the projections to the ﬁrst & third, and second & fourth coordinates respectively. Given E ∈ E and F ∈ F we let −1 −1 E ⊗ F B π13 (E) ∩ π24 (F) = {(x1, y1, x2, y2) | (x1, x2) ∈ E, (y1, y2) ∈ F }. With this notation, D ∈ E ⊗ F if and only if it is contained in E ⊗ F for some E ∈ E and F ∈ F . In other words, E ⊗ F is the coarse structure generated by taking the closure under subsets of the family {E ⊗ F | E ∈ E, F ∈ F }. Using the explicit description of the coarse structure generated by a family of relations (Lemma 2.2.9), one can also show that hEi | i ∈ Ii ⊗ hE j | j ∈ Ji = hEi ⊗ F j | i ∈ I, j ∈ Ji. Given any E ⊆ X × X and F ⊆ Y × Y, we have:

E ⊗ F = (E ⊗ ∆Y ) ◦ (∆X ⊗ F) = (∆X ⊗ F) ◦ (E ⊗ ∆Y ) (1) The above equation follows directly from the deﬁnitions, but it turns out to be very important. We will apply it frequently, since it allows us to decompose products of relations into simpler products. As an immediate application, it implies that hEi | i ∈ Ii⊗hE j | j ∈ Ji is generated by {Ei ⊗∆Y | i ∈ I}∪{∆X ⊗F j | j ∈ J}.

To express the product coarse structure in terms of controlled partial coverings, note that C(E ⊗ F ) = d partial covering of X × Y π1(d) ∈ C(E), π2(d) ∈ C(F ) . Equivalently, C(E ⊗ F ) is the system of partial coverings obtained by taking the closure under reﬁnements of C(E) × C(F ), where a × b = {A × B | A ∈ a, B ∈ b}.

It is immediate to check that X × Y B (X × Y, E ⊗ F ) is a product of X = (X, E) and Y = (Y, F ) in the category Coarse. This is one of the main reasons we use Definition 2.3.1 instead of Roe’s definition of coarse maps (Roe’s requirement that coarse maps be proper makes it impossible to define projections).

28 2.6 Equicontrolled maps

In this section we introduce the notion of equicontrolled maps. This will prove to be essential in our study of coarse groups. Observe that if I is some index set and fi : X → Y for i ∈ I is a family of functions, then they can be seen as sections of the function f : X × I → Y deﬁned by f (x, i) B fi(x).

Deﬁnition 2.6.1. Let (I, C) be a coarse space. A family of controlled maps fi : (X, E) → (Y, F ) with i ∈ I is (I, C)-equicontrolled if f : (X, E) × (I, C) → (Y, F ) is controlled. We say that the fi are equicontrolled 5 if they are (I, Emin)-equicontrolled.

The following statement follows easily from the deﬁnition, but we wish to spell it out explicitly:

Lemma 2.6.2. Let fi : (X, E) → (Y, F ) be a family of maps where i ∈ I is some index set. The following are equivalent:

(i) the fi are equicontrolled; S (ii) for every controlled set E of X the union i∈I( fi × fi)(E) is controlled in Y; S (iii) for every controlled partial covering a of X the union i∈I fi(a) is a controlled partial covering of Y.

Proof. (i) ⇔ (ii): the function f : (X, E)×(I, Emin) → (Y, F ) is controlled if and only if ( f × f )(E⊗∆I) ∈ F for every E ∈ E. On the other hand, we have: [ [ ( f × f )(E ⊗ ∆I) = ( f × f )(E ⊗ {i}) = ( fi × fi)(E). i∈I i∈I

(i) ⇔ (iii): again, f : (X, E) × (I, Emin) → (Y, F ) is controlled if and only if f (a × iI) is a controlled S partial covering of Y for every a ∈ C(E). The conclusion follows because f (a × iI) = ∈ f (a × {i}) = S i I i∈I fi(a).

Vice versa, any function f : X × Y → Z can be seen as a family of functions in two distinct ways: x f B f (x, - ): Y → Z with x ranging in X, and fy B f ( - , y): X → Z with y ranging in Y. The following lemma is very useful and deceptively simple to prove:

Lemma 2.6.3. Let (X, E), (Y, F ) and (Z, D) be coarse spaces. A function f : (X, E) × (Y, F ) → (Z, D) is controlled if and only if both families x f and fy are equicontrolled.

Note that the above statement asks that x f are (X, Emin)-equicontrolled (as opposed to (X, E)-equicontrolled, in which case the statement would be trivial).

Proof. One implication is obvious: if f is controlled with respect to E ⊗ F then it is a fortiori controlled with respect to (E ⊗ Emin) and (Emin ⊗ F ). For the other implication, (1) states that we can write E ⊗ F as (E ⊗ ∆Y ) ◦ (∆X ⊗ F). We therefore see that

( f × f )(E ⊗ F) ⊆ ( f × f )(E ⊗ ∆Y ) ◦ ( f × f )(∆X ⊗ F) is controlled. This concludes the proof because the relations E ⊗ F generate E ⊗ F .

5 Families of (I, Emin)-equicontrolled functions are called “uniformly bornologous” in [BDM08].

29 3 Properties of the category of coarse spaces

3.1 Monomorphisms and epimorphisms

We sketch the proofs of the results of this subsection only for sake of completeness, as they are proved in detail in [DZ17]. Recall that a morphism f : X → Y in a category is monic (or, a monomorphism) if given any two morphisms g1, g2 : Z → X such that f ◦ g1 = f ◦ g2 then g1 = g2. Dually, f : X → Y is epic (or, an epimorphism) if given any two morphisms g1, g2 : Y → Z such that g1 ◦ f = g2 ◦ f then g1 = g2.

Lemma 3.1.1. Let [ f ]:(X, E) → (Y, F ) be a coarse map.

(i) [ f ] is a monomorphism if and only if f is a coarse embedding,

(ii) [ f ] is an epimorphism if and only if f is coarsely surjective.

Sketch of proof. (i): it is clear that when f is a coarse embedding [ f ] is a monomorphism. Lemma 2.4.8 shows that the converse implication is true in a strong sense (it is enough to check trivially coarse spaces). (ii): Once again, it is clear that when f is coarsely surjective [ f ] is an epimorphism. On the other hand, let i1, i2 : Y → Y t Y be the inclusions of Y into each component of Y t Y, and consider Y t Y with the coarse structure D generated by the partial coverings i1(a1), i2(a2) and (i1 ∪ i2)( f (iX)) B {{i1 ◦ f (x), i2 ◦ f (x)} | x ∈ X} where a1, a2 ∈ C(F ). Then [i1 ◦ f ] = [i2 ◦ f ], but we claim that [i1] = [i2] only holds if f is coarsely surjective. In fact, let b B (i1 ∪ i2)( f (iX)) ∪ iY1tY2 . Then for every a1, a2 ∈ C(F ) the partial covering St(i1(a1) ∪ i2(a2), b) is a refinement of St(b, i1(a1 ∪ a2) ∪ i2(a1 ∪ a2)). It follows from Lemma 2.2.9 that any controlled partial covering in C(D) must be a refinement of St(b, i1(a) ∪ i2(a)) for some a ∈ C(D). One can then show that if (i1 ∪ i2)(iY ) is in C(D) then iY is a refinement of St( f (iX), a) and hence f is coarsely surjective.

Together with Lemma 2.3.9 this implies the following:

Corollary 3.1.2 ([DZ17]). The category Coarse is balanced (i.e morphisms that are both monic and epic are isomorphisms).

Remark 3.1.3. Two remarks on the proof of Lemma 3.1.1:

1. The coarse space (Y t Y, D) is the coequalizer of the morphisms [i1 ◦ f ] and [i2 ◦ f ] going from X to Y q Y, where q denotes the coproduct. See Appendix A.1 for more on this.

2. The proof that we sketched above uses the axiom of choice to choose a function. This would have been unnecessary if we had deﬁned Coarse in terms of BinRel (the same remark holds for the proof of Lemma 2.3.9 as well).

3.2 Pull-back and push-forward

Coarse structures are well behaved under pull-back6 but less so under push-forward:

Deﬁnition 3.2.1. Let f : X → Y be any function and F a coarse structure on Y. The pull-back of F under f is the coarse structure

f ∗(F ) B {E ∈ X × X | f × f (E) ∈ F }.

6This subsection describes pull-backs and push-forwards of structures. We are not describing pull-backs and push-outs of commutative diagrams.

30 It is easy to verify that the pull back of a coarse structure is indeed a coarse structure. Note also that ∗ ∗ if f1, f2 : X → (Y, F ) are close functions, then f1 (F ) = f2 (F ). It follows from the deﬁnition that f : (X, f ∗(F )) → (Y, F ) is a controlled map. In fact, f : (X, E) → (Y, F ) is controlled if and only if E ⊆ f ∗(F ). It is also clear that f : (X, f ∗(F )) → (Y, F ) is a coarse embedding. Lemma 2.3.9 implies the following: Corollary 3.2.2. If f : X → (Y, F ) is a coarsely surjective function, then [ f ] is a coarse equivalence between (X, f ∗(F )) and (Y, F ).

∗ ∗ ∗ The pull-back is well-behaved under composition: ( f2 ◦ f1) (F ) = f1 ( f2 (F )). We implicitly used the notion of pull-back when deﬁning products: the product coarse structure on (X1, E1) × (X2, E2) is the ∗ ∗ intersection of the pullbacks of the projections π1(E1) ∩ π2(E2). In turn, this implies that the pull-back is ∗ ∗ ∗ also well behaved with respect to taking products: ( f1 × f2) (F1 ⊗ F2) = ( f1 (F1)) ⊗ ( f2 (F2)).

Push-forwards are somewhat less well-behaved. Given a function f : (X, E) → Y, the natural deﬁnition for the push-forward of E under f is f∗(E) B h f × f (E) | E ∈ Ei. Note that it is important to take the coarse structure generated by { f × f (E) | E ∈ E}, as this set need not contain the diagonal nor be closed under composition. In this monograph we will not explicitly use push-forwards: they only appear implicitly.

3.3 Controlled thickenings, asymptoticity and indexed partial covering Coarse containments and asymptoticity are key concepts in the theory of coarse spaces. The notation introduced in the following definitions will be used extensively in the sequel. Let X = (X, E) be a coarse space. Definition 3.3.1. A controlled thickening of a subset A ⊆ X is a ‘star neighbourhood’ St(A, a) where a ∈ C(E) is a controlled partial covering. A set B ⊆ X is coarsely contained in A (denoted B 4 A) if it is contained in a controlled thickening of A. Equivalently, B 4 A if B ⊆ E(A) for some E ∈ E. Let E, F ∈ E, and A ⊂ X. Since E(F(A)) = (E ◦ F)(A), it follows that coarse containment is a preorder on subsets of X. This preorder induces an equivalence relation: Definition 3.3.2. Two subsets A, B ⊆ X are asymptotic (denoted A B) if A 4 B and B 4 A. We denote the equivalence class of A by [A]. If there is risk of confusion with regard to the coarse structure being used, we will include it in the notation and write B 4E A, B E A and [A]E. Example 3.3.3. If we identify points x ∈ X with singletons {x} ⊆ X, then defines an equivalence relation on X. It follows from the definition, that x x0 if and only if they belong to the same coarsely connected component of (X, E) (Definition 2.2.3).

Example 3.3.4. If (X, d) is a metric space and Ed is the induced coarse structure, then A B if and only if they are at finite Hausdorff distance. In particular, the notion of ‘being asymptotic’ is coherent with the standard terminology for infinite geodesic rays in metric spaces [BH13, DK18]. In a few places we will need to use an extension of the notion of coarse containment to compare different partial coverings. When doing this, rather than considering actual partial coverings—i.e. sets of subsets—it is more convenient to work with indexed families of subsets. Definition 3.3.5. Let I be a fixed index set. An I-indexed partial covering of a set X is a family of non-empty subsets aI = (Ai ⊆ X | i ∈ I, Ai , ∅). If (X, E) is a coarse space, an indexed partial covering aI is controlled if it is a controlled partial covering when ignoring the indexing. We denote by CI(E) the set of all I-indexed controlled partial coverings of (X, E).

31 Deﬁnition 3.3.6. An indexed partial covering aI is equi-coarsely contained in bI (denoted aI 4I bI) if there exists a controlled covering c ∈ C(E) such that Ai ⊂ St(Bi, c) for every i ∈ I (i.e. aI is an ‘indexed reﬁnement’ of St(bI, c)). If aI 4I bI and bI 4I aI we say that the indexed partial coverings are equi-asymptotic (denoted aI I bI).

Remark 3.3.7. One can deﬁne a preorder 4 on (non-indexed) partial coverings by letting b 4 b0 if b is a reﬁnement of St(b0, a) for some controlled covering a ∈ C(E). We prefer to introduce indexed coverings and 4I because this other notion turned out to be somewhat awkward to use.

Any function f : X → Y induces a function fI on I-indexed partial coverings fI(aI) B ( f (Ai) | i ∈ I). Obviously, if f : (X, E) → (Y, F ) is controlled fI sends controlled partial covers to controlled partial covers fI : CI(E) → CI(F ). The converse is also true, provided that I is large enough. The following is lemma is very easy, but extremely useful.

Lemma 3.3.8. Let (X, E), (Y, F ) be coarse spaces and I a ﬁxed index set.

0 0 1. If two functions f, f : X → (Y, F ) are close then fI(aI) I fI (aI) for every partial covering a of X.

2. If f : (X, E) → (Y, F ) is controlled and aI, bI are I-indexed partial coverings of (X, E) such that aI 4I bI, then fI(aI) 4I fI(bI) in (Y, F ). In particular, controlled functions preserve equi-asymptotic classes of I-indexed partial coverings.

0 0 0 Proof. By deﬁnition, if f and f are close then the partial covering ( f ∪ f )(iX) = {{ f (x), f (x)} | x ∈ X} 0 0 0 is controlled. Noting that f (Ai) ⊆ St( f (Ai),and ( f ∪ f )(iX)) for every i ∈ I proves that f (aI) 4I f (aI). 0 Similarly f (aI) 4I f (aI) which proves the claim. For the second statement, let c be a controlled covering of (X, E) so that Ai ⊆ St(Bi, c) for every i ∈ I. Then, f (Ai) ⊆ f (St(Bi, c)) ⊆ St( f (Bi), f (c)) for every i ∈ I. The partial covering f (c) is controlled on (Y, F ) because f is a controlled map. Therefore fI(AI) 4I fI(BI).

When I consist of a single index I = {i}, the I-indexed partial coverings are just sets, I coincides with and fI is just f . In particular, Lemma 3.3.8 implies that controlled maps preserve coarse containment and asymptoticity of sets. Remark 3.3.9. Note that there is a natural correspondence between the set of equivalence classes of equi-asymptotic I-indexed controlled partial coverings and the set of coarse maps from the trivially coarse space (I, Emin) to X = (X, E):

CI(E)/I ←→ MorCrs((I, Emin), X). Namely, each function g: I → X deﬁnes a trivially controlled partial covering (g(i) | i ∈ I); and to any controlled partial covering aI we can associate a function by choosing a point ai ∈ Ai for every i ∈ I. Lemma 3.3.8 implies that these functions factor through to the respective quotients by the equivalence relations of equi-asymptoticty and closeness, and are inverses of one another. Remark 3.3.10. Both statements in Lemma 3.3.8 are valid for partial coverings that are not necessarily controlled. This is the main reason why we introduced indexed partial coverings instead of using MorCrs((I, Emin), X) (we will make use of non-controlled partial coverings to study cosets in coarse groups). Note also that Lemma 3.3.8(1) is also valid for functions that are not controlled. Finally, the special case |I| = 1 shows that Lemma 3.3.8 holds also for asymptotic sets (as opposed to coverings).

3.4 Coarse subspaces, images and quotients When X = (X, E) is a coarse space, it is unnatural to work with points in X because any two points x, x0 ∈ X are indistinguishable in Coarse (as long as X is coarsely connected). The same goes for subsets

32 of X. On the other hand, it is sometimes necessary to work with “coarse subspaces”. It turns out that the categorical approach to deﬁning subobjects works rather very well in this setting. Recall that a monomorphism in Coarse is a coarse embedding (Subsection 3.1) and recall the categorical deﬁnition of subobject (see e.g. [ML13]):

Deﬁnition 3.4.1. In a category C, let f : A → B and f 0 : A0 → B be two monics with the same codomain. Then f factors through f 0 if there exists h: A → A0 such that f = f 0 ◦ h. We say that f and g are equivalent if f factors through g and g factors through f (this is an equivalence relation). A subobject of an object B in C is an equivalence class of monics into B.

f A B f 0 h A0

Remark 3.4.2. A subobject in a category C is not an object in C. However, it does determine an object up to isomorphism: any representative f : A → B of a subobject of B does determine the object A. Equivalent monics determine isomorphic objects. Applying Definition 3.4.1 to the category Coarse provides us with an abstract notion of coarse subspace. According to this definition, a subspace of X = (X, E) is not a subset of X with a coarse structure. Fortunately, the abstract notion of coarse subspace can also be made concrete. Let X = (X, E) and Y = (Y, F ) be coarse spaces and [ f ]: Y → X be monic (i.e. a coarse embedding). If Y0 = (Y0, F 0) and [ f 0]: Y0 → X is an equivalent monic, then f (Y) f 0(Y0). That is, f (Y) and f 0(Y0) are asymptotic subsets of X (Definition 3.3.2). Vice versa, suppose X = (X, E) is a coarse space and Y ⊆ X a subset. Then Y B (Y, E|Y ) is a coarse space and the inclusion ι: Y ,→ X gives a monic [ι]: Y → X. It is easy to see that if Y0 Y is a subset asymptotic to Y and ι0 : Y0 ,→ X is the inclusion, then [ι]: Y → X and [ι0]: Y0 → X are equivalent monics. We thus defined two maps:

image Y → X monic /∼ Y ⊆ X / inclusion

It is obvious that taking the image of the inclusion yields the identity. On the other hand, we see that a monic [ f ]: (Y, F ) → (X, E) is equivalent to the inclusion f (Y) ,→ X, because f is a coarse embedding (Lemma 3.1.1) and hence induces a coarse equivalence between (Y, F ) and ( f (Y), E| f (Y)). We have shown that there is a natural correspondence between subobjects of X (in the sense of monics) and asymptotic-equivalence classes of subsets of X.

Deﬁnition 3.4.3. A coarse subspace of a coarse space X = (X, E) is an equivalence class of coarse embeddings Y ,→ X. Equivalently, it is an equivalence class [Y] of subsets of X up to E-asymptoticity. Coarse subspaces are denoted with the inclusion symbol Y ⊆ X.

Using this terminology, we can deﬁne the coarse image of a coarse map.

Deﬁnition 3.4.4. Let f : X → Y be a coarse map, say f = [ f ]. The coarse image of f is the coarse subspace Im( f) ⊆ Y determined by the asymptotic equivalence class of Im( f ). That is,

Im( f) = Im([ f ]) B [Im( f )].

The coarse image f(Z) of a coarse subsbpace Z = [Z] ⊆ X is deﬁned as [ f (Z)] or, equivalently, as the coarse image of the composition of f and the inclusion Z ,→ X.

33 Note that for Deﬁnition 3.4.4 to be well-posed it is essential that coarse maps are equivalence classes of controlled maps. That is, the fact that [ f (Z)] does not depend on the choice of representative Z for Z follows from Lemma 3.3.8 and crucially depends on the assumption that f is controlled.

It turns out that the situation for coarse quotients is somewhat simpler than for coarse subsets. For the sake of coherence, we will still use the categorical approach. Dually to Definition 3.4.1, one can define an equivalence relation on epic morphisms in a category C and define a quotient object as an equivalence class of epimorphisms. Unlike the notion of subobject, quotient objects in Coarse admit a simple description. Given a controlled function f : (X, E) → (Y, F ), the pull-back f ∗(F ) is a coarse structure containing E (Subsec- tion 3.2). Since f ∗(F ) is invariant under taking close maps, it follows that if we are given a diagram

Y , F f1 ( 1 1) (X, E) g

f2 (Y2, F2)

∗ ∗ where g = [g] is an isomorphism in the category of coarse spaces, then f1 (F1) = f2 (F2). In particular, the pull-back gives a well-deﬁned map

pull-back {coarse quotients} −−−−−−−→{E0 coarse structure on X | E ⊆ E0}.

This map is clearly surjective because id: (X, E) → (X, E0) gives an epimorphism in Coarse whenever E ⊆ E0. More interestingly, it is also injective. In fact, if [q]: (X, E) → (Y, F ) is an epimorphism in Coarse then q is coarsely surjective (Lemma 3.1.1). It follows from Corollary 3.2.2 that q: (X, q∗(F )) → (Y, F ) is an isomorphism and hence [q]:(X, E) → (Y, F ) is equivalent to [id]: (X, E) → (X, q∗(F )). Definition 3.4.5. A coarse quotient of a coarse space X = (X, E) is an equivalence class of coarse surjections X → Y. Equivalently, it is a coarse space (X, E0) where E ⊆ E0. If R is any family of relations on X, we denote by X/R the coarse quotient (X, hE ∪ Ri). A coarse map f : F → Y = (Y, F ) factors through X/R if and only if ( f × f )(R) ∈ F for every R ∈ R. Note that the coarse quotient (X, E0) is an actual object in Coarse (and different choices of E0 define different coarse quotients). This should be contrasted with subspaces: a coarse subspace is only determined up to some equivalence relation and it is therefore only an equivalence class of objects in Coarse.7 This is the reason why we say that the situation for quotients is simpler. Note however that coarse subspaces do define a unique coarse space up to isomorphism. For this reason we will generally handle them as if they were objects in Coarse. For example, we can then say that (the coarse subspace determined by) the coarse image of f : X → Y is a coarse quotient of X. In other words, we use the following: Convention 3.4.6. When we write Y ⊆ X, the symbol Y is used to denote both the equivalence class of subsets [Y] and the coarse space Y = (Y, F ) which embeds into X (in the latter case Y will generally be a subset of X and F = E|Y ).

3.5 Containments and intersections of coarse subspaces Recall coarse subspaces have two equivalent deﬁnitions as equivalence classes of monics or isomorphism classes of objects, see the discussion just before Deﬁnition 3.4.3. The equivalence relation used for

7 Considering pointwise subsets Y ⊂ X equipped with the restrition E|Y is not a satisfactory alternative to the deﬁnition of coarse subspace. This is because it does not capture the essence of “coarseness”. We do want to say that two asymptotic subsets of (X, E) are “the same” coarse subspace.

34 defining coarse subspaces is induced from a preorder (Defintion 3.3.1). In particular, that preoder defines a partial order on the set of coarse subspaces of a coarse space X. Namely, we make the following definition:

Deﬁnition 3.5.1. Given coarse subspaces Y = [Y] and Z = [Z] of X, we say that Y is coarsely contained in Z if Y 4 Z. We denote coarse containements by Y ⊆X Z (we will drop the subscript X from the notation). Equivalently, Y ⊆X Z if the monomorphism Y ,→ X factors through Z ,→ X.

As announced, ⊆X deﬁnes a partial order: it follows from the deﬁnition that Y ⊆X Z and Z ⊆X Y then Y = Z are the same coarse subspace of X.

Lemma 3.5.2. Let X be a coarse space and Y ⊆ X a coarse subspace. Then,

(i) any coarse subspace Z ⊆ X with Z ⊆X Y uniquely determines a coarse subspace Z ⊆ Y;

(ii) any coarse subspace Z ⊆ Y uniquely determines a coarse subspace Z ⊆ X with Z ⊆X Y.

Proof. Let X = (X, E). (i): Realize the coarse subspace Y ⊆ X as a coarse space Y = (Y, F ) by choosing a representative Y ⊆ X with [Y] = Y and equipping it with the restriction of the coarse structure E|Y . Also choose a representative Z ⊆ X so that [Z] = Z. By assumption, there exists a controlled covering a so that Z ⊆ St(Y, a). It follows that Z is in fact contained in the a-neighbourhood of Z0 B Y ∩ St(Z, a). Hence 0 0 0 Z E Z are asymptotic subsets of X. Now Z is a subset of Y, so [Z ]E|Y is a genuine subspace of Y. This coarse subspace is uniquely determined in the sense that if we make a diﬀerent choice of representatives Y0, Z0 and controlled covering a0 we obtain a natural coarse isomorphism f : Y → Y0 such that f(Z) = Z0. (ii): Let ι: Y → X and ι0 : Z → X be monics representing the coarse subspaces. The composition ι ◦ ι0 : Z → X determines a coarse subspace of X that is coarsely contained in Y ⊆ X. Diﬀerent choices of representatives yield equivalent coarse subspaces.

Definition 3.5.1 defines containment among coarse subspaces “living” in some larger coarse space X. However, once a coarse subspace Y ⊆ X is identified as an actual coarse space Y = (Y, F ), then every coarse subspace Z ⊆ X that is coarsely contained in Y uniquely determines a coarse subspace of Y.

The partial ordering on coarse subspaces can be used to deﬁne intersections as meets. However, we wish to warn the reader that taking intersections of two or more coarse subspaces is a rather delicate matter. We use the following:

Deﬁnition 3.5.3. Let Y1,..., Yn be coarse subspaces of X. If it exists, their coarse intersection is a largest common coarse subspace (denoted Y1 ∩ · · · ∩ Yn). In other words, Y1 ∩ · · · ∩ Yn is a coarse subspace of X such that (Y1 ∩ · · · ∩ Yn) ⊆ Yi for each i = 1,..., n and so that if Z ⊆ Yi for each i = 1,..., n then Z ⊆ Y1 ∩ · · · ∩ Yn. Remark 3.5.4. The coarse intersection may be undeﬁned. However, when it exists it is unique.

Lemma 3.5.5. Let Yi be representatives of Yi = [Yi], i = 1,..., n. A subset Z ⊆ X is a representative for the intersection Y1 ... Yn if and only if it satisﬁes the following:

• there is a controlled covering a so that Z ⊆ St(Yi, a) for every i = 1,..., n;

• for every controlled covering a there is a controlled covering b such that

St(Y1, a) ∩ · · · ∩ St(Yn, a) ⊆ St(Z, b).

35 Proof. Let Z B [Z]. The first condition is equivalent to saying that Z ⊆ Yi for every i = 1,..., n. Now, fix a controlled covering a. Then the coarse subspace [St(Y1, a) ∩ · · · ∩ St(Yn, a)] is coarsely contained in Yi for each i = 1,..., n. If Z is indeed equal to the coarse intersection of the Yi’s, it follows that St(Y1, a) ∩ · · · ∩ St(Yn, a) 4 Z, which is precisely the meaning of the second condition. 0 0 Vice versa, assume that the second condition holds. Let Z ⊆ X be a set so that [Z ] 4 [Yi] for each 0 Sn i = 1,..., n. In other words, Z ⊆ St(Yi, ai) for some controlled covering ai. The finite union a B i=1 ai 0 is again a controlled covering, and Z ⊆ St(Y1, a) ∩ · · · ∩ St(Yn, a). Since the latter is contained in St(Z, b), we see that [Z0] ⊆ Z. Hence Z is the coarse intersection.

Notice a coarse intersection may be much larger than a set-wise intersection of two representatives of coarse subspaces. For example, in (R, E|-|) we have [2Z] = [2Z + 1] = [R], so that the coarse intersection [2Z] ∩ [2Z + 1] = [R], even though 2Z and 2Z + 1 are disjoint subsets. The next example shows the coarse intersection may indeed not be deﬁned in general. 2 Example 3.5.6. Equip R with the Euclidean metric and let X B (R, Ed). Let A ⊂ X be the x-axis so (1) ∈ | (1) − (1) | → ∞ (2) y = 0. Choose an increasing sequence bk R with bk bk+1 . Choose a second sequence bk , (1) (2) (1) | (1) − (2)| | (2) − (1) | so that bk < bk < bk+1 and both bk bk and bk bk+1 go to inﬁnity. Recursively, keep choosing (n) sequences bk so that

(n) (n+1) (n) • bk < bk < bk+1,

| (n) − (n+1)| | (n+1) − (n) | • both bk bk and bk bk+1 go to infinity. S Now let Bn B {(bk, n) | k ∈ N} and B B n∈N Bn. Then the coarse intersection [A] ∩ [B] does not exist. In fact, for any fixed N ∈ N, let ZN be the intersection of the (closed) N-neighbourhoods of A and S B. Then ZN contains Bn for every n ≤ 2N, and it is contained in a neighbourhood of ≤ Bn. However, n 2N S the sets Bn are defined so that B2N+1 is not contained in any bounded-radius thickening of n≤2N Bn. Hence ZN+1 ZN. It follows that it is not possible to find a largest coarse subspace of X that is coarsely contained in both A and B. In a similar spirit, it is also possible to define the coarse preimage of a coarse map:

Deﬁnition 3.5.7. Let f : X → Y be a coarse map and Z ⊆ Y a coarse subspace. If it exists, the coarse preimage of Z is a largest coarse subspace of X whose coarse image is coarsely contained in Z (denoted f −1(Z)). In other words, f −1(Z) is a coarse subspace of X such that f( f −1(Z)) ⊆ Z and so that if f(Z0) ⊆ Z then Z0 ⊆ f −1(Z).

Remark 3.5.8. Just as for coarse intersection, the coarse preimage may not exist and if it exists it is unique. Note that the coarse intersection (Y1 ∩ Y2) ⊆ Y1 can be seen as the coarse preimage of Y2 under the coarse embedding Y1 ,→ X. The following result follows easily from the deﬁnitions. Its proof is left as an exercise.

Lemma 3.5.9. If f : X → Y is a coarse equivalence and g: Y → X is a coarse inverse, then for every coarse subspace Z ⊆ Y the coarse preimage f −1(Z) exists and it coarsely coincides with the coarse image g(Z).

Remark 3.5.10. One could also define a coarse union of subspaces in much the same manner as intersections. These are behave better than coarse intersections in that finite coarse unions are always well-defined. What one would expect to be true of finite unions holds also in the coarse category. However, in the sequel we will never have occasion to use coarse unions.

36 4 Coarse Groups

Recall the category Coarse has binary product and terminal objects. Given such a category, one may study its group objects (see below for a deﬁnition). For example, the usual groups are the group objects in the category of sets Set, while topological groups are group objects in the category of topological spaces, Top. As announced, coarse groups are the group objects of Coarse. This is the main object of study of this monograph.

Convention 4.0.1. In the sequel we will work with both groups (in the usual sense) and coarse groups. To avoid confusion, we will always specify the category we are working on by calling the former set-groups.

Historical Remark. As far as we are aware, this work is the first to define and study coarse groups. However, many examples of coarse groups are obtained by equipping set-groups with appropriate coarse structures (we call these coarsified set-groups). Various authors have investigated how set-groups interact with coarse structures: as coarsified set-groups or via coarse actions (Definition 6.1.1). In that context, a number of elementary results discussed here have already been discovered and we recover them here as a special instances of more general statements. Among relevant literature, we should certainly include [DP19, DZ20, NR12, PP19, Pro19, Fri88, PP20, Pro18, Ros17].

4.1 The diagrammatic approach

We introduce a few pieces of notation:

Notation 4.1.1. Given coarse maps f1 : X1 → Y1 and f2 : X2 → Y2 we denote their product by

f1 × f2 : X1 × X2 → Y1 × Y2

(if f1 = [ f1] and f2 = [ f2] then f1 × f2 = [ f1 × f2]).

Given f1 : X → Y1 and f2 : X → Y2 we let ( f1, f2) B ( f1 × f2) ◦ [∆]: X → Y1 × Y2, where ∆: X → X × X is the diagonal embedding. By composition, a coarse mapping of the terminal object x: {pt} → X induces a “constant mapping” x of any other coarse space Y → {pt} −→ X (the coarse image of x can be thought of as a “coarse point” in X). We abuse notation, and denote this composition by x: Y → X.

The astute reader will notice that the commutative diagrams in the following deﬁnition of a coarse group are analogous to the usual group theoretic axioms. The important diﬀerence is that here the diagrams are taken in the coarse category and hence the relevant functions commute only up to closeness.

Deﬁnition 4.1.2. A coarse group is a group object in the Coarse category. That is, the datum of a coarse E × → { } → −1 → space G = (G, ) and morphisms ∗G : G G G, eG : pt G and [-]G : G G such that the

37 following diagrams commute:

id × ∗ G × G × G G G G × G

∗G × idG ∗G

∗ G × G G G (♥) (Associativity)

−1 (id , e ) idG, [-] G G G G × G G G G × G

idG −1 eG (eG, idG) ∗G [-]G , idG ∗G

G × G G G × G G ∗G ∗G (Identity) (Inverse)

−1 We call eG the unit and [-]G the inversion function. When no confusion can arise we will denote the group operations by ∗, e and [-]−1 respectively.

Given a coarse group (G, ∗, e, [-]−1), the unit and inverse functions are (coarsely) determined by ∗. This can be proved by hand or, more conceptually, by examining sets of morphisms in Coarse. Namely, given any coarse space X, we can deﬁne an operation on set of coarse maps MorCrs(X, G) as the composition ( f,g) ∗ f g B X −−−−→ G × G −→ G. This is the analogue of the pointwise multiplication of functions. A simple diagram chase proves the following:

Lemma 4.1.3. The set MorCrs(X, G) equipped with the binary operation is a set-group, where e ∈ −1 MorCrs(X, G) is the unit and the inverse of an element f ∈ MorCrs(X, G) is the composition [-] ◦ f.

Corollary 4.1.4. If G is a coarse group, then e and [-]−1 are uniquely determined by ∗.

e e0 Proof of the corollary. Suppose {pt} −→ G and {pt} −→ G are two units for G. By Lemma 4.1.3 0 MorCrs({pt}, G) is a set-group and e, e are both units in MorCrs({pt}, G). In a set-group the unit is unique and therefore e = e0. Now suppose [-]−1 : G → G is an inversion map, i.e. a controlled function such that ∗ ◦ ([-]−1, id) = −1 −1 ∗◦(id, [-] ) = e. Since the composition [-] ◦ f is the set-group inverse of f for every f ∈ MorCrs(G, G), −1 the coarse map [-] is the unique set-group inverse of id ∈ MorCrs(G, G). That is, it is a uniquely deﬁned function up to closeness.

If G is coarsely connected, all the controlled maps {pt} → G are close and the ﬁrst part of Corol- lary 4.1.4 becomes trivial. The set MorCrs({pt}, G) can be thought of as the set of coarsely connected components of G. It follows from Lemma 4.1.3 that MorCrs({pt}, G) is a set-group. This implies that every coarse group can be decomposed into a set-group and a coarsely connected coarse group.

38 Notation 4.1.5. In view of Corollary 4.1.4, we can lighten the notation and denote coarse groups as (G, ∗). More often than not, we will further simplify the notation by omitting ∗ and simply write G. On the contrary, when dealing with concrete examples it is often convenient to ﬁx some representatives ∗, e, (-)−1 for the corresponding coarse functions. In this case we will often make them explicit using the notation G = (G, E, [∗], [e], [(-)−1]). This will happen most notably in Subsections 4.2 and 4.3.

It is interesting to apply Lemma 4.1.3 to trivially coarse spaces (I, Emin). Recall that MorCrs((I, Emin), G) is nothing but the set of functions from the set I into G considered up to closeness is a group. Lemma 4.1.3 implies that this is a set-group. If I is a finite set then MorCrs((I, Emin), G) can be naturally identified with the direct power MorCrs({pt}, G). If I is infinite the situation is more delicate.

4.2 The point-set approach It is useful to spell out explicitly some consequences of the diagrams (♥) on the level of functions among sets and I-indexed partial coverings (we will use the notation from Subsection 3.3). Recall that we identify functions x: {pt} → X with points x ∈ X (Convention 2.4.5). We often tacitly use this convention. Given a coarse space (G, E) and a function on ∗: G × G → G, we wish to describe more concretely what it means for ∗ to be controlled, coarsely associative etc. Note that we are not assuming that G is a group, just a coarse space. In this section we study if we can make (G, E, ∗) into a coarse group. For any pair of points x, y ∈ G we will write x ∗ y instead of ∗(x, y). The main technical result of this subsection is proved using indexed partial coverings. Given two I-indexed partial coverings aI, bI of G, we deﬁne aI ∗I bI as the I-indexed partial covering

aI ∗I bI B (Ai ∗ Bi | i ∈ I) = (∗(Ai × Bi) | i ∈ I). A fundamental observation which will enable us to provide various concrete examples of coarse groups is a direct consequence of Lemma 2.6.3. That is, given a function ∗: G × G → G we can see it as a family of left multiplication functions g∗ = ∗(g, - ): G → G and as a family of right multiplication functions ∗g = ∗(- , g): G → G. Then Lemma 2.6.3 implies: Corollary 4.2.1. Let G be a set with a coarse structure E. A multiplication function ∗: (G, E) × (G, E) → (G, E) is controlled if and only if both the families of left multiplications g∗ and right multiplications ∗g are equicontrolled. Now we rephrase Corollary 4.2.1 more explicitly in terms of controlled entourages. Given two relations E, F on G, we let E ∗ F B (e1 ∗ f1, e2 ∗ f2) (e1, e2) ∈ E, ( f1, f2) ∈ F) = (∗ × ∗)(E ⊗ F). (2) It follows by the deﬁnition that ∗: (G, E) × (G, E) → (G, E) is controlled if and only if E ∗ F ∈ E for every E, F ∈ E. In particular, the left multiplications g∗: (G, E) → (G, E) are equicontrolled if and only if ∆G ∗ E ∈ E for every E ∈ E. Note also that ∆G ∗ E is the relation obtained by taking the union of the “translations of E under G”: [ [

∆G ∗ E = (g, g) ∗ E = {(g ∗ e1, g ∗ e2) (e1, e2) ∈ E}. g∈G g∈G This suggests the following nomenclature: Deﬁnition 4.2.2. Given a set G and a multiplication function ∗: G × G → G, a coarse structure E on G is equi-left-invariant if ∆G ∗ E ∈ E for every E ∈ E. Equivalently, E is equi-left-invariant if and only if the left multiplications g∗: (G, E) → (G, E) are equicontrolled. Equi-right-invariance is deﬁned analogously, and we say that E is equi-bi-invariant if it is both equi-left-invariant and equi-right-invariant.

39 We can thus rephrase Corollary 4.2.1 as:

Corollary 4.2.3. Let G be a set with a coarse structure E. A multiplication function ∗: (G, E) × (G, E) → (G, E) is controlled if and only if E is equi-bi-invariant.

Recall also that the left mutiplications g∗ are equicontrolled with respect to E if and only if the covering S{g ∗ a | g ∈ G} is controlled whenever a ∈ C(E). That is, E is equi-left-invariant if and only if the coverings g ∗ a ∈ C(E) are ‘uniformly’ controlled as g ∈ G varies.

Lemma 3.3.8 allows us to rephrase the diagrams (♥) in terms of indexed partial coverings. In the following we are somewhat abusing notation by talking about the diagrams (♥) for functions that are not necessarily controlled (and hence do not deﬁne morphisms in Coarse). Here we say that “a diagram commutes” if the compositions of functions along diﬀerent paths are close to one another.

Lemma 4.2.4. Let (G, E) be a coarse space and ∗: G × G → G, e: {pt} → G and (-)−1 : G → G be ﬁxed functions (not necessarily controlled). The relevant diagrams (♥) commute if and only if for every choice of I and I-indexed partial coverings aI, bI, cI we have

aI ∗I (bI ∗I cI) I (aI ∗I bI) ∗I cI (Associativity)

(e ∗I aI) I aI I (aI ∗I e) (Identity) −1 −1 ∗ ◦ (idG, (-) ) I(aI) I e I ∗ ◦ ((-) , idG) I(aI) (Inverse) where e also denotes the constant I-indexed partial covering (e | i ∈ I) (we are identifying the function e with its image e(pt) ∈ G).

Proof. Necessity follows immediately from Lemma 3.3.8(1). Suﬃciency is obtained by choosing I = G and considering the covering (x | x ∈ G).

Remark 4.2.5. Note that Lemma 4.2.4 does not say anything about compositions of those identities. For −1 example, it does not claim that (aI ∗I aI ) ∗I aI I aI. This is because we are not yet assuming that the −1 −1 functions be controlled. Letting aI be the image of a under (-)I , one might also be tempted to say that −1 −1 (Inverse) implies that (aI ∗I aI ) I e I (aI ∗I aI), but this is also not necessarily true. For one, it is certainly necessary to restrict to controlled partial coverings, and it turns out that it is also necessary to assume that the function ∗ be controlled (see the proof of Claim 1 of Theorem 4.2.6). On the other hand, if ∗ and (-)−1 are controlled then we can say much more. In fact, when this is the −1 case Lemma 3.3.8(2) implies that ∗I and (-)I preserve equi-asymptoticity and hence deﬁne functions

−1 ∗I : CI(E)/I × CI(E)/I → CI(E)/I and (-)I : CI(E)/I → CI(E)/I .

Applying both statements of Lemma 3.3.8, we can deduce that if (G, E, [∗], [e], [(-)−1]) is a coarse group C E ∗ −1 the data ( I( )/I , I, (-)I , e) is a set-group. In fact, since we already remarked that there is a natural correspondence between CI(E)/I and MorCrs((I, Emin), (G, E)), the above is an alternative description for the group MorCrs((I, Emin), G). The following is the main technical result of this subsection. Its proof is elementary but delicate, as it deals with functions that are not a priori controlled. Extra care is required when managing compositions and close functions. This proof is the main reason we introduce I-indexed partial coverings. Given arbitrary (possibly non-controlled) maps among coarse spaces, we say that they make a diagram commute up to closeness if the maps obtained as compositions along diﬀerent paths on the diagram are close. In particular, a diagram of controlled maps commutes up to closeness if and only if the induced diagram in Coarse commutes.

40 Theorem 4.2.6. Let (G, E) be a coarse space and ∗, e, (-)−1 functions that make the diagrams (♥) commute up to closeness. If ∗ is controlled then (-)−1 is also controlled and hence G = (G, E, [∗], [e], [(-)−1]) is a coarse group. Proof. Let I be an arbitrary index set. Using the notation from Subsection 4.2, we need to show that for −1 every I-indexed controlled partial covering aI the image partial covering aI is also controlled. −1 −1 Claim 1. For every I-indexed controlled partial covering aI of G we have (aI ∗I aI ) I e I (aI ∗I aI). Proof of Claim. Note that for every A ⊆ G we can write A ∗ A−1 set-wise as

[ −1 [ −1 −1 A ∗ a ⊆ St a ∗ a , {A ∗ g | g ∈ G} = St ∗ ◦ (idG, (-) )(A) , ∗(A, iG) . a∈A a∈A

−1 −1 In particular, if aI is an I-indexed controlled partial covering then aI ∗I aI = (Ai ∗ Ai | i ∈ I) is a −1 reﬁnement of St ∗ ◦ (idG, (-) ) I(aI) , ∗(aI, iG) . Since ∗ is controlled, the functions ∗g are equicontrolled by Lemma 2.6.3. If a is any controlled partial covering of G, it follows that also ∗(a, iG) is a controlled partial covering (Lemma 2.6.2). By deﬁnition of 4, we have −1 −1 St ∗ ◦ (idG, (-) ) I(aI) , ∗(aI, iG) 4I ∗ ◦ (idG, (-) ) I(aI) and the latter is equi-asymptotic to the constant partial covering e by Lemma 4.2.4. The proof that −1 e I aI ∗I aI is analogous, and it uses that the left multiplication functions are equi-controlled. This proves the claim.

−1 To ﬁnish the proof of Theorem 4.2.6, we will now show that the partial covering aI is equi-asymptotic −1 −1 to a partial covering consisting of singletons. This implies that aI is controlled and hence (-) is −1 controlled as well. For every i ∈ I choose gi ∈ Ai and let gI B ({gi} | i ∈ I). Note that aI ∗I gI I e (this can be seen directly or as a consequence of Claim 1). We can thus use (Identity), (Associativity), Claim 1 and once more (Identity) to obtain the following chain of equivalences:

−1 −1 −1 −1 −1 −1 −1 −1 aI I aI ∗I e I aI ∗I (aI ∗I gI ) I (aI ∗I aI) ∗I gI I e ∗I gI I gI . Again, in the above it is crucial that ∗ is controlled. This concludes the proof. Remark 4.2.7. Theorem 4.2.6 means that if we are given a coarse multiplication ∗ and a function (-)−1 that is a ‘candidate inversion’ in Coarse then (-)−1 is automatically controlled and hence [-]−1 is an actual coarse inversion map. This is not true in other categories. For example, if (G, ∗, e, (-)−1) a set-group and G has a topology that makes ∗ continuous, it is not necessarily the case that (-)−1 is continuous (and hence G need not be a topological group).8. Corollary 4.2.8. If (G, ∗, (-)−1, e) is a set-group and E is a coarse structure on G, then (G, E, [∗], [e], [(-)−1]) is a coarse group if and only if E is equi-bi-invariant. As a special case, we see: Corollary 4.2.9. If (G, ∗, e, (-)−1) is a group and G is equipped with a bi-invariant metric d, then −1 (G, Ed, [∗], [e], [(-) ]) is a coarse group. Remark 4.2.10. An alternative point of view on Corollary 4.2.9 is that the coarsiﬁcation of a metric group is a coarse group. We refer to AppendixB for more on this.

8Set-groups equipped with topologies so that the multiplication is continuous are called paratopological groups. Paratopo- logical groups need not be topological groups. One such example is the group of real numbers equipped with the Sorgenfrey topology. Other examples can be taken by considering groups of self-homeomorphisms equipped with the compact-open topology (one such example is given in [Dij05])

41 We can use Corollary 4.2.9 to provide our ﬁrst explicit examples of non-trivial coarse groups. In fact, it is relatively easy to come up with examples of groups admitting unbounded bi-invariant metrics. When equipped with the associated coarse structure, these will hence be unbounded coarse groups. Example 4.2.11. If G is an abelian coarse group, every left-invariant metric is bi-invariant. To deﬁne a left-invariant metric on a group it is enough to specify a length function | - |: G → R≥0 such that • |g| = 0 if and only if g = e;

• |g−1| = |g| for all g ∈ G;

• |hg| ≤ |g| + |h| for all g, h ∈ G.

To any length function is associated a left invariant metric defined by d(g, h) B |g−1h|. One case of special interest is when the group G is finitely generated: any choice of generating set S defines a word length

± ± |g|S B min{n | g = s1 ··· sn , si ∈ S } and hence a left-invariant word distance. Example 4.2.12. Refining Example 4.2.11, we can also define bi-invariant metrics on non-abelian groups. Namely, if a length function is invariant under conjugation then the induced left-invariant metric is actually bi-invariant. One way to obtain conjugation invariant length functions is to consider the word length associated with some (infinite) conjugation-invariant generating set. These metrics are also called cancellation metrics (see below) and they are a useful source of examples and will play a role in various parts of this monograph. A nice study of cancellation metrics is carried out in [BGKM16]. A set S ⊆ G normally generates G if the normal closure S B {gsg−1 | s ∈ S, g ∈ G} is a generating set of G. Let S a normally generating set of G, the cancellation length | - |× on G is the word length function associated with the (usually infinite) set S . The name “cancellation length” is justified as follows. Assume that S is a generating set G = hS i. For every word w in the alphabet S we can say that its cancellation length is equal to the number of letters that is necessary to cancel in order to get a word that represents the trivial element e ∈ G. Then it is easy to show that the cancellation length |g|S equals the smallest cancellation length of a word w representing g ∈ G [BGKM16, Proposition 2.A]. By construction, the cancellation length | - |× is invariant under conjugation and it thus defines a bi-invariant metric d× on G, which we call the cancellation metric d×. (Notice that when S is infinite G does not have bounded geometry when equipped with this metric.) Let F2 = ha, bi be the free group on two generators and let d× be the cancellation metric associated with the generating set S = {a, b}. One interesting feature of the coarse group (F2, Ed× ) is that it is not coarsely abelian. That is, the controlled maps (g, h) 7→ g ∗ h and (g, h) 7→ h ∗ g are not close as functions n n (F2 × F2, Ed× ⊗ Ed× ) → (F2, Ed× ). To prove this it is enough to that show that d([a , b ], ∅) = 2n. By removing every occurrence of a and a¯ from the commutator [an, bn], we get the word bnb¯n which reduces to the empty word. So d([an, bn], ∅) ≤ 2n. To see d([an, bn], ∅) ≥ 2n one needs to show that this path is optimal, and it is not hard to check it by hand. We give more detailed examples of groups with bi-invariant metrics and cancellation metrics in Sections8 and 11.

4.3 Making sets into coarse groups This is a somewhat technical subsection that will be useful later. It is useful to reverse our point of view: we first fix some set G (not necessarily a group), functions ∗: G × G → G and (-)−1 : G → G and an element e ∈ G. Our goal is to characterise the coarse structures E on G that make (G, E, [∗], [e], [(-)−1]) into a coarse group. That is, to find coarse structures E so that the multiplication and inversion are controlled functions and the diagrams (♥) commute.

42 In general, if {Ei}i∈I are coarse structures on a set X and f : X → X is a function that is controlled T when X is equipped with Ei for each i ∈ I, then f is controlled also when X is equipped with ∈ Ei. T T T i I Moreover, ( i∈I Ei) ⊗ ( i∈I Ei) = i∈I(Ei ⊗ Ei) as coarse structures on X × X. From these observations it −1 follows that whenever {Ei}i∈I are coarse structures on G that make (G, ∗, e, (-) ) into a coarse group, then T so does their intersection i∈I Ei (commutativity of the diagrams (♥) is clearly preserved under taking intersections). It is also clear that letting E = Emax makes G into a coarse group. We can thus give the following:

Deﬁnition 4.3.1. Given a set G, functions ∗: G × G → G, (-)−1 : G → G and an element e ∈ G we let Egrp Egrp ∗ −1 Egrp ∗ −1 min B min(G, , e, (-) ) be the minimal group coarse structure on G, i.e. (G, min, [ ], [e], [(-) ]) is a Egrp coarse group. Equivalently, min is the intersection of all the coarse structures that make G into a coarse group.

∗ −1 Egrp E { } Remark 4.3.2. If (G, , e, (-) ) is an actual group, then min = min = ∆G is the minimal coarse structure. That is, set-groups are trivially coarse groups when seen as trivially coarse spaces. E ∗ −1 Egrp By deﬁnition, a coarse structure that makes (G, , e, (-) ) into a coarse group must contain min. E Egrp ∗ −1 ♥ Vice versa, if is a coarse structure containing min then , (-) and e make the diagrams ( ) commute for (G, E). It then follows that such an E makes G into a coarse group if and only if ∗ is controlled. In other words, we can rephrase this as an extension of Corollary 4.2.9:

Corollary 4.3.3. A coarse structure E makes (G, ∗, e, (-)−1) into a coarse group if and only if it contains Egrp min and it is equi-bi-invariant. Egrp It is possible to describe min fairly explicitly as the coarse structure generated by some set of relations. Before doing so, it is convenient to introduce some extra notation. Given a relation E on G and g, h ∈ G, it will be convenient to write g - E → h if (g, h) ∈ E. Note that if x - E → y - F → z then x - E◦F → z. We will also use the shorthand notation g ← E → h to mean that (g, h) belongs to either E or Eop (← E → is not an equivalence relation). Using this notation, in order to prove that E belongs to a coarse structure E it is enough to show that there exist E0,..., En ∈ E such that for every (x, y) ∈ E there are z1,..., zn ∈ G with

x ← E0 → z1 ← E1 → · · · ← En−1 → zn ← En → y.

Deﬁne relations on G as follows:

Eas B (g1 ∗ (g2 ∗ g3), (g1 ∗ g2) ∗ g3) g1, g2, g3 ∈ G −1 −1 Einv B g ∗ g , e g ∈ G ∪ (g ∗ g, e) g ∈ G Eid B (g ∗ e, g) g ∈ G ∪ (e ∗ g, g) g ∈ G .

By deﬁnition, the above are the relations that make the diagrams (♥). Recall that the product of two relations is deﬁned as E ∗ F = {(e1 ∗ f1, e2 ∗ f2) | (e1, e2) ∈ E, ( f1, f2) ∈ F}. We can now explicitly Egrp describe min: Lemma 4.3.4. Given a set with multiplication, unit and inversion functions (G, ∗, e, (-)−1), the minimal coarse structure making it into a coarse group is

Egrp h ∗ ∗ ∗ ∗ i min = ∆G Eas , (∆G Einv) ∆G , ∆G Eid , Eid . E h ∗ ∗ ∗ ∗ i E ⊆ Egrp Proof. Let B ∆G Eas , (∆G Einv) ∆G , ∆G Eid , Eid . It is clear that min because Eas, Egrp ♥ Einv and Eid are contained in min so that the diagrams ( ) commute, and the relations obtained by Egrp ∗ multiplying them by ∆G must be in min in order for to be controlled.

43 It remains to show the other containment. That is, we need to show that E makes G into a coarse group. Note that Eas ∈ E because

g1 ∗ (g2 ∗ g3) ← Eid → (e ∗ (g1 ∗ (g2 ∗ g3))) ← ∆G∗Eas → (e ∗ ((g1 ∗ g2) ∗ g3)) ← Eid → (g1 ∗ g2) ∗ g3.

An analogous argument shows that Einv ∈ E as well. By Theorem 4.2.6, all it remains to prove is that ∗: (G, E) × (G, E) → (G, E) is controlled. In order to do so, we need to show that ∆G ∗ F and F ∗ ∆G belong to E whenever F is one of ∆G ∗ Eas, ∆G ∗ Einv, ∆G ∗ Eid or Eid (see the discussion in Subsection 2.5). We will prove in detail that ∆G ∗ (∆G ∗ Eas) ∈ E and (∆G ∗ Eas) ∗ ∆G ∈ E; the other cases are similar (and simpler). The former is easy to prove, because h1 ∗ h2 ∗ (g1 ∗ (g2 ∗ g3)) ← Eas → (h1 ∗ h2) ∗ (g1 ∗ (g2 ∗ g3))

← ∆G∗Eas → (h1 ∗ h2) ∗ ((g1 ∗ g2) ∗ g3) ← Eas → h1 ∗ h2 ∗ ((g1 ∗ g2) ∗ g3) for every h1, h2, g1, g2, g3 ∈ G. The proof of the latter is slightly more involved: h2 ∗ (g1 ∗ (g2 ∗ g3)) ∗ h1 ← Eas → h2 ∗ (g1 ∗ (g2 ∗ g3)) ∗ h1 ← ∆G∗Eas → h2 ∗ g1 ∗ ((g2 ∗ g3) ∗ h1) ← ∆G∗(∆G∗Eas) → h2 ∗ g1 ∗ (g2 ∗ (g3 ∗ h1))

← ∆G∗Eas → h2 ∗ ((g1 ∗ g2) ∗ (g3 ∗ h1)) ← ∆G∗Eas → h2 ∗ ((g1 ∗ g2) ∗ g3) ∗ h1 ← Eas → h2 ∗ ((g1 ∗ g2) ∗ g3) ∗ h1.

Note that on the third line it was essential that we already knew ∆G ∗ (∆G ∗ Eas) ∈ E. Analogous computations for Einv and Eid shows that ∗ is controlled and thus concludes the proof. Having studied Egrp , the next step is to study the minimal coarse structure that contains a given set of min grp relations. More precisely, let R ⊂ P(G × G) be any set of relations. We deﬁne ER to be the minimal grp −1 coarse structure under which (G, ER , [∗], [e], [(-) ]) is a coarse group and that contains R: \ grp n −1 o ER = E R ⊆ E, (G, E, [∗], [e], [(-) ]) is a coarse group grp −1 grp (notice that ER depends on the choice of representative ∗, e, (-) ). It turns out that ER is easy to R Egrp ∗ R { ∗ | ∈ R} R ∗ { ∗ | ∈ R} describe in terms of and min. Let E B E R R and E B R E R . We can prove the following: Lemma 4.3.5. Let R be any set of relations on the set G. Then Egrp h ∗ R ∗ Egrp i h ∗ R ∗ Egrp i R = (∆G ) ∆G , min = ∆G ( ∆G) , min . h ∗R ∗ Egrp i h ∗ R∗ Egrp i ∈ Egrp Proof. To begin with, it is easy to see that (∆G ) ∆G, min = ∆G ( ∆G), min , because Eas min op and for every relation E ⊆ G × G we have containments (∆G ∗ E) ∗ ∆G ⊆ Eas ◦ ∆G ∗ (E ∗ ∆G) ◦ Eas op and ∆G ∗ (E ∗ ∆G) ⊆ Eas ◦ (∆G ∗ E) ∗ ∆G ◦ Eas. We proceed as in the proof of Lemma 4.3.4: let E h ∗ R ∗ Egrp i Egrp ⊆ E B ∆G ( ∆G), min . It then follows that R and—using Theorem 4.2.6—it is enough to show that ∗:(G, E) × (G, E) → (G, E) is controlled to conclude the proof of the lemma. We ﬁrst show that the left multiplication g∗ is equicontrolled, i.e. that ∗: (G, Emin) × (G, E) → (G, E) ∗ ∈ Egrp ⊆ Egrp ∈ Egrp is controlled. We already know that ∆G E min R for every E min, it is hence enough to show that ∆G ∗ [∆G ∗ (R ∗ ∆G)] ∈ E for every R ∈ R. This is readily done: op ∆G ∗ (∆G ∗ (R ∗ ∆G)) ⊆ Eas ◦ (∆G ∗ ∆G) ∗ (R ∗ ∆G) ◦ Eas op ⊆ Eas ◦ ∆G ∗ (R ∗ ∆G) ◦ Eas ∈ E. ∗ E h ∗R ∗ Egrp i To prove that the right multiplication g is equicontrolled it is enough to write B (∆G ) ∆G, min and use the same argument. By Lemma 2.6.3 it follows that ∗: (G, E) × (G, E) → (G, E) is controlled.

44 Remark 4.3.6. Given a set of relations R on G we can also ask what is the smallest equi-left-invariant Eleft R Egrp ⊆ Eleft Egrp ⊆ R coarse structure R containing . If we already know that min R (e.g. because min ), then it is easy to check that Eleft h ∗ R Egrp i R = ∆G , min op op (the key point being that ∆G ∗ (∆G ∗ E) ⊆ Eas ◦ (∆G ∗ ∆G) ∗ E ◦ Eas ⊆ Eas ◦ ∆G ∗ E ◦ Eas for every relation E ⊆ G × G). E Example 4.3.7. Recall that the coarse structure of finite off-diagonal fin is the minimal coarsely connected Egrp h ∗ ∗ | ⊆ × i coarse structure. If G is a set-group, Lemma 4.3.5 implies that fin = ∆G E ∆G E G G finite (the order of multiplication does not matter, as G is a set-group). The latter is also equal to the coarse structure generated by the set ∆G ∗ (A × {e}) ∗ ∆G with A ⊆ G finite. Notice that (∆G ∗ (A × {e}) ∗ ∆G)(e) = S{ ∗ ∗ −1 | ∈ } Egrp g A g g G . It follows that the coarse group (G, fin ) is bounded if and only if there exists a Egrp finite set A whose conjugates cover G. Since min is the minimal coarsely connected equi-bi-invariant coarse structure on G, this shows that the set-groups that do not admit unbounded coarsely connected equi-bi-invariant coarse structures are precisely the groups having only finitely many conjugacy classes. Egrp For Lemma 4.3.4 it is truly necessary to add Eid in the set of generators of min (as opposed to keeping only ∆G ∗ Eid). This can be easily seen by considering an function ∗ that is not surjective. For the sake of concreteness, here is an example: Example 4.3.8. For example, let (G, ∗) be a set-group. Consider the set G00 = G × {±1} and extend the functions ∗, (-)−1 as (x, ±1) ∗ (y, ±1) B (x ∗ y, +1) and (x, ±)−1 B (x−1, +1). That is, G00 is obtained by “doubling” all the elements in G and extending the multiplication function in such a way that its image × { } Egrp 00 ∗ E ⊗ P {± } is always contained in G +1 . It is not hard to show that min(G , ) is min ( 1 ). On the other ∗ × { } × × { } Egrp hand, any relation of the form E F is contained in (G +1 ) (G +1 ), this show that min we cannot be generated using only “product relation”.

4.4 Adapted and cancellative operations Example 4.3.8 is admittedly artiﬁcial, but it does serve to point at some technical diﬃculties that can arise when studying general coarse groups. These may compound to make proofs very technical and complicated. For this reason it can be useful to assume some ‘decency’ condition on the operation functions, under which they behave more like group operations.

Deﬁnition 4.4.1. Let G be a set. A choice of functions ∗: G × G → G, (-)−1 : G → G and e ∈ G are adapted if g ∗ e = e ∗ g = g and g ∗ g−1 = g−1 ∗ g = e for every g ∈ G.

The assumption that functions are adapted can simplify technical statements. For example, Lem- mas 4.3.4 and 4.3.5 can be rephrased as:

∗ −1 Egrp h ∗ i Corollary 4.4.2. Let G be a set. If , e and (-) are adapted operations on G then min = ∆G Eas . If a family R of relations on G contains Eas, then

grp left ER = h∆G ∗ (R ∗ ∆G)i = h(∆G ∗ R) ∗ ∆Gi and ER = h∆G ∗ Ri.

It is easy to see that we can always assume that the operations are adapted if G is a coarse group:

Lemma 4.4.3. If (G, E, ∗, e, [-]−1) is a coarse group we can ﬁnd adapted representatives ∗, e, (-)−1 for the coarse operations.

Proof. Choose representatives ∗, e, (-)−1 for ∗, e and [-]−1. Deﬁne g ∗0 e B g and g ∗0 h B g ∗ h if h , e. Then ∗0 is close to ∗ and hence it is also a representative for ∗. Similarly, we can also assume that e ∗ g = g.

45 We now wish to define g ∗0 g−1 B e for every g ∈ G. However, we need do this in a way that preserves the modifications done in the previous paragraph, as we are already requiring that g ∗0 e = g. By modifying the representative (-)−1 if necessary, we can always assume that g−1 = e if and only if 0 0 g = e. For example, we can let e(−1) B e and set g(−1) B g whenever g−1 = e. This modification to (-)−1 ensures that the two modifications to ∗0 are compatible. Finally, we need to impose that g−1 ∗0 g = e. The modification to (-)−1 implies that this does not conflict with the requirement that e ∗0 g = g. It is also compatible with the requirement that g ∗0 g−1 = e, because these two modifications only overlap in the case where (g−1)−1 = g and in this case they are compatible as g−1 ∗0 g = e = g−1 ∗0 (g−1)−1.

For cardinality reasons, it is not always possible to assume that (-)−1 is an involution:

Example 4.4.4. Let G B A0 t A1 t A3 where the Ai are three sets of diﬀering cardinalities. Take the coarse structure hA0 × A0, A1 × A1, A2 × A2i, so that G is coarsely equivalent to the set {0, 1, 2} with the trivial coarse structure. Equip G with coarse group operations making it isomorphic (as a coarse group) −1 to (Z/3Z, Emin). Since |A1| , |A2| the inversion (-) cannot be bijective, and hence it is not an involution. This example also shows that the multiplication functions g∗ and ∗g cannot be assumed to be bijections.

Deﬁnition 4.4.5. Let G be a set. A choice of functions ∗: G × G → G, (-)−1 : G → G and e ∈ G are cancellative if they are adapted and the left and right multiplication functions g∗: G → G and ∗g : G → G are bijective for every g ∈ G, with inverses given by g−1 ∗ and ∗g−1 respectively. When ∗, e and (-)−1 are cancellative, it is much easier to perform algebraic manipulations on G. This is because g∗h1 = g∗h2 if and only if h1 = h2, and similarly for h1 ∗g = h2 ∗g. The cancellation law implies that the left and right inverse of an element g ∈ G are uniquely determined. The adaptedness condition further implies that the left and right inverse coincide. It follows that (-)−1 is uniquely determined by ∗ −1 −1 −1 and it is an involution (hence a bijection). Note however that (g1 ∗ g2) need not equal g2 ∗ g1 .

As already noted, the operations on a coarse group need not have cancellative representatives. However, it is always possible to construct a kindred coarse group that does admit such representatives. This is constructed by taking a sets of reduced formal multiplications of elements of G:

Theorem 4.4.6. Let (G, E, [∗], [e], [(-)−1]) be any coarse group. Then there exists a coarse group −1 (Gˇr, Eˇ, [∗ˇ], [eˇ], [(-) ]) and a controlled function Eval: Gˇr → G that induces a coarse equivalence and so that ∗ ◦ (Eval × Eval) = Eval ◦ ∗ˇ and (-)−1 ◦ Eval = Eval ◦ (-)−1.

−1 Proof. We can assume that the representatives ∗, e, (-) are adapted on G. We will deﬁne the set Gˇr as a subset of the set of formal compositions of multiplications of elements of G and their inverses. To be precise, let W(G) be the set of ﬁnite formal words in the alphabet

g | g ∈ G t g¯ | g ∈ G t [ , ] ,? .

Let Gˇ be the subset of W(G) recursively deﬁned as follows:

• Gˇ contains the empty word ∅,

• Gˇ contains the 3-letters words [g] and [¯g] for every g ∈ G,

• if w1 and w2 are words in Gˇ, then [w1 ? w2] is a word in Gˇ.

Let now Gˇr ⊂ Gˇ be the subset of reduced words, i.e. Gˇr consists of those words that do not contain [g]] ? [[g¯] nor [g¯]] ? [[g] as subword for any g ∈ G. There is a natural reduction map Red: Gˇ → Gˇr: this

46 is defined by recursively replacing subwords of the form [w1 ? [g]] ? [[g¯] ? w2] and [w1 ? [g¯]] ? [[g] ? w2] with [w1 ? w2], where w1, w2 denote any two words in Gˇ. We will now define some cancellative functions on Gˇr. The multiplication function ∗ˇ : Gˇr × Gˇr → Gˇr is defined as the composition

concatenation reduction Gˇr × Gˇr −−−−−−−−−−→ Gˇ −−−−−−−→ Gˇr

(w1, w2) 7→ [w1 ? w2] 7→ Red([w1 ? w2]).

−1 The unite ˇ is the empty string ∅ and the inverse function (-) : Gˇr → Gˇr is deﬁned recursively:

• ∅−1 B ∅,

• [g]−1 B [¯g] and [¯g]−1 B [g], −1 −1 −1 ˇ • [w1 ? w2] B [w2 ? w1 ] for every [w1 ? w2] is in Gr. −1 By construction, the inversion (-) is an involution of Gˇr and the left and right multiplication functions w∗ˇ and∗ ˇw are bijections with inverses w−1 ∗ˇ and∗ ˇw−1 . Furthermore, there is a natural surjective −1 function Eval: Gˇr → G, which is obtained by replacing g¯, [ , ] ,? with g , ( , ) , ∗ and evaluating the resulting expression. The empty string ∅ is sent to e. ∗ We can now equip Gˇr with the pull-back coarse structure Eˇ B Eval (E). Since Eval is surjective, −1 it deﬁnes a coarse equivalence [Eval]: (Gˇr, Eˇ) → (G, E). Importantly, since ∗, e, (-) are adapted, it follows easily from the deﬁnition that the diagrams

Eval×Eval Eval Gˇr × Gˇr G × G Gˇr G

∗ˇ ∗ (-)−1 (-)−1 Eval Eval Gˇr G Gˇr G commute (as functions among sets). Given E, F ∈ Eˇ, it is then immediate to check that ∗ˇ × ∗ˇ(E ⊗ F) is in Eˇ = Eval∗(E), i.e. ∗ˇ is controlled with respect to Eˇ. We similarly deduce that (-)−1 is controlled and the diagrams (♥) commute (up to closeness).

Remark 4.4.7. It is worth making a few remarks:

1. The conditions on the map Eval imply that [Eval]: (G, E) → (Gˇ, Eˇ) is an isomorphism of coarse groups (Subsection 5.1). That is, Theorem 4.4.6 states that every coarse group is isomorphic to a coarse group admitting cancellative representatives.

2. One might be optimistic that Theorem 4.4.6 can be improved to show that every coarse group is isomorphic to a set-group equipped with an equi-bi-invariant coarse structure. However, this should not be the case. In Section 11 we construct some examples of coarse groups which we conjecture are not isomorphic to any “coarsiﬁed set-group”.

3. Theorem 4.4.6 can be very useful to prove “purely coarse” statements (i.e. statements that are invariant under coarse equivalence/isomorphism). For example, suppose we wish to prove that a certain property P1 for a coarse group G implies P2. If both P1 and P2 are invariant under isomorphisms of coarse groups, we can apply Theorem 4.4.6 to reduce the proof to the case where G has cancellative operations. In this monograph we will not use often this strategy: since we are mainly dealing with rather elementary properties we found it more instructive to sort out the technical details explicitly. The approach via reduction to cancellative operations is going to be more helpful when dealing with more advanced statements.

47 Recall that a bounded set for a given coarse structure E on G is a subset B ⊆ G such that B × B ∈ E. The following results will be used later on. Lemma 4.4.8. Let ∗, e, (-)−1 be cancellative on the set G. For any equi-left-invariant coarse structure E and E ∈ E there exists a E-bounded set B ⊆ G with e ∈ B such that E ⊆ ∆G ∗ (B × {e}).

Proof. Enlarging E if necessary, we can assume that ∆G ⊆ E. Let B B (∆G∗E)(e). The set B is E-bounded because ∆G ∗ E ∈ E. To prove the lemma it will be enough to show that E(g) × {g} ⊆ (g, g) ∗ (B × {e}) for every g ∈ G. −1 Since the operations are adapted, e ∈ B and for every g ∈ G we have g ∗ E(g) ⊆ (∆G ∗ E)(e). Since g∗ is the inverse of g−1 ∗, it follows that E(g) ⊆ g ∗ B and E(g) × {g} ⊆ (g, g) ∗ (B × {e}). −1 Corollary 4.4.9. If ∗, e, (-) are cancellative on the set G, there exists a set Bas ⊆ G with e ∈ Bas and Egrp h ∗ × { } i h ∗ × i min = ∆G (Bas e ) = ∆G (Bas Bas) . Egrp h ∗ i Egrp Proof. By Corollary 4.4.2, min = ∆G Eas . It is then enough to choose Bas as a min-bounded set with ∆G ∗ Eas ⊆ ∆G ∗ Bas.

Corollary 4.4.10. Given G and Bas ⊆ G as in Corollary 4.4.9, then

g1 ∗ (g2 ∗ g3) ∈ ((g1 ∗ g2) ∗ g3) ∗ Bas −1 (g1 ∗ g2) ∗ g3 ∈ (g1 ∗ (g2 ∗ g3)) ∗ Bas Proof. By deﬁnition,

g1 ∗ (g2 ∗ g3) ∈ Eas((g1 ∗ g2) ∗ g3) ⊆ (∆G ∗ (Bas × {e})((g1 ∗ g2) ∗ g3) = ((g1 ∗ g2) ∗ g3) ∗ Bas. This proves the ﬁrst containment. For the other one, we have

op (g1 ∗ g2) ∗ g3 ∈ Eas(g1 ∗ (g2 ∗ g3)) ⊆ (∆G ∗ ({e} × Bas)(g1 ∗ (g2 ∗ g3)) −1 and the latter is equal to (g1 ∗ (g2 ∗ g3)) ∗ Bas .

4.5 Coarse groups are determined locally The main result of this subsection is the coarse analogue of the fact that the topology of a topological group is determined by the set of open neighbourhoods of the identity. Interestingly, we will later see that this fact lies at the heart of the fundamental observation of geometric group theory (the Milnor–Švarc Lemma). ∗ −1 Egrp As in the previous section, let (G, , e, (-) ) be a set with operations and min be the minimal coarse structure making it into a coarse group. For any coarse structure E on G let Ue(E) be the set of bounded neighbourhoods of the identity e ∈ G

Ue(E) B {U | U ⊆ G bounded, e ∈ U}

(we are once again identifying e: {pt} → G with its image). The set Ue(E) is a partial cover of G (note however it is generally not controlled). For every U ∈ Ue(E) we have a relation U × U ∈ E. We let Ue(E) be the set of such relations: Ue(E) B {U × U | U ∈ U(E)}. Remark 4.5.1. The word “neighbourhood” is used quite loosely. If B ⊆ G is any bounded set contained in the same coarsely connected component of e, then U = B ∪ {e} is a bounded neighbourhood of identity. In particular, if (G, E) is coarsely connected then every bounded set is ‘morally’ a bounded neighbourhood of e. Note also that the assumption e ∈ U is made out of convenience, but it is somewhat unnatural within the coarse category.

48 Recall that a coarse structure E on G is equi-left-invariant if ∆G ∗ E ∈ E for every E ∈ E (Defini- tion 4.2.2). The following lemma shows that equi-left-invariant coarse structures are determined by their bounded sets: Proposition 4.5.2. Given (G, ∗, e, (-)−1) as above and an equi-left-invariant coarse structure E such that Egrp ⊆ E min , then E h ∗ U E Egrp i = ∆G e( ) , min . Moreover, if the operations are cancellative then E = h∆G ∗ Ue(E)i. E0 h ∗ U E Egrp i E0 ⊆ E E ⊆ E0 Proof. Let B ∆G e( ) , min , it is clear that . We thus have to prove that . Let E ∈ E be fixed. Note that if g1 ← E → g2 then −1 −1 (g1 ∗ g2) ← ∆G∗E → (g1 ∗ g1) ← Eid → e. −1 It follows that the set U B {g1 ∗ g2 | (g1, g2) ∈ E} ∪ {e} is a E-bounded neighbourhood of e ∈ G and hence U × U ∈ Ue(E). We can now write −1 g2 ← Eid◦Eas → g1 ∗ (g1 ∗ g2) ← ∆G∗(U×U) → g1 ∗ e ← Eid → g1, and deduce that E ∈ E0. For the second statement, it follows from Corollary 4.4.9 that E = h∆G ∗ Ue(E) , ∆G ∗ (Bas × Bas)i for some bounded neighbourhood of the identity Bas. By definition, Bas × Bas ∈ Ue(E), so E = h∆G ∗ Ue(E)i. Obviously, an analogous statement holds if E is equi-right-invariant. We thus have the following. Corollary 4.5.3. If (G, E, [∗], [e], [(-)−1]) is a coarse group, then E h ∗ U E Egrp i hU E ∗ Egrp i = ∆G e( ) , min = e( ) ∆G , min . Equivalently, in terms of controlled coverings we have C E h{i ∗ | ∈ U E } ∪ C Egrp i h{ ∗ i | ∈ U E } ∪ C Egrp i ( ) = G U U e( ) ( min) = U G U e( ) ( min) , where iG ∗ U = {g ∗ U | g ∈ G} and hai denotes the smallest set of partial coverings containing a and which is closed under (C0)–(C2). E Egrp Proposition 4.5.2 and its corollary imply that any equi-left-invariant coarse structure containing min is completely determined by its bounded sets (as opposed to entourages). This fact makes it much easier to check whether two equi-left-invariant coarse structures on G are equal. Notice also that Corollary 4.5.3 does not hold for arbitrary coarse structures. For example, the coarse structures with compact fibers as in Example 2.2.7 all have the same bounded sets (i.e. relatively compact subsets of X) but differ as coarse structures.

Example 4.5.4. Let G be a finitely generated set-group with finite generating set S , and let dS be the associated word metric (Example 4.2.11). Bounded sets in G are sets of finite diameter and, since S is finite, this means that the bounded sets are precisely the finite subsets of G. This immediately implies E Eleft that dS does not depend on the choice of the finite generating set. Morever, recall that we defined fin as the minimal left invariant coarse structure so that all the finite sets are bounded (Remark 4.3.6). Since all E E ⊆ Eleft the dS -bounded sets are finite, d fin . By minimality we see that this is actually an equality. If S is a finite normally generating set, let d× be the cancellation metric associated with S (i.e. the Egrp word metric with respect to the set S of all the conjugates of S , see 4.2.12), and fin the minimal E equi-bi-invariant coarse structure containing fin. Since S is finite, it follows by the equi-bi-invariance Egrp E that S is fin -bounded. It also follows from the definition that d× -bounded sets are contained in finite ± ± ··· Egrp E ⊆ Egrp products S S , and are hence fin -bounded (see also Subsection 4.6). This shows that d× fin and, by minimality, these two coarse structures must be equal. In particular, Ed× does not depend on the choice of the finite generating set S .

49 If the operations ∗, e, (-)−1 are cancellative, it is further possible to use Proposition 4.5.2 to describe controlled entourages and partial coverings in terms of products with bounded sets: Corollary 4.5.5. If (G, E, [∗], [e], [(-)−1]) is a coarse group with cancellative operations, then E = h∆G ∗ Ue(E)i = hUe(E) ∗ ∆Gi. In particular, a partial cover a is controlled if and only if there exists a bounded neighbourhood of identity e ∈ U ⊆ G such that a is a reﬁnement of iG ∗ U and U ∗ iG. −1 Corollary 4.5.6. Let (G, E, [∗], [e], [(-) ]) be a coarse group with cancellative operations and aI, bI any two I-indexed partial covers. Then aI 4I bI if and only if there exists a bounded neighbourhood of identity e ∈ U ⊆ G such that Ai ⊆ Bi ∗ U for every i ∈ I.

Proof. One implication is clear, because Bi ∗ U ⊆ St(Bi, iG ∗ U) and iG ∗ U is a controlled covering of G. For the other implication, let c be a controlled covering such that that Ai ⊆ St(Bi, c) for every i ∈ I. By 0 0 Corollary 4.5.5, we may assume that c = iG ∗ U for U ∈ Ue(E). Therefore, for every i ∈ I we have [ 0 0 0 −1 0 Ai ⊆ St Bi, iG ∗ U = St Bi, {g ∗ U | g ∈ G} = g ∗ U ∃x ∈ Bi s.t. e ∈ x ∗ (g ∗ U ) , where we used that the operations are cancellative for the last equality. We further rewrite this as

0 [ −1 0 −1 0 St Bi, iG ∗ U = x ∗ (x ∗ (g ∗ U )) | g ∈ G, e ∈ x ∗ (g ∗ U ) . x∈Bi Consider the bounded set

0 [ 0 0 U B St(e, iG ∗ (iG ∗ U )) = g1 ∗ (g2 ∗ U ) | g1, g2 ∈ G s.t. e ∈ g1 ∗ (g2 ∗ U ) . Then we see that

0 [ 0 0 St Bi, iG ∗ U ⊆ x ∗ (g1 ∗ (g2 ∗ U )) | g1, g2 ∈ G s.t. e ∈ g1 ∗ (g2 ∗ U ) = Bi ∗ U, x∈Bi thus proving the claim. The assumption that the operations on G are cancellative is truly necessary for Corollaries 4.5.5 and 4.5.6. Again, this can be seen by considering coarse groups where the multiplication map ∗ is not surjective (Example 4.3.8). Remark 4.5.7. It is interesting to compare Corollary 4.5.3 with Lemma 4.3.5. Speciﬁcally, it is a curious fact that if we already know that Ue(E) is the family of relations obtained from the bounded neighbourhoods of e in a coarse group (G, E) then the coarse structure Egrp is generated by Egrp and Ue(E) min ∆G ∗ Ue(E) (as opposed to having to take (∆G ∗ Ue(E)) ∗ ∆G). As we shall soon see, this is related with the fact that the family of neighborhoods Ue(E) is closed under conjugation.

4.6 Sets of neighbourhoods of the identity Let G be a set with operations ∗, e, (-)−1. We showed in Subsection 4.3 that every family of relations R grp induces an equi-bi-invariant coarse structure ER that makes G into a coarse group. In particular, if we are given any partial cover a of G we can let R = {A × A | A ∈ a} to obtain a equi-bi-invariant coarse structure for which all the elements of a are bounded. We showed in Subsection 4.5 that such a coarse grp structure is determined by the set Ue(ER ) of bounded neighbourhoods of the identity. Here we compare grp Ue(ER ) with the original partial covering a. The aim of this subsection is to characterise those partial covers u which are the collection of bounded neighbourhoods of the identity for some coarse structure E making G into a coarse group. That is, letting Ru B {U × U | U ∈ u}, we wish to ﬁnd those u so that that u = U Egrp e Ru

50 (if u = U (E) and (G, E) is a coarse group then we necessarily have E = Egrp). We will manage to do so e Ru under the assumption that the operations are cancellative. E Egrp U E Let be a coarse structure on G containing min and let e( ) be the set of bounded neighbourhoods of e ∈ G. Then

(U0) e ∈ U for every U ∈ Ue(E);

(U1) if U ∈ Ue(E) and e ∈ V ⊂ U then V ∈ Ue(E).

If the operations are cancellative, by Corollary 4.4.9 there exists a Bas ⊆ G with e ∈ Bas and such that Egrp h ∗ × i min = ∆G (Bas Bas) . It follows that

(U2) Bas ∈ Ue(E).

Assume now that E is also equi-left-invariant and let U ∈ Ue(E). Then (∆G ∗ (U × U))(e) ∈ Ue(E) (it contains e because the operations are adapted). Since the operations are cancellative we can write:

[ [ −1 (∆G ∗ (U1 × U2))(e) = g ∗ U1 | g ∈ G, e ∈ g ∗ U2 = g ∗ U1 | g ∈ U2 . (3)

Since e ∈ U and the operations are well adapted, g ∈ g ∗ U for every g ∈ G. Therefore we have:

−1 (U3) if U ∈ Ue(E) then U ∈ Ue(E).

−1 If U1, U2 are E-bounded then U1 × U2 ∈ E. Applying (3) to U1 × U2 ∈ E we get

(U4) if U1, U2 ∈ Ue(E) then U1 ∗ U2 ∈ Ue(E). Finally, if E is equi-bi-invariant we see as in (3) that for every U with e ∈ U

[ −1 g ∗ (U ∗ g ) | g ∈ G ⊆ (∆G ∗ ((U × U) ∗ ∆G))(e) whence

S −1 (U5) if U ∈ Ue(E) then {g ∗ (U ∗ g ) | g ∈ G} ∈ Ue(E). u Egrp It turns out that the above conditions imply that is the set of {U×U | U∈u})-bounded neighbourhoods of the identity:

Proposition 4.6.1. Assume that ∗, e and (-)−1 are cancellative operations on the set G. Let u be a partial left left cover satisfying (U0)–(U4) and let Ru {U × U | U ∈ u}. Then u = U E , where E is the minimal B e Ru Ru R Egrp equi-left invariant coarse structure containing u and min. If u also satisfies (U5) then u = U Egrp (in which case Eleft = Egrp). e Ru Ru Ru Proof. We start with the first equality. It is clear that u ⊆ U Eleft, so it remains to check the other e Ru containment. left grp grp We know that E = h∆ ∗Ru , E i (see Remark 4.3.6). By Corollary 4.4.9, E = h∆ ∗(B ×B )i. Ru G min min G as as left Let Ru Ru ∪ {B × B }, then E is generated by the family of relations ∆ ∗ Ru . ,as B as as Ru G ,as From (3) and (U0)–(U4) we deduce that (∆G ∗ R)(e) ∈ u for every R ∈ Ru,as. To prove our claim it will be enough to show that the family of relations in Eleft such that E(e) ∪ {e} ∈ u is a coarse structure. Ru Such family is obviously closed under taking unions with the diagonal ∆G and is closed under taking subsets because u satisfies (U1). All that remains to show is that if E, F ∈ E are relations such that E(e) and F(e) are in u then (E ◦ F)(e) is in u as well (Lemma 2.2.9). Note that (E ◦ F)(e) = S{E(g) | g ∈ F(e)}. As in the proof of Lemma 4.4.8, we see that E(g) ⊆ g ∗ ((∆G ∗ E)(e)) and hence (E ◦ F)(e) ⊆ F(e) ∗ ((∆G ∗ E)(e)).

51 Once again, it follows from (3) and (U0)–(U4) that (∆G ∗ E)(e) ∈ u, hence (E ◦ F)(e) ∈ u by (U4).

For the second equality, assume now that u also satisﬁes (U5). Again one containment is clear. The proof of the other containment follows the same steps of before: the coarse structure Egrp is generated by Ru ∆G ∗ (Ru,as ∗ ∆G) for the same set of relations Ru,as. We need to show that conditions (U0)–(U5) imply that ∆G ∗ (R ∗ ∆G) (e) ∈ u for every R ∈ Ru,as. As for (3), we have:

[ [ −1 ∆G ∗ ((U1 × U2) ∗ ∆G) (e) = g1 ∗ (U1 ∗ g2) | e ∈ g1 ∗ (U2 ∗ g2) = g1 ∗ (U1 ∗ g2) | g2 ∗ g1 ∈ U2 .

−1 −1 −1 We can rewrite the latter as union of (g2 ∗ (g2 ∗ g1)) ∗ (U1 ∗ g2) ⊆ (g2 ∗ U2 ) ∗ (U1 ∗ g2). Applying Corollary 4.4.10 three times, we also have

−1 −1 −1 −1 −1 (g2 ∗ U2 ) ∗ (U1 ∗ g2) ⊆ g2 ∗ U2 ∗ (U1 ∗ g2) ∗ Bas −1 −1 −1 ⊆ g2 ∗ ((U2 ∗ U1) ∗ g2) ∗ Bas ∗ Bas −1 −1 −1 ⊆ g2 ∗ ((U2 ∗ U1) ∗ g2) ∗ Bas ∗ Bas ∗ Bas .

S −1 −1 If U1, U2 ∈ u, then the union ∈ g ∗ ((U ∗ U1) ∗ g2) is in u by (U4) and (U5). Applying (U4) again, g2 G 2 2 we deduce that (∆G ∗ (U1 × U2))(e) is in u. This proves that ∆G ∗ (R ∗ ∆G) (e) ∈ u for every R ∈ Ru,as. The rest of the proof is now precisely as before.

5 Coarse homomorphisms, subgroups and quotients

5.1 Coarse homomorphisms

Let G = (G, E) and H = (H, F ) be coarse groups with coarse multiplication ∗G and ∗H respectively. A coarse homomorphism is a map such that multiplying in the domain and then applying the map is asymptotically close to first applying the map and then multiplying in the image. We define this formally with another diagram: Definition 5.1.1. A coarse homomorphism is a coarse map f : G → H such that the following diagram commutes: f× f G × G H × H

∗G ∗H f G H.

We denote the set of coarse homomorphism by HomCrs(G, H) ⊆ MorCrs(G, H).

Concretely, if ∗G = [∗G] and ∗H = [∗H], then a controlled function f : G → H gives rise to a coarse homomorphism [ f ]: G → H if and only if f (x ∗G y) and f (x) ∗H f (y) are uniformly close (i.e. there is an F ∈ F such that f (x ∗G y) ← F → f (x) ∗H f (y) for every x, y ∈ G). Example 5.1.2. If f : (G, E) → (H, F ) is a bounded map sending G into the coarsely connected component of (H, F ) containing eH, then [ f ] = eH is trivially a coarse homomorphism. A second trivial source of examples is obtained by considering set-homomorphisms between coarse groups arising by coarsifying set-groups. More precisely, suppose G and H are set-groups equipped with equi-bi-invariant coarse structures and let f : G → H be a set-group homomorphism. If f is a controlled map, then [ f ] is of course a coarse homomorphism, since the diagram in Deﬁnition 5.1.1 commutes at the level of sets.

52 Notice that in this case it is particularly simple to verify whether f is controlled. By Corollary 4.5.3, the coarse structures E and F are equal to h∆G ∗ Ue(E)i and h∆G ∗ Ue(E)i respectively. Furthermore, since f is a set-homomorphism, f × f (∆G ∗ E) is contained in ∆H ∗ ( f × f (E)). It follows that f is controlled if and only if it sends bounded neighbourhoods of eG into bounded neighbourhoods of eH. Example 5.1.3. A less trivial example of coarse homomorphisms is given by R-quasimorphisms of set-groups. Let G be a set-group, a function φ: G → R is a R-quasimorphism of G if there is a constant D such that |φ(xy) − φ(x) − φ(y)| ≤ D for every x, y ∈ G (the smallest such constant is called the defect of φ and is denoted by D(φ)). Since R is abelian, (R, Ed) is a coarse group (here d is the Euclidean metric). If E is any coarse structure making G into a coarse group and φ: (G, E) → (R, Ed) is a controlled map, then φ is a coarse homomorphism if and only if it is an R-quasimorphism. It is well-known that there exists R-quasimorphisms that are not close to any set-homomorphism. As a matter of fact, there is a very beautiful connection between the theory of R-quasimorphisms that are not close to homomorphisms and that of low-dimensional bounded cohomology. We refer to [Cal09, Fri17] for more on this. For concreteness we report a family of examples for the free group −1 −1 F2 = ha, bi due to Brooks. Namely, given any word w in {a, b, a , b } we consider the counting funtion Cw : G → N assigning to an element g ∈ G the number of times that the word w appears in the reduced word representing g. The associated Brooks quasimorphism is deﬁned as φw(g) B Cw(g) − Cw−1 (g). Choosing the word appropriately, it is not hard to show that the brooks quasimorphisms can give rise to R-quasimorphisms that are not close to any homomorphism [Bro81, Mit84].9

The above example shows coarse homomorphisms [ f ]: (G, E) → (R, Ed) need not be close to set-homomorphisms. Unsurprisingly, this fact is not speciﬁc to (R, Ed). Below we give an example with

[ f ]: (F2, Ed× ) → (F2, Ed× ), where d× is the cancellation metric as in Example 4.2.12. In Section 12 we will provide many more examples using quasimorphisms in the sense of Hartnick–Schweitzer [HS16].

Example 5.1.4. An interesting example of a coarse homomorphism [ f ]: (F2, Ed× ) → (F2, Ed× ) that is not close to a set-homomorphism is given by Fujiwara–Kapovich [FK16, Section 9.2]. Consider F2 = ha, bi with the cancellation metric. Following [FK16], choose two “appropriate" words, say u = a2ba2 and 2 2 v = b ab . Set H F2 to be the free group in hu, vi. Deﬁne f : F2 → H by

f (w) = fu,v(w) = t1 ··· tn

−1 −1 where ti ∈ {u, u , v, v } in order of appearance in w. For example, with the choices of u, v above we see:

2 2 2 2 2 2 −2 −1 −2 2 −2 −1 −2 −1 −1 fu,v(a ba ba b ab a ) = uuv and fu,v(ba b a b ab a b ) = u v .

It is easy to see that f is a coarse homomorphism. We explain why it is not close to a set-group homomorphism. Notice that f acts as the identity on the cyclic subgroups hui and hvi. So the image of f is unbounded. However, f sends the subgroups generated by hai and hbi to the identity. Thus any homomorphism close to f must send hai and hbi to bounded subgroups, but this is possible only for the trivial homomorphism. Alternatively, let φ: F2 → Z be the set-homomorphism such that φ(u) = 1 ∈ Z and φ(v) = 0. The astute reader will notice the composition f ◦ φ is the Brooks quasimorphism for u. If f was close to a homomorphism, so would be the composition f ◦ φ. Lastly, conjugation gives a ﬁrst example of coarse-homomorphisms of coarse groups that are not necessarily set-groups:

9One way to prove this is by using the fact that every quasi-morphism is close to a unique homogeneous quasimorphism (see Lemma 5.1.15 and Remark 5.1.16). In particular, if φ is close to a homomorphism φ¯ then the latter must be given by the formula ¯ φ(gk) φ(g) = limk→∞ k . Letting w = ab we see that the last formula does not deﬁne a homomorphism, hence φw is not close to any homomorphism.

53 Example 5.1.5. Let G = (G, E) be a coarse group and g ∈ G a fixed element. Then it is immediate to −1 check that the coarse conjugation gc: G → G given by h 7→ (g ∗ h) ∗ g defines a coarse homomorphism, and hence an automorphism of G. On the other hand, if g lies in the same coarsely connected component of e—i.e. the singleton {(e, g)} is a controlled entourage—then ∆G ∗ {(e, g)} is controlled and hence the map h 7→ h ∗ g is close to idG. Analogously, also h 7→ g ∗ h is close to idG and hence gc defines the trivial coarse automorphism [gc] = [idG]. In particular, this shows that if G is coarsely connected then all the conjugation automorphisms are coarsely trivial. More generally, we already remarked that the set of coarsely connected components of G coincides with MorCrs({pt}, G), which is a set-group by Lemma 4.1.3. It is then easy to see that the set of conjugation automorphisms of G can be naturally identified with the group of inner automorphisms of MorCrs({pt}, G).

Naturally, compositions of coarse homomorphisms are coarse homomorphisms. We can thus deﬁne a category of coarse groups Deﬁnition 5.1.6. The category of coarse groups (denoted CrsGroup) is the category with objects that are coarse groups and morphisms that are coarse homomorphisms:

MorCrsGroup(G, H) B HomCrs(G, H). Remark 5.1.7. The following elementary properties of coarse homomorphisms can be easily checked by hand or a diagram chase: 1. If a coarse homomorphism f : G → H is a coarse equivalence, the coarse inverse f −1 : H → G is also a coarse homomorphism. It follows that such an f is an isomorphism of coarse groups.

2. Recall that the set of coarse maps into a coarse group MorCrs(X, G) admits a natural group operation (see the discussion before Lemma 4.1.3). If f : G → H is a coarse homomorphism, the composition with f induces a group homomorphism MorCrs(X, G) → MorCrs(X, H).

3. A coarse homomorphism f : G → H must send eG to eH and it must respect group-inversion: −1 −1 f ◦ [-]G = [-]H ◦ f. The following observation is rather useful:

−1 −1 Lemma 5.1.8. Let [ f ]: (G, E, [∗G], [eG], [(-)G ]) → (H, F , [∗H], [eH], [(-)H ]) be a coarse homomorphism. Then the pull-back coarse structure f ∗(F ) (Deﬁnition 3.2.1) is equi-bi-invariant on (G, ∗) and −1 hence makes (G, ∗G, eG, (-)G ) into a coarse group.

Proof. For notational convenience, we let mG B ∗G and mH B ∗H. This is because we want to show that ∗ ∗ ∗ ∗ ∗ ∗ ∗ mG : (G × G, f (F ) ⊗ f (F )) → (G, f (F )) is controlled by proving that f (F ) ⊗ f (F ) ⊆ mG( f (F )) ∗ and we want to avoid writing ∗G. ∗ ∗ ∗ By the deﬁnition of pull-back, mG : (G × G, mG( f (F ))) → (G, f (F )) is controlled. Since f ◦ mG is close to mH ◦ ( f × f ), we have

∗ ∗ ∗ ∗ ∗ ∗ mG( f (F )) = ( f ◦ mG) (F ) = (mH ◦ ( f × f )) (F ) = ( f × f ) (mH(F )). ∗ Since mH is controlled, F ⊗ F ⊆ mH(F ). It follows that ∗ ∗ ∗ ∗ ∗ f (F ) ⊗ f (F ) = ( f × f ) (F ⊗ F ) ⊆ mG( f (F )) ∗ and hence mG is controlled. This shows that f (F ) is equi-bi-invariant on (G, ∗). The diagrams (♥) commute also when G is equipped with the coarse structure f ∗(F ) because ∗ ∗ −1 E ⊆ f (F ). It then follows from Theorem 4.2.6 that f (F ) makes (G, ∗G, eG, (-)G ) into a coarse group.

54 Corollary 5.1.9. Let f : G → H be a coarse homomorphism. Let G = (G, E) and H = (H, F ) and f be a representative for f. The following are equivalent:

(i) f is a coarse embedding;

(ii) f is proper (Deﬁnition 2.3.6);

−1 (iii)f (U) is bounded for every U ⊂ H bounded neighbourhood of eH. Proof. The only non-trivial implication is (iii) ⇒ (i). By Lemma 5.1.8 f ∗(F ) makes G into a coarse group, and by Proposition 4.5.2 we have

E h ∗ × | E ∪ Egrp i = ∆G (U U) U is a -bounded neighbourhood of eG min ∗ F h ∗ × | ∗ F ∪ Egrp i f ( ) = ∆G (U U) U is a f ( )-bounded neighbourhood of eG min .

Since we may assume that f (eG) = eH, condition (iii) implies that a neighbourhood of eG is E-bounded if and only if it is f ∗(F )-bounded. It follows that E = f ∗(F ) and hence f is a coarse embedding.

Example 5.1.10. Given two coarse groups G, H, Lemma 5.1.8 can be used to construct an intermediate coarse group GH characterised by the property that every coarse homomorphism G → H factors through GH. Namely, if G = (G, E) and H = (H, F ) we let \ n ∗ o EG→H B f (F ) [ f ]:(G, E) → (H, F ) coarse homomorphism .

It follows from Lemma 5.1.8 that EG→H is equi-bi-invariant and hence GH B (G, EG→H) is a coarse group. By construction, E ⊆ EG→H and therefore [idG]: G → GH is a coarse homomorphism. It is also clear that [ f ]: G → H is a coarse homomorphism if and only if [ f ]: GH → H is a coarse homomorphism. This shows that coarse homomorphisms factor through GH:

f=[ f ] G H.

[idG] [ f ] GH

The coarse group GH can be used to quantify the lack or abundance of coarse homomorphisms from G to H. For example, if G = (G, Emin) and (H, Emin) are trivially coarse groups, then GH is bounded if and only if the only set-group homomorphism G → H is the trivial one. If G is a finitely generated L set-group and H is the direct sum of all the finite symmetric groups n∈N S n then G is residually finite if and only if E(G,Emin)→(H,Emin) = Emin. Example 5.1.11. Clearly, the converse of Lemma 5.1.8 does not hold. That is, the fact that f ∗(F ) makes G into a coarse group does not imply that [ f ]: (G, f ∗(F )) → (H, F ) is a coarse homomorphism.To see this it is enough to choose two coarse groups (G, E), (H, F ) and a coarse embedding f : (G, E) ,→ (H, F ). Then f ∗(F ) = E and hence makes G into a coarse group. However there is no reason why f should be a coarse homomorphism. Concretely, equip R and R2 with the Euclidean metric and consider the embedding given by the logarithmic spiral it t 7→ log(1 + |t|) exp log(1 + |t|) (or any other coarse embedding embedding that is not close to a linear embedding).

The following is another useful observation that interacts nicely with Lemma 5.1.8.

55 −1 Lemma 5.1.12. Given a set with mutiplication, unit and inversion functions (G, ∗G, eG, (-)G ), a coarse −1 group H = (H, F , [∗H], [eH], [(-)H ]) and a function f : G → H, the following are equivalent: −1 −1 (i) The composition f ◦ ( - ∗G - ) is close to ∗H ◦ ( f × f ); the composition f ◦ (-)G is close to (-)H ◦ f ; and f (eG) is close to eH. Egrp → F (ii) The function f :(G, min) (H, ) is controlled and [ f ] is a coarse homomorphism. Proof. It is clear that (ii) implies (i): the ﬁrst closeness condition is included in the deﬁnition of coarse homomorphism and the other two are easy consequences (Remark 5.1.7). For the other implication, we Egrp h ∗ ∗ ∗ ∗ i know by Lemma 4.3.4 that min = ∆G Eas , (∆G Einv) ∆G , ∆G Eid , Eid where Eas B (g1 ∗G (g2 ∗G g3), (g1 ∗G g2) ∗G g3) g1, g2, g3 ∈ G −1 −1 Einv B g ∗G g , eG g ∈ G ∪ (g ∗G g, eG) g ∈ G Eid B (g ∗G eG, g) g ∈ G ∪ (eG ∗G g, g) g ∈ G .

It is thus enough to check that f × f sends those generating relations into F . For convenience, we will × ∈ F ∗ Egrp only show that f f (Eas) : checking ∆G G Eas and the other generators of min is just as easy but notationally awkward. Let Fas be the analog of Eas for H. By assumption, there is a controlled set F ∈ F such that f (g1 ∗G g2)- F → f (g1) ∗H f (g2) for every g1, g2 ∈ G. We then see that

f (g1 ∗G (g2 ∗G g3)) - F → f (g1) ∗H f (g2 ∗G g3)

- ∆H∗H F → f (g1) ∗H ( f (g2) ∗H f (g3))

- Fas → ( f (g1) ∗H f (g2)) ∗H f (g3) op op - (F ∗H∆H)◦F → f ((g1 ∗G g2) ∗G g3).

This shows that f × f (Eas) ∈ F . Corollary 5.1.13. Let G and H = (H, F ) be a set and a coarse group as in Lemma 5.1.12, and let f : G → H be a function satisfying item (i) of the same lemma. Then the pull-back f ∗(F ) is equi-bi-invariant on (G, ∗G). Egrp → F Proof. By Lemma 5.1.12 [ f ]: (G, min) (H, ) is a coarse homomorphism and we can hence apply Lemma 5.1.8.

We can combine the results described thus far to prove a criterion that makes it easier to check whether a function deﬁnes a coarse homomorphism:

−1 −1 Proposition 5.1.14. Let G = (G, E, [∗G], [eG], [(-)G ]) and H = (H, F , [∗H], [eH], [(-)H ]) be coarse groups. If a function f : G → H satisﬁes

(i) the composition f ◦ (- ∗G -) is close to ∗H ◦ ( f × f ),

(ii)f (U) eH for every bounded neighborhood of the identity U ∈ Ue(E), then f is controlled and [ f ]: G → H is a coarse homomorphism.

−1 −1 Proof. We claim that the composition f ◦ (-)G is close to (-)H ◦ f . Since G is a coarse group, there is a −1 bounded neighborhood U of eG so that g ∗G g ∈ U for every g ∈ G. By the hypotheses on f , there are controlled sets F1, f2 ∈ F such that

−1 eH - F1 → f (g1 ∗ g1 ) and f (g2 ∗G g3)- F2 → f (g2) ∗H f (g3)

56 for every g1, g2, g3 ∈ G. Hence for every g ∈ G

−1 −1 eH - F1 → f (g ∗ g )- F2 → f (g) ∗H f (g ).

Since H is a coarse group, we have some Fid, Fas, Finv ∈ F so that

−1 −1 f (g) - Fid → f (g) ∗H eH −1 −1 - ∆H∗H(F1◦F2) → f (g) ∗H f (g) ∗H f (g ) −1 −1 - Fas → f (g) ∗H f (g) ∗H f (g ) −1 - Finv∗H∆H → eH ∗H f (g ) −1 - Fid → f (g ) for every g ∈ G. This proves our claim. We now see that f satisﬁes condition (i) of Lemma 5.1.12. By Corollary 5.1.13, the pull-back f ∗(F ) is an equi-bi-invariant coarse structure on G. By hypothesis, we know that E-bounded neighborhoods of ∗ ∗ eG are also f (F )-bounded. It then follows from Proposition 4.5.2 that E ⊆ f (F ) or, in other words, that f :(G, E) → (H, F ) is controlled.

If a coarse homomorphism is a coarse embedding or, equivalently, it is proper (Corollary 5.1.9) then it is a monomorphism in Coarse (Lemma 3.1.1). A fortiori, it is a monomorphism in CrsGroup as well. Similarly, coarsely surjective coarse homomorphisms are epimorphisms in CrsGroup. However, one should be careful. We will construct a monic coarse homomorphisms that is not proper. It is well-known that every R-quasimorphism of a set-group G is close to a homogeneous quasimorphism (i.e. a quasimorphism φ: G → R such that φ(gk) = kφ(g) for every k ∈ Z and g ∈ G. See e.g. [Cal09, Section 2.2]). The same proof (sketched below for convenience) proves the following:

−1 Lemma 5.1.15. Let (G, E, ∗, (-) ) be a coarse group, (V, k·k) a Banach space and Ek·k the associated metric coarse structure on V. Let [ f ]: (G, E) → (V, Ek·k) be a coarse homomorphisms. Then f is close to a map f¯ such that:

(♠) f¯(g ∗ g) = 2 f¯(g) for every g ∈ G.

0 1 −1 −1 Sketch of Proof. Replacing f with f (g) B 2 ( f (g) − f (g )), we can assume that f (g ) = − f (g) for every g ∈ G. Let D B supg,h∈Gk f (g ∗ h) − f (g) − f (h)k, this supremum is ﬁnite because f is a coarse homomorphism. Given g ∈ G, let g0 B g and deﬁne by induction gk+1 B gk ∗ gk ∈ G. The obvious inductive argument shows that f (g ) f (g ) D k k+1 − k k ≤ . 2k+1 2k 2k+1 ¯ f (gk) ♠ ¯ It follows that the function f (g) B limk→∞ 2k is close to f . Condition ( ) holds trivially for f .

Remark 5.1.16. If (G, ∗) is a set-group, we can continue the proof of Lemma 5.1.15 to show that f¯ is a homogeneous coarse homomorphism. Namely, for any g ∈ G we can simply write f¯(g) = 2n n k 2n 2n 2n limn→∞ f (g )/2 . For every k ∈ N, write (g ) as the k-fold product g ∗· · ·∗g to deduce by induction on k that n n k f (gk2 ) − k f (g2 )k ≤ (k − 1)D.

It immediately follows that f¯(gk) = k f¯(g) for every k ∈ Z.

57 Note that the argument in Remark 5.1.16 fails completely if G is a coarse group, because the element “gk” is not well deﬁned: since ∗ is not associative it is absolutely necessary to specify the order of multiplication of the various copies of g. Once such an order is set, we are no longer entitled to write n n n (gk)2 = g2 ∗ · · · ∗ g2 . In particular, when multiplying k elements of G the order of multiplication might conceivably change the end result up to an error that grows linearly with k. As a matter of fact, we do not know the answer to the following:

Question 5.1.17. Let (G, E) be a coarse group, (V, k·k) a Banach space and [ f ]: (G, E) → (V, E·) a coarse homomorphism. Must there exist a representative f¯ ∈ [ f ] and a constant R ≥ 0 such that the k-fold products satisfy k f g ∗ (g ∗ · · · (g ∗ g)) − f ((g ∗ g) ∗ · · · g) ∗ g k ≤ R for every g ∈ G and k ∈ N? What if V = R?

We can use Lemma 5.1.15 to give an example of a monic coarse homomorphism that is not proper: 2 Example 5.1.18. Consider the Hilbert space ` (N) with the coarse structure Ek-k2 induced by the natural 2 2 0 ` -distance. Let v1, v2,... be the natural orthonormal basis of ` (N) and let k-k2 be the (non-complete) 2 0 1 norm on ` (N) obtained by rescaling the `2-norm in such a way that kvnk2 = n for every n ∈ N. Let 0 2 2 Ek k0 be the coarse structure induced by k-k , then both (` (N), Ek k ) and (` (N), Ek k0 ) are abelian coarse - 2 2 - 2 - 2 groups. 2 2 The identity map idk k →k k0 : (` (N), Ek k ) → (` (N), Ek k0 ) is a controlled homomorphism and hence - 2 - 2 - 2 - 2 2 [idk k →k k0 ] is a coarse homomorphism. We claim that it is monic. Let [ f ], [ f ]: (G, E) → (` (N), Ek k ) - 2 - 2 1 2 - 2 be two coarse homomorphisms. Using Lemma 5.1.15, we can assume that f1(g ∗ g) = 2 f1(g) and f2(g ∗ g)) = 2 f2(g) for every g ∈ G. If there exists a g ∈ G such that f1(g) , f2(g), by recursively taking products 0 gk+1 B gk ∗gk we obtain a sequence of elements such that k f1(gk)− f2(gk)k2 → ∞. That is, we just showed that two coarse homomorphisms [ f ], [ f ] such that the composition [idk k →k k0 ◦ f ], [idk k →k k0 ◦ f ] are 1 2 - 2 - 2 1 - 2 - 2 2 Ek k0 -close must be Ek k -close to start with. - 2 - 2 2 2 0 On the other hand idk k →k k0 : (` (N), Ek k ) → (` (N), Ek k0 ) is not proper, because k-k -bounded sets - 2 - 2 - 2 - 2 2 need not be k-k2-bounded. 2 2 Remark 5.1.19. In Example 5.1.18, idk k →k k0 : (` (N), Ek k ) → (` (N), Ek k0 ) is a monic that is (set-wise) - 2 - 2 - 2 - 2 surjective. In particular, it is also an epimorphism in CrsGroup. Since it is not an isomorphism, this shows that CrsGroup is not a balanced category.

5.2 Coarse subgroups Recall that in Subsection 3.4 we showed that there is a natural correspondence between equivalence classes of monic coarse maps Y ,→ X and equivalence classes of subsets of X (here X = (X, E) and the equivalence relation on the subsets of X is E-asymptoticity). This correspondence associates a coarse map f with its coarse image Im( f) = [Im( f )]. Using this correspondence deﬁnes coarse subspaces of X as equivalence classes of monic coarse maps (categorical approach) or, equivalently, as equivalence classes of subsets of X (set-theoretic approach). For coarse groups the situation is more complicated. Example 5.1.18 shows that in CrsGroup there exist inequivalent monomorphisms that have the same coarse image (namely, the maps

2 2 2 2 [idk k →k k0 ]:(` (N), Ek k ) → (` (N), Ek k0 )) and [idk k0 →k k0 ]:(` (N), Ek k0 ) → (` (N), Ek k0 )). - 2 - 2 - 2 - 2 - 2 - 2 - 2 - 2 The categorical approach to deﬁning coarse subgroups will hence produce a diﬀerent result from the set-theoretic approach. We decided to use the latter, as it represents more faithfully the idea of a “coarse subgroup”.

58 Deﬁnition 5.2.1. A coarse subgroup of a coarse group (G, E) is a coarse subspace H = [H] ⊆ G such −1 −1 that (H ∗G H) ⊆ H and H ⊆ H (i.e. such that H ∗G H 4 H and H 4 H). We denote coarse subgroups by H ≤ G.

The following proposition shows that a coarse subgroup uniquely determines a coarse group (up to isomorphism). We will hence be entitled to use the (somewhat imprecise) convention that H denotes both a coarse group and an equivalence class of coarse subsets of G.

−1 Proposition 5.2.2. Let G = (G, E, [∗G], [eG], [(-)G ]) be a coarse group. If H ⊆ G is such that H∗G H 4 H −1 and (H)G 4 H, then H B (H, E|H) admits coarse group operations under which the inclusion [ι]: H ,→ G is a monic coarse homomorphism. The coarse group operations on H are uniquely determined as coarse maps. Furthermore, if H0 H 0 0 0 and H B (H , E|H0 ) then H and H are isomorphic coarse groups (in fact, the inclusions H ,→ G and H0 ,→ G are equivalent monics in CrsGroup).

Proof. Since the inclusion ι: H → G is a coarse embedding (and hence monic in Coarse), it is clear that if there exists a coarse multiplication ∗H : H × H → H such that the following diagram commutes

∗ H × H H H ι×ι ι (4) ∗ G × G G G then ∗H is unique. To construct ∗H, let amult be a controlled covering of G such that H ∗G H ⊆ St(H, amult). For every pair (h1, h2) ∈ H × H choose an h¯ ∈ H such that h1 ∗G h2 ∈ St(h¯, amult) and let h1 ∗H h2 B h¯. It is easy to check by hand that ∗H : (H, E|H) × (H, E|H) → (H, E|H) is a controlled map. Alternatively, we can also note that the composition ι ◦ ∗H : H × H → G is close to ∗G ◦ (ι × ι) to deduce that ι ◦ ∗H is controlled and hence conclude that ∗H is controlled as well (because ι is a coarse embedding). We thus showed that the newly deﬁned ∗H makes Diagram (4) commute. Diagram (4) allows us to deduce associativity of ∗H from the associativity of ∗G. Namely, we have the following equality of coarse maps H × H × H → G:

ι ◦ ∗H ◦ (idH × ∗H) = ∗G ◦ (idG × ∗G ◦ (ι × ι × ι))

= ∗G ◦ (∗G × idG) ◦ (ι × ι × ι) = ι ◦ ∗H ◦ (∗H × idH).

That is, the controlled maps ι ◦ ∗H ◦ (idH ×∗H) and ι ◦ ∗H ◦ (∗H × idH) are close (one can also show it explicitly by making a judicious use of star-neighbourhoods). Since H = (H, E|H) is equipped with the subspace coarse structure, it follows immediately that ∗H ◦ (idH ×∗H) and ∗H ◦ (∗H × idH) are close functions H × H × H → (H, E|H), thus proving associativity of ∗H. −1 −1 Similarly, if ainv is a controlled covering of G such that (H)G ⊆ St(H, ainv) we can deﬁne (h)H by ¯ −1 ¯ −1 −1 −1 choosing a h ∈ H such that (h)G ∈ St(h, ainv). Again, (-)H is controlled and ι ◦ (-)H is close to (-)G |H. −1 We can further let eH B ∗H(h, (h)H ) for an arbitrarily chosen h ∈ H. An argument analogous to the −1 above will show that ∗H, eH and [-]H ) also satisfy (Identity) and (Inverse), and thus make H into a coarse group. The inclusion ι: H → G is a coarse homomorphism by construction. Finally, it is simple to verify that if H0 H are asymptotic subsets that are equipped with coarse 0 0 group operations that make the inclusions [ι]: (H, E|H) → (G, E) and [ι ]: (H , E|H0 ) → (G, E) into coarse 0 0 homomorphisms, then H = (H, E|H) and H = (H , E|H0 ) are isomorphic as coarse groups. One way to do so, is to recall that the inclusions [ι]: H → G and [ι0]: H0 → G are equivalent monics in Coarse i.e. they have a commuting coarse equivalence H → H0. Using monicity of [ι] and [ι0], a simple diagram chase shows that this coarse equivalence is indeed a coarse group isomorphism.

59 Corollary 5.2.3. The set of coarse subgroups of G coincides with set of coarse images of coarse homomorphisms into G. Proof. By Proposition 5.2.2 a coarse subgroup H ≤ G is the coarse image of a coarse homomorphism (the inclusion H ,→ G). Vice versa, if f : H → G is a homomorphism of coarse groups then, by deﬁnition, f ◦ ∗H and ∗G ◦ ( f × f) are close (see the diagram in Deﬁnition 5.1.1). Since Im( f ◦ ∗H) ⊆ Im( f ), we deduce that Im( f ) ∗G Im( f )) 4 Im( f ). That is, the coarse Im( f) is coarsely closed under multiplication. −1 The coarse containment (Im( f ))G 4 Im( f ) is analogous. When it exists, the coarse intersection of coarse subgroups is a coarse subgroup.

Lemma 5.2.4. If H1,..., Hn are coarse subgroups of G and the coarse intersection Z B H1 ∩ ··· ∩ Hn (Deﬁnition 3.5.3) exists, then Z is coarse subgroup of G and of each Hi for i = 1,..., n.

Proof. Let Z and Hi be representatives of Z and Hi respectively. By hypothesis we have Z 4 Hi for all i = 1,..., n. Since Z × Z 4 Hi × Hi and ∗ is controlled, it follows from Lemma 3.3.8 that Z ∗ Z 4 Hi ∗ Hi and the latter is contained in a controlled thickening of Hi because it is a coarse subgroup. As this is true for every i = 1,..., n, it follows by the deﬁnition of coarse intersection that Z ∗ Z 4 Z. −1 Similar arguments show that also eG and Z are coarsely contained in Z. This proves that Z is indeed a coarse subgroup of G. As explained in Subsection 3.5, once Hi is realized as a coarse space (and hence a coarse group), the coarse subspace Z ⊆ G uniquely determines a coarse subspace of Hi. It is immediate that this is a coarse subgroup of Hi. We already remarked that the coarse intersection of two coarse subspaces may not exist (Exam- ple 3.5.6). However, one may wonder whether the intersection of coarse subgroups always exists. The following example shows that this is not the case.

Example 5.2.5. Let G B (`2(N), Ek-k2 ), where k - k2 is the standard `1-norm and let v0, v1,... be the 1 standard orthonormal basis. Let wn B v2n + n v2n+1 and deﬁne the vector subspaces

H B hwn | n ∈ NiR and K B hv2n | n ∈ NiR.

Then [H] and [K] are coarse subgroups of G and we claim that their intersection is not well deﬁned. In fact, for any ﬁxed C > 0 the intersection of the C-neighbourhoods of H and K in `2(N) is the set   X X X 2   2 an  IC B anwn an ∈ R, a < ∞, ≤ C ⊂ `2(N).  n n2  n∈N n∈N n∈N 

It prove the claim it is hence enough to note that IC+1 is not contained in a bounded neighbourhood of IC: for each n ≥ 1 the former contains the vector n(C + 1)wn, and its nearest-point projection in In is nCwn which is at distance nkwnk from it. The following observation is simple, but useful for producing examples of coarse subgroups: Lemma 5.2.6. Let G = (G, E, [∗], [e], [(-)−1]) be a coarse group and H ⊂ G some subset. If there exists a bounded set A ⊂ G containing e and such that both H ∗ H and H−1 are contained in H ∗ A then [H] is an coarse subgroup. If G is a set-group, the converse is also true. Proof. Since ∗ is controlled, it follows from (Identity) that

H ∗ A H ∗ e H and hence H ∗ H 4 H (see Lemma 4.2.4). For the same reason, we also have H−1 4 H. Thus [H] is a coarse subgroup. The fact that for set-groups the converse is also true follows immediately from Corollary 4.5.6.

60 Recall that a subset A of a set-group G is a k-approximate subgroup for some k ∈ N if A = A−1 and AA ⊂ KA, where K ⊂ G has cardinality at most k. That is, A is symmetric and the product of A with itself is covered by k translates of A.

Corollary 5.2.7. If (G, ∗) is a set-group and E makes it into a coarsely connected coarse group G, then every approximate subgroup H ⊂ G determines a coarse subgroup [H].

We recommend the recent monograph [CHT20] for a survey of geometric approximate groups and their actions.

5.3 Coarse quotients In analogy with the previous subsection, we will take a set-theoretic approach to the deﬁnition of coarse quotient groups. We hence give the following:

−1 Deﬁnition 5.3.1. A coarse quotient of a coarse group G = (G, E, [∗]E, [e]E, [(-) ]E) is a coarse group 0 −1 G = (G, E , [∗]E0 , [e]E0 , [(-) ])E0 obtained by equipping the same set G with a larger equi-bi-invariant coarse structure E0 ⊇ E. Here we used the notation [∗]E to emphasize that we are using the same set-functions to deﬁne the operations in the coarse quotient group, we have just enlarged the coarse structure. We decorated [-] with the coarse structure to draw attention to the fact that the closeness equivalence relation on functions depends on the coarse structure. Hereafter we will suppress the coarse structure decoration from the group operations.

At this point it is worth going back to our first years of undergraduate studies and remembering that the quotient of a set-group G is defined by fixing a normal subgroup N C G and noting that the set of cosets G/N admits a group operation. Then if f : G → Q is a surjective homomorphism then ker( f ) is a normal subgroup of G and Q is a quotient group. By the first isomorphism theorem, f induces an isomorphism between Q and the quotient G/ ker( f ). The situation in the coarse setting is completely analogous. Definition 5.3.1 provides us with a concrete object (set with operations and coarse structure), just as G/N provided us with a concrete group. Moreover, if Q = (Q, F ) is a coarse group and [ f ] = f : G → Q is a coarsely surjective coarse homomorphism, then G B (G, f ∗(F )) is a coarse group by Lemma 5.1.8 and hence it is a coarse quotient of G. Further, f descends to a natural coarse homomorphism f = [ f ]: G → Q which is an isomorphism because f is coarsely surjective (Corollary 3.2.2). Thus whenever we are given a coarsely surjective coarse homomorphism G → Q we are entitled to say that Q is a coarse quotient of G. Later on we will also show that coarse quotients admit a description that is more reminiscent of coset spaces ([ref]). Note that, in some sense, Lemma 5.1.8 is the coarse analogue of the first isomorphism theorem. The following result can also be seen as an isomorphism theorem. It is almost a tautology, but it is good to write it explicitly:

Proposition 5.3.2. Let G = (G, E) be a coarse group and R a family of relations on the set G. The minimal coarse structure E(R) such that E ∪ R ⊆ E(R) and that makes G into a coarse group is equal to

E(R) = h(∆G ∗ R) ∗ ∆G, Ei = h∆G ∗ (R ∗ ∆G), Ei. (5)

The coarse quotient group G/hhRii B (G, E(R)) has the property that a coarse homomorphism [ f ] = f : G → H = (H, F ) descends to G/hhRii if and only if f × f (R) ∈ F for every R ∈ R.

E R Egrp h ∗ R ∗ Egrp i Proof. It follows from Lemma 4.3.5 that ( ) must contain R = (∆G ) ∆G, min and hence h∆G ∗ (R ∗ ∆G), Ei ⊆ E(R). The same argument as in Lemma 4.3.5 shows that h∆G ∗ (R ∗ ∆G), Ei is equi-bi-invariant, and hence the above containment is an equality. A similar argument works for

61 h∆G ∗ (R ∗ ∆G), Ei as well. In particular the coarse structures h(∆G ∗ R) ∗ ∆G, Ei and h∆G ∗ (R ∗ ∆G), Ei are indeed equal. E h ∗E ∗ Egrp i Alternatively, one can also show that (5) holds by noting that = (∆G ) ∆G, min by Lemma 4.3.5 and hence h ∗ R ∗ Ei h ∗ R ∗ ∗ E ∗ Egrp i h ∗ R ∪ E ∗ Egrp i (∆G ) ∆G, = (∆G ) ∆G, (∆G ) ∆G, min = (∆G ( )) ∆G, min .

Applying Lemma 4.3.5 once again we see that E(R) = h(∆G ∗ R) ∗ ∆G, Ei. Let f : G → H be a coarse homomorphism. If f × f (R) ∈ F for every R ∈ R, then R ∪ E ⊆ f ∗(F ) and hence E(R) ⊆ f ∗(F ) by Lemma 5.1.8 and minimality of E(R). It follows that f : (G, E(R)) → (H, F ) is controlled and hence yields a coarse homomorphism. On the other hand, if f descends to a controlled map then it must send the relations R ∈ R into F . Definition 5.3.3. Given a coarse group G = (G, E) and any set R of relations on G, we say that G/hhRii B (G, E(R)) is the quotient of G by R. The quotient coarse group G/hhRii is not to be confused with the quotient coarse space G/R. Rather, the above definition has a strong analogy with group presentations. The group G/hhRii can be thought of as the group obtained by adding the relations R to “a presentation” of G. One case of interest is that of quotients of the form G/hh{A × A}ii where A ⊆ G is some arbitrary subset. Loosely speaking, this is the quotient of G by a sort of “coarse normal closure of A”. Note however that there is an important difference between this notion of quotient and actual group presentations: Namely, if G is a set-group and A ⊂ G is some subset, the set-group obtained by adding A to a presentation of G is the set-group quotient G/hhAii where the normal closure hhAii is the subgroup consisting of all finite products of conjugates of elements in A and their inverses. In particular, hhAii itself becomes trivial in the quotient. This is not the case for coarse quotients. Namely, even though each finite product of elements in A will be at bounded distance from e in the coarse quotient G/hh{A × A}ii, this bound is not uniform and hence it may be that the set of all the finite products of elements of A is not bounded. For instance, consider G = (R, E|-|) and let A = B(0; 1) be the unit ball around 0. Since A generates R as a set-group, the quotient R/hhAii is trivial. However, A is bounded and hence the coarse quotient G/hh{A × A}ii is still G! Similar observations will also play a role when discussing quotient coarse actions and coarse coset spaces (Subsections 6.5 and 6.6). The above example of a coarse quotient behaving very differently from a set-group quotient may be somewhat unsatisfying. In fact, we use that A {e} to deduce that the coarse quotient G/hh{A × A}ii is simply G. This suggests that it should always be possible to find a (coarse) subgroup [H] ≤ G so that G/hh{A × A}ii = G/hh{H × H}ii. The next example shows that this is not the case (this is related to the fact that coarse homomorphisms need not have a coarse kernel, see Section7). 2 Example 5.3.4. As in Example 5.1.18, let G be the Hilbert space ` (N) with the coarse structure Ek-k2 2 0 induced by the ` -distance, v1, v2,... the natural orthonormal basis and let k-k2 be the Hilbert norm 1 0 defined by kv k . Let A Bk k0 (0; 1) ⊂ G be the unit ball for the norm k-k . Then it is easy to n 2 B n B - 2 2 check that G/hh{A × A}ii is nothing but (` (N), Ek k0 ). However, we claim that there is no coarse subgroup 2 - 2 [H] ≤ G such that G/hh{A × A}ii = G/hh{H × H}ii. This can be deduced from the fact that the epimorphism G → G/hh{A × A}ii is also a monomorphism in CrsGroup (as explained in Example 5.1.18). In fact, assume for contradiction that G/hh{A × A}ii = G/hh{H × H}ii for some coarse subgroup [H] ≤ G. Since the quotient map G → G/hh{A × A}ii is not an isomorphism, we see that H cannot be bounded.The composition of monomorphisms f : H ,→ G → G/hh{A × A}ii is itself a monomorphism. However, H is bounded in G/hh{A × A}ii = G/hh{H × H}ii and hence the compositions

id f e f H −−−→H H −→ G/hh{A × A}ii H −→ H −→ G/hh{A × A}ii coincide. Since f is a monomorphism, idH = e and hence H is bounded, which is a contradiction.

62 We conclude this section by pointing out that—unlike the case of coarse subgroups—we do not know for certain whether the categorical and set-theoretical approach to the deﬁnition of coarse quotient group are inequivalent. This is because we do not know the answer to the following:

Question 5.3.5. If q: G → Q is an epimorphism in CrsGroup, is it an epimorphism in Coarse? That is, are epimorphisms of CrsGroup necessarily coarsely surjective?

We suspect that the answer to the above question should be negative. On the other hand, if it was positive it would imply that categorical and set-theoretical approaches to deﬁning quotient subgroups coincide. This would make this theory somewhat easier to handle. Remark 5.3.6. The easiest way to show that epimorphisms in Group are surjective is by noting that the image of a homomorphism q: G → Q is a subgroup of Q and then considering the amalgamated product Q ∗q(G) Q. If q(G) , Q, the two inclusions of Q into Q ∗q(G) Q show that q is not an epimorphism. The diﬃculty in adapting this proof to the context of coarse group is that there does not appear to be a good analogue for the amalgamated product. More generally, it is not clear to us whether CrsGroup admits free products and/or free objects.

6 Coarse actions

In this section we study coarse actions of coarse groups. One of the main points is to illustrate the connection between coarse group actions and geometric group theory. More precisely, geometric group theory can be described as the study of coarse actions of trivially coarse groups.

6.1 Basic deﬁnitions Let G = (G, E) be a coarse group and Y = (Y, F ) any coarse space.

Deﬁnition 6.1.1. A (left) coarse action is a coarse map α: G × Y → Y such that the following diagrams commute id ×α (e,id ) G × G × Y G G × Y Y Y G × Y

∗×idY α α idY G × Y α Y Y In analogy with the notation α: G y Y for set-group actions, we denote coarse actions by α: G y Y (or [α]: (G, E) y (Y, F ) if we want to specify that the controlled map α is a representative for the coarse map [α]).

Example 6.1.2. Here are some elementary examples of left coarse actions.

1. The trivial coarse action is the coarse action deﬁned by g · y = y for every g ∈ G, y ∈ Y (i.e. it is the projection to G × Y → Y).

2. The multiplication operation ∗: G × G → G is a left coarse action of G on itself.

3. More generally, let G = (G, E) and let F ⊇ E be any coarser coarse structure on G. It is an easy exercise to show that the multiplication map ∗ deﬁnes a left coarse action of (G, E) on (G, F ) if and only if F is equi-left-invariant (Deﬁniton 4.2.2). Despite being elementary, we will later see that this is the prototypical example of a coarse action.

ﬁb Example 6.1.3. Recall a topological space X is endowed with two natural coarse structures Ecpt ⊆ Ecpt whose bounded sets are the relatively compact sets (Example 2.2.7). If G is a topological group, the

63 family Ke of relatively compact subsets of G containing the identity element satisfies the conditions left (U0)–(U4) of Subsection 4.6. It follows that there exists a unique equi-left-invariant coarse structure Ecpt left left fib whose set of bounded neighbourhoods of the identity Ue(Ecpt ) is Ke. We see that that Ecpt ⊆ Ecpt ⊆ Ecpt left 10 and, by equi-left-invariance, [∗]:(G, Emin) y (G, Ecpt ) is a coarse action. In this specific case we can left also describe very explicitly the coarse structure Ecpt . Namely,

left −1 Ecpt B {E ⊆ G × G | ∃K ⊆ G compact s.t. ∀(g1, g2) ∈ E, g2 g1 ∈ K}.

Its controlled partial covers are refinements of covers of the form iG ∗ K with K ∈ Ke. As for usual set-group actions, it is often convenient to write coarse actions as binary operations. That is, once a coarse action [α]: (G, E) y (Y, F ) is fixed we prefer to write g · y instead of α(g)(y). We will only use this notation when no confusion can arise. Recall that for a fixed F ∈ F we use the shorthand 0 0 y ← F → y if (y, y ) is in F or Fop. Using this notation, commutativity of the diagrams in Definition 6.1.1 takes the familiar form:

g1 · (g2 · y) ← F → (g1 ∗ g2) · y and e · y ← F → y for some ﬁxed F ∈ F and every choice of g1, g2 ∈ G and y ∈ Y. It is also convenient to extend the binary notation to relations on G and Y: given E ⊂ G × G and F ⊂ Y × Y we let

E · F B {(e1 · f1, e2 · f2) | (e1, e2) ∈ E, ( f1, f2) ∈ F} = α × α(E ⊗ F)

(this is convention is analogous to the deﬁnition of E1 ∗ E2). Example 6.1.4. The construction of Example 6.1.3 has a natural extension to actions of topological groups on topological spaces. Namely, if α: G y Y is a continuous action of a topological group left on a topological space, we can consider the coarse structure Fcpt whose controlled partial covers are reﬁnements of coverings of the form {g(K) | g ∈ G, K compact}. Equivalently,

left Fcpt B {F ⊆ Y × Y | ∃K ⊆ Y compact s.t. ∀(y1, y2) ∈ F, ∃g ∈ G with y1, y2 ∈ g(K)}.

With this coarse structure, α becomes a coarse action of the trivially coarse group [α]: (G, Emin) y left (Y, Fcpt ). Example 6.1.5. Let α: G y Y be a set-group action. Then the diagrams of Definition 6.1.1 trivially commute for any choice of coarse structures on G and Y. It follows that such an α defines a coarse action [α]:(G, E) y (Y, F ) if and only if it is controlled. As a special case, it is interesting to note that [α]: (G, Emin) y (Y, F ) is a coarse action if and only if ∆G · F ∈ F for every F ∈ F . Recalling Definition 4.2.2, we can thus say that α defines a coarse action of the trivially coarse group (G, Emin) on (Y, F ) if and only if the coarse structure F is equi-left-invariant under α. If (Y, d) is a metric space and Y is equipped with the metric coarse structure Ed, then it is easy to check that [α] is a coarse action if and only if it is an action by uniform coarse equivalences. That is, there must be (increasing, unbounded) controlled functions ρ−, ρ+ : [0, ∞) → [0, ∞) such that

ρ−(d(y1, y2)) ≤ d(g · y1, g · y2) ≤ ρ+(d(y1, y2)) for every pair of points y1, y2 ∈ Y and g ∈ G (the upper bound follows directly from checking that the functions α(g) are equicontrolled. The lower bound is obtained by observing that α(g−1) is the inverse

10 left ﬁb Note that this is an action of the trivially coarse group (G, Emin). In general, the coarse structures Ecpt, Ecpt and Ecpt do not make G into a coarse group.

64 of α(g)). Moreover, if (Y, d) is a geodesic metric space this only happens if α is an action by uniform 1 quasi-isometries, i.e. there must be constants L, A such that L d(y1, y2)−A ≤ d(g·y1, g·y2) ≤ Ld(y1, y2)+A for every pair of points y1, y2 ∈ Y. More generally, a quasi-action of a set-group G on a metric space (X, d) is a map α from G to the set of quasi-isometries of X such that the maps α(g, - ) are uniform quasi isometries and

α(gh, -) ≈ α(g, -) ◦ α(h, -) are close functions for every g, h ∈ G (i.e. the diagram in Definition 6.1.1 commutes up to closeness). As Egrp E above, we see that every quasi-action is a coarse action of the trivially coarse group (G, min) y (X, d). Remark 6.1.6. Special examples of coarse actions—especially quasi-actions—have been studied in many guises by various authors. Some examples include the works of Manning [Man06] and Mosher- Sageev-Whyte [MSW03, MSW11] on quasi-actions on trees; and the works of Eskin-Fisher-Whyte [EF10, EFW12, EFW13]. An important motivation comes from the theory of quasi-isometric rigidity. In fact, quasi-actions are often implicit in the study of such rigidity statements. One more source of examples is given by Cornulier’s study of near actions [Cor19]. A near bijection on a set is a bijection between two cofinite subsets, and the set of near bijections is the near symmetric group of a set. Cornulier defines balanced near actions as homeomorphisms into the near symmetric group. It is easy to see that these are coarse actions of trivially coarse set-groups. Cornulier gives an extensive list of well-explained examples and references, which is an excellent resource. Remark 6.1.7. Let (G, ∗, e, (-)−1, E) be a coarse group and (Y, F ) a coarse space. The set of functions α: G × Y → Y that define a coarse action [α]: (G, E) y (Y, F ) decreases if E is replaced with some coarser coarse structure E0 ⊃ E. This counterintuitive behaviour can be appreciated with an extremal example: it is natural to expect that the bounded coarse group (G, Emax) only admits trivial coarse actions (i.e. such that α(g) stays uniformly close to idY as g ranges in G). For a more sophisticated explanation see also Section 14.

Deﬁnition 6.1.8. Let α: G y Y and α0 : G y Y0 be coarse actions. A coarse map f : Y → Y0 is coarsely G-equivariant (or simply coarsely equivariant when G is clear from the context) if the following diagram commutes: G × Y α Y

idG× f f 0 G × Y0 α Y0. That is, if α = [α], α0 = [α0], f = [ f ] and Y0 = (Y, F 0), then f is coarsely equivariant if there exists 0 0 F ∈ F such that f (α(g)(y)) ← F → α (g)( f (y)) for every g ∈ G and y ∈ Y. Two coarse actions α: G y Y and α0 : G y Y0 are isomorphic if there exists a coarsely equivariant coarse equivalence f : Y → Y0 (the coarse inverse of f is automatically coarsely equivariant).

Example 6.1.9. The additive group R acts on (R, Ed) by α: x 7→ x + α and coarsely acts on (Z, Ed) by α: n 7→ bn + αc. The inclusion Z ,→ R is coarsely equivariant (but not set-theoretically equivariant), so these actions are isomorphic. The above formulae deﬁne coarse actions of both the trivially coarse group (R, Emin) and the metric coarse group (R, Ed). In either case, the inclusion gives an isomorphism of coarse actions. In fact, the notion of coarse equivariance only depends on the coarse structures of the target spaces. Example 6.1.10. Let (G, E) be a coarse group and F ⊇ E an equi-left-invariant coarse structure, so that [∗]: (G, E) y (G, F ) is a coarse action. For a ﬁxed g ∈ G, let F ∗ (g, g) B h{F ∗ (g, g) | F ∈ F } ∪ Ei be the “right translate” of F . Here we added E to the set of generators because we need E ⊆ F ∗ (g, g) and this is not necessarily the case e.g. if ∗g is not surjective. Since E ⊆ F ∗ (g, g), associativity of [∗]

65 implies that ∆G ∗ (F ∗ (g, g)) is in F ∗ (g, g) for every F ∈ F and hence the coarse structure F ∗ (g, g) is equi-left-invariant. We thus obtain a new coarse action [∗]: (G, E) y (G, F ∗ (g, g)). However, F ∗ (g, g) is defined so that the right multiplication ∗g : (G, F ) → (G, F ∗ (g, g) is controlled and, by associativity, ∗g−1 : (G, F ∗ (g, g)) → (G, F ) is controlled as well. These two controlled maps are coarsely equivariant and they are (coarse) inverses of one another. They hence determine an equivariant coarse equivalence, i.e. they give an isomorphism between the two coarse actions (G, E) y (G, F ) and (G, E) y (G, F ∗ (g, g)). Similarly, we can define the coarse structure (g, g) ∗ F , however this will be trivially equal to F itself because F is (equi)-left-invariant. In particular, F ∗ (g−1, g−1) = (g, g) ∗ F ∗ (g−1, g−1) (we can drop the parentheses from the notation because ∗ is coarsely associative). So we can view right translate of F as conjugation. We say that F and (g, g) ∗ F ∗ (g−1, g−1) are conjugate equi-left-invariant coarse structures. If the set {g, e} is F -bounded then F and (g, g) ∗ F ∗ (g−1, g−1) coincide (see Example 5.1.5). A connected coarse structure is preserved by conjugation. However, disconnected equi-left-invariant coarse structures might not be preserved under conjugation. For example, let F2 = ha, bi be the 2-generated free group and let F be the coarse structure determined by the ∞-metric where d(w, wan) = |n| and 0 −1 0 d(w, w ) = ∞ if w w < hai (the coarse space (F2, F ) can be seen as a disconnected union of lines indexed over the set of cosets F2/hai). The set {e, a} is F -bounded and hence {b, ab} is (F ∗ (b, b))-bounded, but {b, ab} is not F -bounded, so F , F ∗ (b, b). If α: G y Y is a coarse action and [Z] = Z ⊂ Y is a coarse subspace, we can define G · Z as S [ g∈G g · Z] = [α(G × Z)] or, equivalently, as the coarse image of the composition

id ×ι α G × Z −−−−→G G × Y −−→ Y where ι: Z → Y is the inclusion. Deﬁnition 6.1.11. A coarse subspace Z ⊆ Y is coarsely G-invariant if G · Z = Z. That is, if α = [α] and Z = [Z], then Z is coarsely invariant if and only if α(G × Z) Z. The same proof of Proposition 5.2.2 shows that if Z ⊆ Y is coarsely G-invariant then there is a coarse action a|z : G y Z such that the inclusion Z ,→ Y is coarsely equivariant. Further, the coarse action a|z is uniquely deﬁned up to coarsely equivariant coarse equivalences. This action is the restriction of α to Z. Concretely, if Z = [Z] and α = [α] we can construct a controlled map a|Z : G × Z → Z by choosing a|Z(g, z) ∈ Z uniformly close to a(g, z) ∈ Y.

6.2 Coarse action by conjugation A coarse action of special interest is given by conjugation. Let G = (G, E) be a coarse group. We already remarked that conjugation by a fixed g ∈ G defines a coarse automorphism [gc]: G → G (Example 5.1.5). This can be improved to show that conjugation defines a left coarse action: Definition 6.2.1. The action by conjugation of a coarse group G on itself is the left coarse action c: G y G where c = [c] and c: G × G → G is defined by c(g, x) B g ∗ x ∗ g−1 (up to closeness, this definition is independent of the order of multiplication in g ∗ h ∗ g−1). If c is controlled, it follows easily from the coarse associativity of ∗ that c is indeed a left coarse action. To prove that c is controlled one needs to show that (c × c)(E ⊗ ∆G) and (c × c)(∆G ⊗ E) are − controlled whenever E ∈ E. To prove the former, one shows that if x ← E → y then x 1 ∗ y ← E0 → e for some E0 ∈ E and hence

−1 −1 0 −1 −1 −1 c(g, x) = g ∗ x ∗ g ← - → g ∗ x ∗ e ∗ g ← ∆G∗E ∗∆G → g ∗ x ∗ x ∗ y ∗ g ← - → g ∗ y ∗ g = c(g, y)

(for the sake of brevity we made cavalier use of parentheses and the ← - → notation. Here ← - → is referred to some ﬁnite composition of Eas, Einv, Eid and their multiplications with ∆G). Showing that

66 c × c(∆G ⊗ E) is similar (and simpler). Alternatively, controlledness could be proved by diagram chasing by realizing c as an appropriate composition of diagonal embeddings, permutation of coordinates, taking inverses and left/right multiplications. Of course, the action by conjugation is well-behaved under coarse homomorphism: Lemma 6.2.2. If f : G → H is a coarse homomorphism of coarse groups, then the composition

f×id c G × H −−−−−→H H × H −−→ H is a coarse action and f : G → H is coarsely equivariant between c: G y G and c ◦ ( f × idH): G y H.

Proof. With a very simple diagram chase we see that the composition c ◦ ( f × idH) is a coarse action because f is a coarse homomorphism. The fact that f is equivariant follows immediately from the deﬁnition.

We already pointed out that if G is coarsely connected then [gc] = idG for every ﬁxed g ∈ G. However, this does not mean that the action by conjugation is trivial! In fact, the maps gc need not be uniformly close to the identity. More precisely, it is easy to see that the action by conjugation is trivial if and only if the multiplication coarsely commutes: Deﬁnition 6.2.3. A coarse group G = (G, E) is coarsely abelian if any of the following equivalent statements hold: • the coarse action by conjugation c: G y G is trivial (i.e.there exists E ∈ E such that

−1 g ∗ h ∗ g ← E → g

for every g, h ∈ G);

• there exists E ∈ E such that g1 ∗ g2 ← E → g2 ∗ g1 for every g1, g2 ∈ G. Remark 6.2.4. Recall that the the cancellation metric on the free group (Example 4.2.12) provides an example of a non coarsely abelian coarsely connected coarse group. Obviously, if G is a set-group and E is an arbitrary equi-bi-invariant coarse structure on G then the coarse group (G, E) can very well be coarsely abelian even if G is not abelian. For one, this is the case if E = Emax. A marginally more interesting example is the case G = H × F where H is abelian, F is finite E E Egrp and the coarse structure is connected (e.g. = fin ). Every coarse group G = (G, E) has a coarse abelianization. Using the notation of Definition 5.3.3, this is the coarse quotient G/hhRcommii where Rcomm B {(g1 ∗ g2, g2 ∗ g1) | g1, g2 ∈ G}. Obviously, if G is a set-group and G/[G, G] is its abelianization, the quotient map π factors through the coarse abelianization

[id] [π] (G, Emin) −−→ (G, Emin)/hhRcommii −−→ (G/[G, G], Emin). However, neither of the above maps is an isomorphism of coarse groups in general.

Having established the meaning of coarse conjugation, we can deﬁne the notions of coarse normality and centrality. Deﬁnition 6.2.5. A subset A of a coarse group (G, E) is coarsely invariant under conjugation if c(G, A) A. Equivalently, if A = [A] is coarsely invariant under the coarse action by conjugation (we may also say that is normalized by G). We say that A is in the coarse centralizer of G if it is coarsely invariant under conjugation and the restriction c|A : G y A is the trivial coarse action. A coarse subgroup H ≤ G is coarsely normal if it is coarsely invariant under conjugation. We will denote it by H E G.

67 Explicitly, A ⊆ G is coarsely invariant under conjugation if there is an E ∈ E such that for every g ∈ G 0 0 0 and a ∈ A there is an a = a (g, a) ∈ A with g ∗ a ← E → a ∗ g. The set A is in the coarse centralizer if we can further assume that a0(g, a) = a (i.e. the points in A uniformly coarsely commute with every g in G). In terms of indexed coverings, A is coarsely invariant under conjugation if and only if iG ∗ A G A ∗ iG as G-indexed coverings. Similarly, A is in the coarse centralizer if and only if iG ∗ iA G×A iA ∗ iG as (G × A)-indexed coverings.

6.3 Coarse orbits Let α: G y Y be a coarse action deﬁned by a controlled map α: (G, E) × (Y, F ) → (Y, F ). Recall our convention gα( - ) B α(g, - ) and αy( - ) B α( - , y). Given any point y ∈ Y, consider the set G · y = αy(G) = α(G × {y}). It is easy to see that the asymptotic equivalence class of G · y does not depend on the choice of the representative α for α, and that G · y G · y0 whenever y0 y (both these facts can be deduced from Lemma 3.3.8). We already remarked that the equivalence classes [y] correspond to the coarsely connected component of (Y, F ). Yet, if we view [y] as a coarse subspace of Y, we will also say that [y] is a coarse point in (Y, F ) and denote it by [y] = y ∈ Y. This is mostly done for psychological reasons, so that we can deﬁne coarse orbits as:

Deﬁnition 6.3.1. The coarse orbit (also called G-orbit) of a coarse point y ∈ Y is the coarse subspace G · y B [Im(αy)] ⊆ Y. The associated orbit map is αy = [αy]: G → Y.

Equivalently, a coarse point can be seen as a coarse map y: {pt} → Y, the orbit map αy is the composition id ×y α G × {pt} −−−−→G G × Y −−→ Y and the coarse orbit G · y is the coarse image of αy. Obviously, a coarse action α: G y Y can have at most as many distinct coarse orbits as there are coarsely connected components in Y, since a coarse orbit depends only on the choice of a coarsely connected component [y] = y. In particular, if Y is coarsely connected then the coarse orbit is unique up to coarse equivalence. It is worth pointing out that some care is needed when translating between coarse geometric properties of coarse orbits and properties of α. This is due to the fact that the coarse properties of G · y are—by definition—properties of the coarse space (αy(G), F |αy(G)). The discussion preceding Definition 6.3.1 explains that such properties do not depend on the choice of y ∈ Y, but this does not say whether they hold “uniformly” as y varies. For example, it may well be that all the G-orbits of α are bounded, but α is not the trivial coarse action (i.e. the G-orbits are not “uniformly bounded”). Example 6.3.2. Consider the action by rotation α: R y C, α(t)(z) B eitz. Since it is a set-group action by isometries, α gives a coarse action (R, Emin) y (C, Ed), where d is the metric on C. This coarse action is non-trivial (i.e. it is not close to the projection R × C → C), but it has an unique coarse orbit and this coarse orbit is trivial. It is easy to verify that α does not define a coarse action of the coarse group (R, Ed). This can also be deduced by the fact that α is non-trivial but factors through an S1-action: if α was a coarse action for 1 1 (R, Ed) then it would descend to a coarse action of (S , Ed), but (S , Ed) cannot have non-trivial actions because it is the bounded coarse group. E F F E Finally, note that α is not a coarse action of (R, min) on (C, ) if = fin is the finite off-diagonal structure. This can also be deduced by noting that (C, F ) is coarsely connected and hence a coarse action on (C, F ) ought to have only one coarse orbit. On the other hand, α clearly has uncountably many distinct coarse orbits on (C, F ). Note that G-orbits are G-invariant coarse subspaces because G · (G · y) = (G ∗ G) · y = G · y. In particular, the coarse action α restricts to a coarse G-action on each G-orbit of Y. We already explained at length that the coarse orbit G · y is well defined up to coarse equivalence. The fact that the restriction α|G·y

68 is well-defined implies that G · y is actually well defined up to coarsely G-equivariant coarse equivalence (Definition 6.1.8). Remark 6.3.3. If Z ⊆ Y is G-invariant and z ∈ Z, then the G-orbit of z is coarsely contained in Z. Example 6.3.2 shows that the converse does not hold: since all the coarse orbits of the coarse action [α]: (R, Emin) y (C, Ed) are trivial, every coarse subspace of C coarsely contains the coarse orbit of each of its points. On the other hand, the real axis in C determines a coarse subspace that is not (R, Emin)-invariant. The pull-back of the coarse structure of Y under any orbit map defines a coarse structure on G. That ∗ ∗ is, given [α]: (G, E) y (Y, F ) and y ∈ Y we consider the coarse structure αy(F ) B α( - , y) (F ). Note ∗ that αy(F ) does not depend on the choice of representatives α for α and y for y. Yet, it is notationally convenient to fix such choices. ∗ It follows from the definition of pull-back, that αy : (G, αy(F )) → (Y, F ) is a coarse embedding and ∗ hence induces a coarse equivalence between (G, αy(F )) and its coarse image G · y. Furthermore, we can ∗ show that G coarsely acts on (G, αy(F )) by left multiplication: ∗ ∗ Lemma 6.3.4. The group operation ∗ defines a left coarse action ∗: (G, E) × (G, αy(F )) → (G, αy(F )). ∗ Moreover, the orbit map αy is coarsely equivariant from ∗: G y (G, αy(F )) to α: G y Y. ∗ ∗ Proof. We first need to show that ∗: (G, E) × (G, αy(F )) → (G, αy(F )) is controlled. By definition, this ∗ is the case if and only if the composition αy ◦ ∗: (G, E) × (G, αy(F )) → (Y, F ) is controlled. On the other hand, αy ◦ ∗ is close to α( - , α( - , y)) by definition of coarse action. The latter is controlled because it is the composition of controlled maps:

× ∗ idG αy α (G, E) × (G, αy(F )) −−−−−→ (G, E) × (Y, F ) −−−−−−→ (Y, F ).

It follows that αy ◦ ∗ is controlled as well. ∗ Note that E ⊆ αy(F ) because αy : (G, E) → (Y, F ) is controlled. Commutativity of the diagrams in the deﬁnition of coarse action follows immediately from the relevant diagrams in (♥). This proves that ∗ the left multiplication induces a coarse action on (G, αy(F )). To show equivariance, it is enough to write αy as the composition of α and (idG, y): G → G × Y and note that equivariance follows immediately from the deﬁnition of coarse action:

∗ [idG ×(idG,y)] ∗ [idG ×α] (G, E) × (G, αy(F )) (G, E) × (G, αy(F )) × Y (G, E) × (Y, F )

∗ [∗×idY ] [α] ∗ [(idG,y)] ∗ [α] (G, αy(F )) (G, αy(F )) × Y (Y, F )

(we are also using that commutative diagrams remain coarsely commutative when considered with respect to larger coarse structures).

∗ ∗ The key point in the proof of Lemma 6.3.4 is the remark that ∗: (G, E) × (G, αy(F )) → (G, αy(F )) is controlled. In the language of Subsection 2.6, this can be rephrased by saying that the left multiplications ∗ ∗ g∗: (G, αy(F )) → (G, αy(F )) are (G, E)-equicontrolled, or equivalently that the right multiplications ∗ ∗g : (G, E) → (G, E) are (G, αy(F ))-equicontrolled. It is instructive to see what this means in a more concrete example: Example 6.3.5. Let G be a set-group and α: G y Y a free action by isometries on a metric space (Y, d). Equip Y with the metric coarse structure Ed. Then α deﬁnes a coarse action of the trivially coarse group (G, Emin). If we identify G with any of its orbits αy(G) ⊂ Y, the metric d induces a metric dy on G. The ∗ coarse structure αy(Ed) coincides with the induced metric coarse structure (if the action was not free, dy ∗ would be a pseudo-metric on G, but αy(Ed) would still coincide with the induced coarse structure).

69 ∗ ∗ In this example, the fact that the left multiplications g∗: (G, αy(Ed)) → (G, αy(Ed)) are (G, Emin)-equicontrolled follows trivially from the observation that g∗ preserves the dy. ∗ On the other hand, the fact that the right multiplications ∗g : (G, Emin) → (G, Emin) are (G, αy(Ed))-equicontrolled follows from the observation that ∗g has displacement uniformly bounded in terms of dy(e, g). That is, for every h ∈ G

dy(h, h ∗ g) = d(h · y, hg · y) = d(y, g · y) = dy(e, g).

0 ∗ It follows from the triangle inequality that if (gi, gi ) are pairs of uniformly close elements in (G, αy(Ed)) 0 ∗ then (h ∗ gi, h ∗ gi ) are also uniformly close, i.e. the right multiplication is (G, αy(Ed))-controlled. ∗ The same argument also shows that in general (G, αy(Ed)) is not a coarse group. In fact, if it was a ∗ coarse group then the left multiplications would have to be (G, αy(Ed))-equicontrolled. But this is the case only if dy(g ∗ h, h) is uniformly bounded in terms of dy(g, e), which is usually not the case. Remark 6.3.6. Let c: G × G y G be the coarse action by conjugation. Note that c(G, e) e because gc(e) ← Eid◦Einv → e for every g ∈ G. In other words, the coarse orbit c(G, e) is bounded in G. If G is coarsely connected, it follows that the coarse action by conjugation has only bounded orbits (but we already remarked that it is generally not a trivial coarse action). It would be interesting to study more carefully the coarse geometry of all the coarse orbits of c when G is coarsely disconnected, i.e. to study ∗ the pull-back coarse structures cx(E) as x ∈ G varies. ∗ Note that if [α]: (G, E) y (Y, F ) is any coarse action, the pull-back coarse structure αy(F ) is as in Example 6.1.2(3). That is, it is an equi-left-invariant coarse structure containing E. In other words, Lemma 6.3.4 shows that taking the pull-back allows us to deﬁne a coarsely equivariant coarse map between α and a coarse action by left multiplication of (G, E) on G equipped with a coarser equi-left-invariant coarse structure. Recall in Example 6.1.10 we deﬁned the conjugate—equivalently, the right translate—of an equi-left invariant coarse structure F ⊇ E as

(g, g) ∗ F ∗ (g−1, g−1) = F ∗ (g−1, g−1) = h{F ∗ (g, g) | F ∈ F } ∪ Ei, where g ∈ G is some fixed element. We also remarked that the right multiplications ∗g and ∗g−1 define a coarse equivalence between (G, F ) and (G, (g, g) ∗ F ∗ (g−1, g−1)). In particular, (g, g) ∗ F ∗ (g−1, g−1) ∗ can also be defined as the pull-back of F under ∗g. Now, fix y ∈ Y, g ∈ G and consider αg·y(F ). Since ∗ [α] is a coarse action, we can rewrite αg·y(F ) as

∗ ∗ ∗ ∗ ∗ −1 −1 αg·y(F ) = α(- , g · y) (F ) = α(( - ) ∗ g, y) (F ) = (αy ◦ ∗g) (F ) = (g, g) ∗ αy(F ) ∗ (g , g ).

That is, taking the pull back under different points in the same G-orbit defines conjugate equi-left-invariant coarse structures on G. This discussion is particularly interesting when G · y = Y because it implies that α is isomorphic to an action by left multiplication of G on itself equipped with an appropriate coarse structure, and the latter is well defined up to conjugacy. More precisely, we make the following:

Deﬁnition 6.3.7. A coarse action [α]: (G, E) y (Y, F ) is cobounded if there exists (equivalently, for every) y ∈ Y such that αy(G) is coarsely dense in (Y, F ).

The above is the coarse analogue of transitivity for set-group actions. In analogy with the fact that transitive left actions of a set-group are isomorphic to actions by multiplication on the left cosets of the isotropy group, we can fully describe the family of cobounded coarse actions up to isomorphism:

70 Proposition 6.3.8. Let G = (G, E) be a coarse group. There is a natural correspondence between conjugacy classes of equi-left-invariant coarse structures on G containing E and isomorphism classes of cobounded left coarse actions:

{F equi-left-invariant coarse structures | E ⊆ F }/conj. ←→ {cobounded left coarse actions}/isom.

(in particular, the class of cobounded coarse actions up to isomorphism is a set).

Proof. If F is equi-left-invariant and contains E then (G, E) coarsely acts on (G, F ) (Example 6.1.2). This action is cobounded, as G ∗ eG is coarsely dense in (G, E). We already noted that [∗]: (G, E) y (G, F ) and [∗]:(G, E) y (G, (g, g) ∗ F ∗ (g−1, g−1)) are isomorphic coarse actions (Example 6.1.10). This shows that we have a well-deﬁned map from the LHS to the RHS. The converse map is obtained via pull-back: if [α]: (G, E) y (Y, F ) is a cobounded coarse action and y ∈ Y is any ﬁxed point, then ∗ [α] is isomorphic to [∗]: (G, E) y (G, αy(F )) by Lemma 6.3.4. If z ∈ Y is any other point, then z is ∗ ∗ close to g · y for some g ∈ G because [α] is cobounded. It follows that αz (F ) = αg·y(F ), and the latter ∗ ∗ is conjugated to αy(F ). That is, up to conjugacy, αy(F ) does not depend on the choice of y. Finally, if α: (G, E) y (Y, F ) and α0 : (G, E) y (Y0, F 0) are isomorphic via an equivariant coarse equivalence f :(Y, F ) → (Y0, F 0), then 0 ∗ 0 ∗ ∗ 0 ∗ (α f (y)) (F ) = αy( f (F )) = αy(F ). ∗ This shows that αy(F ) is also invariant under taking isomorphic coarse actions. Corollary 6.3.9. If G = (G, E) is coarsely connected then every isomorphism class of cobounded left coarse actions contains a unique representative of the form ∗: (G, E) y (G, F ) where F ⊇ E is an equi-left-invariant coarse structure.

6.4 The Fundamental Observation of Geometric Group Theory The Milnor–Švarc Lemma states that if set-group G acts properly and cocompactly on a proper length metric space (Y, d) then G is finitely generated and the orbit map is a quasi-isometry between (Y, d) and G equipped with a word metric. Here “properly” means that set {g ∈ G | (g · K) ∩ K , ∅} is finite for every compact K ⊆ Y. Because of its importance, this result is also known as “the Fundamental Observation of Geometric Group Theory”. Although not hard, its proof usually takes several pages. In this subsection we show that in our settings it is an observation about trivially coarse groups, and it is essentially contained in Proposition 4.5.2 and Lemma 6.3.4 (whose proofs further simplify for trivially coarse groups). 11 Let G be a set-group, so that (G, Emin) is a trivially coarse group, and let [α]: (G, Emin) y (Y, F ) be a cobounded coarse action. For any fixed y ∈ Y, Lemma 6.3.4 implies that the orbit map g 7→ g · y is an ∗ equivariant coarse equivalence (G, αy(F )) → (Y, F ). We now give a temporary definition:

Deﬁnition (Temporary). The coarse action [α]: (G, Emin) y (Y, F ) is proper if the set {g ∈ G | (g · B) ∩ B , ∅} is ﬁnite for every bounded set B ⊆ Y. We will also say that a set-group action by isometries on a metric space (Y, d) is metrically-proper if the induced coarse action [α]:(G, min) y (Y, Ed) is proper.

Remark 6.4.1. Metric-properness coincides with topological properness (Subsection 13.3) if the metric space is locally compact. For general metric spaces, metric properness is strictly stronger. ∗ If (Y, F ) is coarsely connected, then α is a proper coarse action if and only if the αy(F )-bounded subsets of G are the finite subsets of G (bounded subsets must be finite because the singleton {y} is bounded in Y, finite sets are bounded because (Y, F ) is coarsely connected). On the other hand, Proposition 4.5.2

11Brodskiy–Dydak–Mitra wrote an analogous proof in the setting of proper cocompact actions by isometries of set-groups [BDM07].

71 shows that an equi-left-invariant coarse structure is determined by the bounded sets of the coarse structure. In particular, if there is an equi-left invariant coarse structure in which bounded sets are the ﬁnite subsets, then it is unique. We will soon see that this is a key point of the Milnor–Švarc Lemma. The other important element concerns the ﬁnite generation. To deal with it we need to introduce a coarse analogue of a “length space”.

Deﬁnition 6.4.2. A coarse space (X, E) is coarsely geodesic if it is coarsely connected and the coarse structure E is generated by a single set E = hE¯i. That is, E ⊂ X × X is in E if and only if it is a subset of the n-fold composition E¯ ◦ · · · ◦ E¯ for some n ∈ N.

Recall that a metric space is a length space if it is path connected and the distance between any two points equals the inﬁmum of the lengths of paths connecting them. If (Y, d) is a length metric space and Ed is the metric coarse structure, then (Y, Ed) is coarsely geodesic because Ed is generated e.g. by the 1-neighbourhood of the diagonal ∆G ⊂ G × G. Finally, note that if (X, E) is coarsely geodesic with E = hEi, and x ∈ X is ﬁxed, then every E-bounded set is contained in a x-section of some composition: E¯ ◦ · · · ◦ E¯(x) (recall that E(z) = {x | (x, z) ∈ E} and that a set is E-bounded if and only if it equals E(z) for some z ∈ X and E ∈ E).

Corollary 6.4.3 (Milnor–Švarc Lemma). If α: G y (Y, d) is a metrically-proper cocompact action by isometries of a set-group, then G is ﬁnitely generated and the orbit map is a coarse equivalence when G is equipped with a word metric.

Proof. Consider the metric coarse structure Ed on Y. The space (Y, Ed) is coarsely connected (this is true for every metric space) and it is coarsely geodesic because (Y, d) is a length space. Since the action is by isometries, it defines a coarse action [α]: (G, Emin) y (Y, Ed). Every bounded set in (Y, Ed) has bounded diameter. Since the action α is proper, then [α] is proper as a coarse action. ∗ It follows that αy(Ed) is the unique equi-left-invariant coarse structure on G whose bounded sets are precisely the finite subsets of G. Every compact set in Y is Ed-bounded, so cocompactness of α implies that [α] is cobounded as a ∗ ∗ coarse action. It follows that the orbit map is a coarse equivalence (G, αy(Ed)) → (Y, Ed). In turn, αy(Ed) is coarsely geodesic. ∗ ¯ ¯ Choose a generator for the coarse structure αy(Ed) = hEi. Since G is a set-group, we can enlarge E ¯ ∗ to make it into a relation of the form E = ∆G ∗ (S × {eG}) for some αy(Ed)-bounded set S (it is enough to let S B (∆G ∗ E¯)(eG)). Since it is bounded, S is finite. Taking the n-fold composition, we have ¯ ¯ n ∗ E ◦ · · · ◦ E(eG) = S . Since (G, αy(Ed)) is coarsely connected, for every g ∈ G there exists n ∈ N with g ∈ S n, and hence S is a finite generating set for G. Finally, the (coarse) action by left multiplication of G on itself equipped with the metric coarse structure E|·|S induced by the word metric |·|S defined by the finite generating set S is proper and ∗ cobounded. By uniqueness, it follows that E|·|S = αy(Ed). That is, the orbit map is a coarse equivalence between (G, E|·|S ) and (Y, Ed) (and it is easy to see that any coarse equivalence between length-metric spaces is a quasi-isometry).

Remark 6.4.4. The standard proof of the Milnor–Švarc Lemma is obtained by studying how the orbit G · y sits in the metric space (Y, d). Using the metric assumptions on Y and the existence of a proper cocompact action by isometries it is possible to deduce that G itself must be ﬁnitely generated and bound the distortion of the orbit map. That is, the G-action is used to control the coarse geometry of Y and the latter is then used to study the coarse geometry of G. The proof that we just presented displays a subtle change of paradigm. Since coarse geometry is well behaved under pull-backs, we can directly use the action to pull-back the coarse structure of Y to G. The advantage of doing this is that we are then left to study a coarse structure on a group and we here we can use uniqueness results ‘for free’.

72 In other words, the standard proof can be seen as a hands-on argument on the metric space Y, where the cobounded proper coarse action is secretly used to move around closed balls. The ‘coarse’ proof establishes a coarse language ‘a priori’ and elucidates that proper cobounded coarse actions are particularly well behaved in this language. It is interesting to note that the Milnor–Švarc Lemma can be split into two separate statements. The first one is implicit in Proposition 4.5.2 and implies that a cobounded coarse action is uniquely determined by its ‘degree of properness’. That is, cobounded coarse actions of G are uniquely determined by the family of subsets of G that have bounded image under the orbit map (see Section 13 for more on this). The second part of the statement is more specialized. It is the observation that if the (unique) space admitting a metrically-proper cobounded G-action is coarsely geodesic then the group is finitely generated. These two statements can then be combined by observing that, for a finitely generated group, the action on the Cayley graph is proper and cobounded and is hence the model of every metrically-proper cobounded action.

6.5 Quotient coarse actions Let α: G y Y be a coarse action and q: Y → Q a coarse quotient of coarse spaces (Section 3.4).

Deﬁnition 6.5.1. A quotient coarse action for α is a coarse action α¯ : G y Q that makes the diagram commute: G × Y α Y

idG×q q G × Q α¯ Q. That is, α¯ is such that the quotient map q: Y → Q is coarsely equivariant.

With this language, Proposition 6.3.8 implies that every cobounded coarse action of G is a coarse quotient of the action by left multiplication of G on itself. Let α: G y Y be deﬁned by a controlled map α: (G, E) × (Y, F ) → (Y, F ). For every relation R ⊆ Y × Y, recall the notation [ ∆G · R B (α × α)(∆G ⊗ R) = (αg × αg)(R). g∈G

Deﬁnition 6.5.2. A family R of relations on Y is coarsely equivariant (or, G-equivariant) if ∆G · R ∈ hF , Ri for every R ∈ R (the deﬁnition does not depend on the choice of representative α ∈ [α]).

Recall that Y/R denotes the quotient coarse space (Y, hF , Ri) and let q = [idY ]: Y → Y/R be the quotient map. Deﬁnition 6.5.2 is designed so that the following holds.

Lemma 6.5.3. A family of relations R is coarsely equivariant if and only if α descends to a quotient coarse action α¯ : G × (Y/R) → Y/R that makes the diagram commute:

G × Y α Y

idG×q q G × (Y/R) α¯ Y/R.

That is, α¯ is a coarse action such that the quotient map q: Y → Y/R is coarsely equivariant.

Proof. Assume that R is G-equivariant. The product coarse structure E ⊗ hF , Ri on G × Y is generated by the relations of the form E⊗F or E⊗R where E ∈ E, F ∈ F and R ∈ R. Hence, α: (G×Y, hE⊗hF , Rii) →

73 (Y, hF , Ri) is controlled if and only if E · R ∈ hF , Ri for every E ∈ E and R ∈ R. Just as in Lemma 2.6.3, it follows from (1) that E · R = (α × α) (E ⊗ ∆Y ) ◦ (∆G ⊗ R) ⊆ (E · ∆Y ) ◦ (∆G · R) and the latter is contained in hF , Ri beacuse ∆G · R is contained in R. Vice versa, if there exists a coarse map α¯ that makes the diagram commute, then α¯ = [α] because q = [idY ]. On the other hand, α is only controlled if R is coarsely equivariant. Corollary 6.5.4. Let R be any set of relations, then the family ∆G ·R B ∆G · R | R ∈ R is coarsely equivariant. Furthermore, hF , ∆G · Ri is the minimal coarse structure containing F and R such that α:(G, E) × (Y, hF , ∆G · Ri) → (Y, hF , ∆G · Ri) is controlled.

Proof. Since [α]: (G, E) y (Y, F ) is a coarse action, there is a F¯ ∈ F such that E1 · (E2 · F) ⊆ op F¯ ◦ (E1 ∗ E2) · F ◦ F¯ for every choice of relations E1, E2 on G and F on Y. In particular, for every R ∈ R ∆G · (∆G · R) ⊆ F¯ ◦ (∆G ∗ ∆G) · R ◦ F¯ ⊆ F¯ ◦ ∆G · R ◦ F¯ ∈ hF , ∆G · Ri and hence the family ∆G ·R is coarsely equivariant. The ‘furthermore’ is clear.

If α: G y Y is a set-group action and R is an equivalence relation on Y, then α factors through the quotient Y/R if and only if R is G-equivariant. Since R is an equivalence relation, the coarse structure hRi coincides with the set of subsets of R. It follows that if we see α as a trivially coarse action [α]: (G, Emin) y (Y, Emin), then an equivalence relation R is G-equivariant if and only if it is coarsely equivariant. When seen as a trivially coarse action, the quotient set-group action G y Y/R is isomorphic to the quotient coarse action of α. More interestingly, quotient coarse actions may be deﬁned also when R is not an equivalence relation and hence there is no set-quotient Y/R.

Example 6.5.5. Let α: G y (Y, d) be an isometric action on a length space and let R1 B {(y1, y2) | d(y1, y2) ≤ 1}. Then R1 is not an equivalence relation, but it is a G-invariant relation on Y and hence induces a quotient coarse action [α]: (G, min) y (Y, hR1i). Since Y is a length metric space, hR1i coincides with the metric coarse structure Ed, i.e. the quotient coarse action is nothing but [α]: (G, min) y (Y, Ed). If the action α is not by isometries, R1 need not be G-invariant. On the other hand, it is easy to see that R1 is coarsely equivariant if and only if the action is by uniform coarse equivalences. This was to be expected, because hRi = Ed and the latter condition determines those actions such that [α]: (G, min) y (Y, Ed) is a coarse action (Example 6.1.5). If the metric space (Y, d) is not a length space, then hR1i need not coincide with Ed. Rather, (Y, hR1i) will be a disconnected union of the 1-connected components of Y equipped with their own metric coarse structures. It is still true that α induces a quotient coarse action if and only if there is a constant C > 0 such that d(g · y1, g · y2) ≤ C whenever d(y1, y2) ≤ 1. Example 6.5.6. The topological example of coarse action given in Example 6.1.4 can be seen as coarse left quotient actions of the set group action G y Y. Namely, the coarse structure Fcpt on Y coincides with h∆G ·Fcpti, where Fcpt is the smallest coarse structure whose bounded sets are all the relatively compact subsets of Y (we generally denote it by Ecpt, we switched to Fcpt to avoid confusion).

6.6 Coarse cosets spaces It is useful to study coarse quotients of the coarse action by left multiplication of a coarse group on itself. By Corollary 6.5.4, all we need to do is to understand the coarse structures of the form hE, ∆G ∗ Ri. It is much easier to describe such a coarse structure under if the operation functions ∗, e, (-)−1 are adapted (we are using notations and deﬁnitions from Section 4.3). For example, we can prove the following:

74 Lemma 6.6.1. Let (G, E) be a coarse group R a set of relations on G and ∗, e, (-)−1 adapted representatives for the coarse operations. Then hE, ∆G ∗ Ri = h∆G ∗ (∆G ∗ R)i ◦ E, where

h∆G ∗ (∆G ∗ R)i ◦ E B {E1 ◦ E2 | E1 ∈ h∆G ∗ (∆G ∗ R)i, E2 ∈ E}.

Proof. We only need to show that hE , ∆G ∗ Ri ⊆ h∆G ∗ (∆G ∗ R)i ◦ E, as the other containment is obvious. By Lemma 2.2.9, hE , ∆G ∗ Ri consists of the relations contained in finite compositions of relations in ∆G ∗ R and E. By induction, it is therefore enough to show that for every E ∈ E and R ∈ R there exists 0 0 E ∈ E such that E ◦ (∆G ∗ R) ⊆ (∆G ∗ (∆G ∗ R)) ◦ E . If a pair (x, y) belongs to E ◦ (∆G ∗ R) then there are (z, w) ∈ R and g ∈ G such that y = g ∗ w and x - E → g ∗ z - ∆G∗R → g ∗ w = y. Since the operations are adapted, we have: −1 −1 x = x ∗ (z ∗ z)- ∆G∗(∆G∗R) → x ∗ (z ∗ w) −1 - E∗∆G → (g ∗ z) ∗ (z ∗ w) −1 - Eas◦(∆G∗Eas) → g ∗ ((z ∗ z ) ∗ w) = g ∗ w = y, thus showing that E ◦ (∆G ∗ R) ⊆ (∆G ∗ (∆G ∗ R)) ◦ (E ∗ ∆G) ◦ Eas ◦ (∆G ∗ Eas). Corollary 6.6.2. If G is a set-group then for any bi-invariant coarse structure E and relations R we have hE, ∆G ∗ Ri = h∆G ∗ Ri ◦ E. Remark 6.6.3. The meaning of Lemma 6.6.1 is best understood by examining the metric analog. If (Y, d) is a metric space and R is an equivalence relation we define a function on the quotient d¯: Y/R × Y/R → R by setting d¯([y], [y0]) to be the infimum of the distances between points in [y] and [y0]. This d¯ may fail to satisfy the triangle inequality. To obtain a distance function on Y/R one would have to define d¯0([y], [y0]) as the infimum of the d¯-length of finite sequences of points connecting [y] to [y0] (and then check whether d¯0([y], [y0]) = 0 if and only if [y] = [y0]). This is analogous to writing hE, Ri as arbitrary finite compositions of relations. On the other hand, if R is preserved by an isometric group action G y (Y, d) that is transitive on each equivalence class of R, then d¯ does automatically satisfy the triangle inequality. Since d¯ is given by an explicit formula, it is much easier to study metric properties of the quotient Y/R. Case in point, the d¯-ball of radius r with centre [y] in Y/R is the set of [y0] that intersect the d-neighbourhood of radius r of [y] in Y. This is an analog of the characterisation of hE, Ri as h∆G ∗ (∆G ∗ R)i ◦ E: it says that a hE, Ri-bounded neighbourhood of a point g ∈ G can be recognised by seeing whether it is an E-bounded neighbourhood of a h∆G ∗ (∆G ∗ R)i-bounded neighbourhood of g. A subclass of special interest among the quotients coarse actions of the left multiplication of G on itself consists of the coarse actions on ‘cosets spaces’. Let G = (G, E) be a coarse group and A = [A] some coarse subset of G. Definition 6.6.4. The (left) coarse cosets space G/A is the coarse quotient

G/A B G/{∆G ∗ (A × A)} = (G, hE , ∆G ∗ (A × A)i)

(this is well deﬁned, as hE , ∆G ∗ (A × A)i does not depend on the choice of representative for A nor ∗). Lemma 6.6.1 can be further improved when specialized to coarse coset spaces: Lemma 6.6.5. Let (G, E) be a coarse group, ∗, e, (-)−1 adapted representatives for the coarse operations and A ⊆ G any subset. Then −1 −1 hE , ∆G ∗ (A × A)i = h∆G ∗ ({e} × (A ∗ A))i ◦ E = E ◦ h∆G ∗ ((A ∗ A)) × {e}i. 0 In particular, E ∈ hE , ∆G ∗ (A × A)i if and only if there exist n ∈ N and E ∈ E such that −1 ∗n 0 E ⊆ (∆G ∗ ({e} × (A ∗ A) )) ◦ E , where (A−1 ∗ A)∗n denotes the n-fold product (A−1 ∗ A) ∗ (A−1 ∗ A) ∗ · · · ∗ (A−1 ∗ A).

75 Proof. As in the proof of Lemma 6.6.1, it is enough to show that

−1 0 E ◦ (∆G ∗ (A × A)) ⊆ (∆G ∗ ({e} × (A ∗ A))) ◦ E

0 for some E ∈ E. If (x, y) ∈ E ◦ (∆G ∗ (A × A)), then there are a1, a2 ∈ A and g ∈ G such that y = g ∗ a2 and x - E → g ∗ a1 - ∆G∗(A×A) → g ∗ a2 = y. Since the operations are adapted,

−1 −1 x = x ∗ e - ∆G∗({e}×(A ∗A)) → x ∗ (a1 ∗ a2) −1 - E∗∆G → (g ∗ a1) ∗ (a1 ∗ a2) −1 - Eas◦(∆G∗Eas) → g ∗ ((a1 ∗ a1 ) ∗ a2) = g ∗ a2 = y as desired. Where h...i ◦ E = E ◦ h...i follows from the fact that coarse sutrctures are closed under op ∗ { }× −1∗ ∗n ∗ { }× −1∗ −1 symmetry E 7→ E . Also note that if x - ∆G ( e (A A) ) → y - ∆G ( e (A A))) → z then z = y ∗ (a1 ∗ a2) −1 ∗(n+1) and hence x - ∆G∗({e}×(A ∗A) ) → z, thus proving the ‘in particular’ statement. Remark 6.6.6. The meaning of Lemma 6.6.5 is perhaps more clear in terms of controlled coverings. Assume for ease of notation that A = A−1, so that (A−1 ∗ A)∗n = A∗2n. A priori, a controlled covering for hE , ∆G ∗ (A × A)i are obtained by taking arbitrarily many iterated star neighbourhoods of iG ∗ A and controlled coverings a ∈ C(E). Instead, Lemma 6.6.5 ensures us that every controlled covering is a reﬁnement of a covering of the form ∗2n St iG ∗ A , a with a ∈ C(E). In particular, hE , ∆G ∗ (A × A)i-bounded sets are coarsely contained into left translates of A∗2n for n large enough.

Example 6.6.7. Let G be a set-group and fix A ⊂ G. Then E ∈ h∆G ∗ (A × A)i if and only if there exists some n ∈ N such that for every (x, y) ∈ E we have x−1 ∗ y ∈ (A−1 ∗ A)∗n. In particular, if A = H is a set-subgroup then ∆G ∗ (H × H) = ∆G ∗ ({e} × H) is an equivalence relation where x ∼ y if and only if xH = yH. Then h∆G ∗ (H × H)i is the coarse structure consisting of all the sub-relations of ∼ and the coarse quotient (G, h∆G ∗ (H × H)i) is naturally isomorphic to (G/H, Emin). That is, if G = (G, Emin) and H = [H], then G/H is (equivariantly) coarsely equivalent to the trivial coarsification of the usual coset space G/H. Similarly, if d is a bi-invariant metric on a set-group G and H < G a subgroup then d defines a quotient metric d¯ on the space of cosets G/H. If G = (G, Ed) is the associated coarse group, it follows easily from Lemma 6.6.5 that G/H is isomorphic to (G/H, Ed¯). By Corollary 6.5.4, the coarse group operation ∗ descends to a quotient coarse action ∗¯ : G y G/A. This should be thought of as the action by left multiplication on the left cosets of A. Recall that for any given a coarse action α: G y Y and a coarse point (i.e. a coarsely connected component) y ∈ Y, we know that the orbit map αy : G → G · y is coarsely equivariant with respect to ∗: G y G and the restriction α|G·y : G y G · y. More precisely, it follows by Lemma 6.3.4 that αy is an isomorphism ∗ between α|G·y : G y G · y and ∗: G y (G, αy(F )) (where F is the coarse structure of Y). By Corollary 6.5.4, hE, ∆G ∗ (A × A)i is the minimal coarse structure containing E and A × A such that ∗ induces a G-action on itself. In particular, if the coarse image αy(A) is bounded in Y, it follows ∗ that hE, ∆G ∗ (A × A)i ⊆ αy(F ). In other words, α|G·y is a quotient coarse action of G y G/A. This can be seen as a version of the Orbit–Stabilizer Theorem. Informally, αy(A) is bounded in Y if and only if “it is contained in the stabilizer of y” and this happens if and only if the coarse action of G on the coarse orbit G · y factors through the action on the left cosets of A. There are a number of caveats to the above informal description. To begin with, A need not fix y −1 unless we know a priori that eG ≺ A. This issue can be fixed by replacing A with a ∗ A for some a ∈ A (for every g ∈ G the cosets spaces G/A and G/g ∗ A) are naturally isomorphic). This fact points to the second caveat: we did not assume that A is a coarse group (hence it need not contain eG). Example 6.5.5 shows that if G is a set-group equipped with a left-invariant length metric then the coarse

76 action (G, Emin) y (G, Ed) is isomorphic to the action on the cosets space (G, Emin)/B1 where B1 is the ball of radius 1 around eG. It is generally impossible to replace B1 with some subgroup of G. Even more importantly, it is generally not possible to uniquely define the stabilizer of y because there need not be a “maximal” coarse subset A that stabilizes y. For instance, let again d be a left-invariant metric on a set-group G but this time assume that the coarse structure Ed cannot be generated by a single relation (i.e. (G, Ed) is not coarsely geodesic). We then see that Ed , h∆G ∗ (A × A)i for every A ⊆ G. This issue is related to the fact that the coarse preimage of a set under a coarse homomorphism need not be well-defined. We will encounter similar difficulties when trying to define coarse kernels of coarse homomorphisms (Section7). On the positive side, one can find reasonable replacements for stabilizers by considering bornologies and proper coarse actions (Section 13).

It is worthwhile pointing out a few properties of coarse coset spaces of coarse subgroups. To start, Lemma 6.6.5 implies the following: Corollary 6.6.8. Let (G, E) be a coarse group, ∗, e, (-)−1 adapted representatives for the coarse operations and H = [H] ⊆ G a coarse subgroup. Then E ∈ hE , ∆G ∗ (H × H)i if and only if it is a subset of either (equivalently, both)

0 0 ∆G ∗ ({e} × H) ◦ E and/or E ◦ ∆G ∗ (H × {e}) .

0 for some E ∈ E. Equivalently, controlled covering are reﬁnements of coverings of the form St iG ∗ H , a with a ∈ C(E). Proof. Since H is a coarse subgroup, {e} × (H−1 ∗ H)∗n is contained in ({e} × H) ◦ E for some E ∈ E. −1 ∗n Hence ∆G ∗ ({e} × (H ∗ H) ) ⊆ (∆G ∗ ({e} × H)) ◦ (∆G ∗ E) and ∆G ∗ E ∈ E. Secondly, there is a coarse analogue of the fact that the cosets of a set-subgroup partition the parent set-group: Lemma 6.6.9. Let [H] be a coarse subgroup of (G, E). If there exists an E-bounded set B ⊆ G such that both B ∩ (g1 ∗ H) and B ∩ (g2 ∗ H) are non-empty then (g1 ∗ H) E (g2 ∗ H). −1 −1 −1 −1 Proof. Note that g1 ∗ (g1 ∗ H) (g1 ∗ g1) ∗ H H and g1 ∗ (g2 ∗ H) (g1 ∗ g1) ∗ H = g ∗ H, where −1 −1 we let g B g1 ∗ g2. Since left multiplication by g1 is a coarse equivalence of (G, E), it is therefore enough to show that H (g ∗ H). −1 By assumption, there exist h1, h2 ∈ H such that h1 (g ∗ h2). Hence g (h1 ∗ h2 ) ∈ (H ∗ H) H. It follows that g ∗ H 4 (H ∗ H) H, as desired (the converse coarse containment is analogous). Corollary 6.6.10. Let H = [H] be a coarse subgroup of G = (G, E) and G/H = (G, D) with D = hE, ∆G ∗ (H × H)i be the coarse cosets space. Then, for all subsets A, B ⊆ G the following equivalences hold: A 4D B ⇔ A 4E H ∗ B ⇔ A 4E B ∗ H ; B is D-bounded ⇔ B 4E g ∗ H for some g ∈ G ⇔ B 4E g ∗ H for every g ∈ B .

Proof. Since the equivalence class [H ∗ B]E does not depend on the choice of representative for [∗], we can assume that the operations are adapted. If A 4D B it follows Corollary 6.6.8 that A ⊆ E ◦ ∆G ∗ (H × {e}) (B) = E(B ∗ H) i.e. A B ∗ H. The converse implication is immediate. To obtain A H ∗ B one should consider 4E 4E E ◦ ∆G ∗ ({e} × H) (B) instead. The second chain of equivalences follows from the ﬁrst, as B is D-bounded if and only if B 4D {g} for some (every) g ∈ B.

77 To conclude this section, it is important to know when ∗¯ makes a coarse coset space G/A into a coarse group. More precisely, If we realize G/A as (G, hE, ∆G ∗(A×A)i) then ∗ still denotes a binary operation on G/A. In general, such ∗ will not be controlled with respect to (G, hE, ∆G ∗(A×A)i)⊗(G, hE, ∆G ∗(A×A)i). When it is controlled, we still denote the induced coarse map by ∗¯ : G/A × G/A → G/A. In this case, it follows from Theorem 4.2.6 that (G/A, ∗¯ ) is automatically a coarse group. In other words, the coarse action ∗¯ deﬁnes a coarse group if and only if it is possible to complete the following commutative diagram: ∗¯ G × G/A G/A

q×idG/A ∃?∗¯ G/A × G/A

Lemma 6.6.11. If A ⊆ G is coarsely invariant under conjugation (Deﬁnition 6.2.5), then G/A is a coarse group. In the notation of Subsection 5.3, we have G/A = G/hhA × Aii = G/hhA × {e}ii.

Proof. Let A ⊆ G be coarsely invariant under conjugation and write R B A × A. We already know that ∗:(G, E) × (G, hE, ∆G ∗ Ri) → (G, hE, ∆G ∗ Ri) is controlled. It follows from (1) that

hE , ∆G ∗ Ri ⊗ hE , ∆G ∗ Ri = hE ⊗ E , ∆G ⊗ (∆G ∗ R) , (∆G ∗ R) ⊗ ∆Gi.

It follows that we only need to check that (∆G ∗ R) ∗ ∆G is hE, ∆G ∗ Ri-controlled. Since A ⊆ G is coarsely invariant under conjugation, there exists E ∈ E such that for every g ∈ G and 0 0 a ∈ A, g ∗ a ← E → a ∗ g for some a ∈ A. Fix aa ¯ ∈ A, then

0 0 (g1 ∗ a) ∗ g2 ← Eas → g1 ∗ (a ∗ g2) ← ∆G∗E → g1 ∗ (g2 ∗ a ) ← Eas → (g1 ∗ g2) ∗ a ← ∆G∗R → (g1 ∗ g2) ∗ a¯.

Since the end result does not depend on a ∈ A, this shows that (∆G ∗ R) ∗ ∆G ∈ hE, ∆G ∗ Ri as desired. Finally, it follows from the universal property of the quotient that [id]: G/hhA × Aii → G/A is controlled. Also, [id] is controlledly proper (hence an isomorphism) because the coarse structure on G/A is the minimal equi-left-invariant coarse structure so that A is bounded.

Theorem 6.6.12. Let H < G be a coarse subgroup. The cosets space G/H endowed with ∗¯ is a coarse group if and only if H is coarsely normal.

Proof. Lemma 6.6.11 implies that G/H is a coarse group whenever H C G. To prove the converse implication, let G = (G, E), choose well adapted representatives ∗, e, (-)−1 for the coarse operations and let H = [H] for some H ⊂ G. By hypothesis, ∗: (G, hE, ∆G ∗ (H × H)i) × (G, hE, ∆G ∗ (H × H)i) → (G, hE, ∆G ∗ (H × H)i) is controlled. In particular, ({e} × H) ∗ ∆G ∈ hE, ∆G ∗ (H × H)i. By Lemma 6.6.5, there exist n ∈ N and E ∈ E such that −1 ∗n ({e} × H) ∗ ∆G ⊆ ∆G ∗ ({e} × (H ∗ H) ) ◦ E. (6) −1 ∗n 0 Since [H] is a coarse subgroup, (H ∗ H) E H. In particular, there exists an E ∈ E such that (H−1 ∗ H)∗n ⊆ E0(H) = {x | ∃h ∈ H, (x, h) ∈ E}. Note that

{e} × (E0(H)) = ({e} × H) ◦ (E0)op. (7)

Since ∆G ⊗ (E1 ◦ E2) = (∆G ⊗ E1) ◦ (∆G ⊗ E2) for every pair of relations E1, E2, then ∆G ∗ (E1 ◦ E2) ⊆ (∆G ∗ E1) ◦ (∆G ∗ E2) and hence (6) and (7) give

0 op 00 ({e} × H) ∗ ∆G ⊆ ∆G ∗ ({e} × H) ◦ ∆G ∗ (E ) ◦ E C ∆G ∗ ({e} × H) ◦ E .

78 It is now easy to complete the proof. Recall that for any relation E, the writing xE denotes the section −1 π1 (x) ∩ E. The set H ∗ g coincides with the section

g(({e} × H) ∗ ∆G) B {x | (g, x) ∈ ({e} × H) ∗ ∆G}, and hence 00 00 00 op H ∗ g ⊆ g((∆G ∗ ({e} × H)) ◦ E ) = g∗H E C (E ) (g ∗ H).

That is, H∗iG 4G iG ∗H as G-indexed controlled coverings and hence [H] is a normal coarse subgroup. Remark 6.6.13. We wish to remark that Theorem 6.6.12 really is an analogue of what happens for set-groups. Namely, if H is a subgroup of G, the only natural “quotient” construction that one has at their disposal is the set of cosets G/H equipped with the action by multiplication G y G/H. When writing cosets as gH with g ∈ G we are implicitly describing the action, as gH is deﬁned as the image under g of the special coset H < G. In general at this point there is nothing else to do. However, if H is normal in G then G/H turns out to have a natural multiplication function that is compatible with the G-action.

7 Coarse kernels

In this section we deﬁne the coarse kernel of a coarse homomorphism and show that the coarse analogs of the Isomorphism Theorems hold true. It is important to note that not all coarse homomorphisms have a well-deﬁned coarse kernel.

7.1 Coarse preimages and kernels Recall from Subsection 3.5 that, given a coarse map f : X → Y and a coarse subspace Z ⊆ Y, the coarse preimage f −1(Z) ⊆ X (Deﬁnition 3.5.7) is a coarse subspace of X such that

• the coarse image of f −1(Z) under f is coarsely contained in Z,

• if f(W) ⊆ Z for some W ⊆ X then W ⊆ f −1(Z).

Further recall that the coarse preimage need not exist in general. However, when it does exist it is unique, and the next lemma shows that it is well-behaved with respect coarse groups.

Lemma 7.1.1. Let f : G → H be a coarse homomorphism of coarse groups and K ≤ H a coarse subgroup. If the preimage f −1(K) exists, then it is a coarse subgroup of G. Moreover, if K is coarsely normal (Deﬁnition 6.2.5) then so is f −1(K).

Proof. Let G = (G, E), H = (H, F ) and fix some representatives K = [K], f = [ f ], f −1(K) = [Z]. Since f is a coarse homomorphism, we see that f (Z ∗G Z) f (Z) ∗H f (Z). By hypothesis, f (Z) 4 K, hence −1 f (Z) ∗H f (Z) 4 K ∗H K K. Since [Z] = f (K), it follows by the definition of coarse preimage that −1 Z ∗G Z 4 Z. The same argument also shows that eG and Z are coarsely contained in Z, hence [Z] is a coarse subgroup. The proof of the statement on normality is very similar: [K] ≤ H is normal if and only if c(H, K) K, −1 where c(h, k) = h ∗H k ∗H h (the order of multiplication does not matter up to closeness). It is easy to check that f (c(G, Z)) c( f (G), f (Z)) 4 c(H, K), hence c(G, Z) 4 Z by definition of coarse preimage.

Remark 7.1.2. In the above proof we preferred to ﬁx representatives for the sake of concreteness. However, since coarse images, subgroups, and preimages are all well-deﬁned, we could have also argued more implicitly. For example, the chain of coarse containments

−1 −1 f f (K) ∗G f (K) ⊆ K ∗H K = K (8)

79 implies that f −1(K) is closed under multiplication. All the coarse containments appearing in (8) can be proved from the deﬁnitions with some diagram chasing.

Of course, one object of special interest is the preimage of the trivial coarse subgroup eH ∈ H.

Deﬁnition 7.1.3. If it exists, the coarse preimage of eH ∈ H under f is the the coarse kernel of f. We denote it by Ker( f) E G. Concretely, given a subset K ⊆ G, its equivalence class [K] is the coarse kernel of f = [ f ] if and −1 only if for every bounded set B ⊆ H that contains eH the preimage f (B) is contained in a controlled thickening of K: f −1(B) ⊆ St(K, a), a ∈ C(E). It follows from Lemma 7.1.1 that, when it exists, the coarse kernel of a coarse homomorphism is a coarsely normal coarse subgroup of G.

Example 7.1.4. Of course, if G = (G, Emin) and H = (H, Emin) are trivially coarse group and f = f : G → H is a (trivially coarse) homomorphism then Ker( f) = ker( f ). More interestingly, if we equip both Egrp G and H with the equi-bi-invariant coarse structure fin the coarse kernel still exists and it is equal to [ker( f )]. In fact, in this case bounded subsets of G and H are precisely the finite sets. Given a finite −1 set B = {h1,..., hn} ⊆ H containing eH, the preimage f (B) is the union of finitely many left cosets −1 −1 g1 ker( f ),..., gn ker( f ), where gi ∈ G is a fixed arbitrary element in f (hi). In other words, f (B) is the set {g1, ··· , gn} ∗G ker( f ), which is a controlled thickening of ker( f ) (and is hence coarsely contained in ker( f )). n n Example 7.1.5. Let G = (R , Ek-k2 ) and H = (R, Ek-k2 ). Then any linear function f : R → R is controlled and defines a coarse homomorphism. It is once again easy to check that Ker([ f ]) = [ker( f )]. Consider now Zn < Rn for some n ≥ 2. The inclusion gives an isomorphism of coarse groups n n n n (Z , Ek-k2 ) (R , Ek-k2 ). In particular, this restricts to an isomorphism between Ker([ f |Z ]) ≤ Z n n−1 and Ker([ f ]) = [ker( f )] ≤ R . The latter is of course isomorphic to (R , Ed) and it is there- n n fore unbounded. As a consequence, if we choose an injective homomorphism f |Zn : Z → R then Ker([ f |Zn ]) , [ker( f |Zn )] = [{0}]. n More generally, if G = (Z , Ek-k2 ) and H = (H, Ed) is a group equipped with a bi-invariant proper metric (i.e. such that closed balls are compact) and f : Zn → H is a homomorphism, then Ker([ f ]) = [ker( f )] if and only if f (Zn) is discrete in H.

The following examples show that the coarse kernel may not exist.

Example 7.1.6. If G is a set-group, G = (G, Emin) is a trivially coarse group and f : G → H a coarse homomorphism, let I ≤ H be the coarsely connected component of Im( f) containing eH. Then f has a coarse kernel if and only if I = eH is bounded in H. In fact, the coarse subgroup I always has a coarse preimage f −1(I) (this is obtained by taking the preimage under f of the coarsely connected component of H). Therefore, if I is bounded then f −1(I) is the coarse kernel of f. On the other hand, if I is not bounded we see that for every bounded set B ⊂ f (G) there is a strictly larger bounded set B ( B0 ⊂ f (G). In turn, this shows that for every subset of G with bounded image there is a strictly larger (and hence non asymptotic) subset with bounded image. This shows that eH does not have a coarse preimage. To give a concrete example, the identity map id: (Z, Emin) → (Z, Ed) does not have a coarse kernel because (Z, Ed) is not bounded.

0 Example 7.1.7. Recall once again Example 5.1.18: G = (`2(N), Ek-k2 ) and H = (`2(N), Ek-k ), where 0 2 k - k2 is the non-complete norm obtained by rescaling the standard unitary base v1, v2,... of `2(N) so 0 1 k k 0 E → E 0 that vn 2 = n . We claim that the identity map idk-k2→k-k : (`2(N), k-k2 ) (`2(N), k-k ) does not have a 0 2 2 coarse kernel. In fact, consider the k-k2-balls around the origin X 2 n f (n) 2o B(0; r) B f ∈ `2(N) ≤ r ⊂ `2(N). n2 n∈N

80 All these balls are bounded neighbourhoods of the identity in (` (N), Ek k0 ). However, they are not 2 - 2 E k k0 asymptotic to one another when seen as subsets of (`2(N), k-k2 ). Since every - 2-bounded neighbourhood of 0 is contined in such a ball for some r large enough, this implies that 0 ∈ `2(N) does not have a well-defined coarse preimage under idk k →k k0 . - 2 - 2 The following example is quite different in flavour:

Example 7.1.8. Let G B (F2, d×) be the coarse group obtained by equipping the non-abelian free group with the coarse structure coming from the cancellation distance d× (this is the bi-invariant metric where the length of a word is the minimal number of letters it is necessary to remove to make it reduce to the trivial word, see Example 4.2.12). We identify F2 with the set of (reduced) words in the letters a and b. Let H = {ak | k ∈ Z} be the subgroup generated by the letter a and consider the coarse quotient group Q B G/hh{H × H}ii (this is the quotient in the sense of Deﬁnition 5.3.3, it is not a coarse cosets space). As usual, we can construct a natural model for Q by considering (F2, D) where

D = Egrp = hE , ∆ ∗ (H × H) ∗ ∆ i, d×,H×H d× G G and the quotient map q: G → Q is realized by the identiy function q = [idF2 ]. We claim that q does not have a coarse kernel. In this specific case, one can show that the coarse structure D coincides with the structure induced a by a different cancellation metric d¯× on F2, namely the cancellation metric associated with the infinite (non-freely) generating set S = {ak | k ∈ Z} ∪ {b±}. For every r ∈ N consider the set

n Xr o n1 m1 n2 m2 nr mr Cr B a b a b ··· a b ni, mi ∈ Z, mi = 0 ⊆ F2. i=1

Note that C is contained in the ball B ¯a (e; r) because for every w ∈ C it is possible to remove all r d× r the occurrences of the letter a using at most r cancellations and the remainder reduces to the trivial Pr word because i=1 mi = 0. In particular, Cr is a D-bounded neighbourhood of the identity. However, n n r n −rn considering words of the form (a b ) a b with n arbitrarily large we see that Cr+1 is not Ed× -coarsely contained in B ¯a (e; r). It follows that for every r ∈ N the ball B ¯a (e, r) is not D-coarsely contained in d× d× B ¯a (e; r + 1). From here it is easy to conclude that the coarse kernel does not exist. d× −1 It is interesting to note that [H] is not coarsely normal in G (the sets gHg reach arbitrarily d×-far from H as g ranges in F2). This immediately implies that [H] cannot be the coarse kernel of the quotient F2 → F2/hh{H × H}ii, for coarse kernels are coarsely normal. Note also that Q is far from being the group quotient under the normal closure of H in F2: such a quotient would just be isomorphic to Z, while a it is simple to observe that (F2, d¯×) is not coarsely abelian.

7.2 The Isomorphism Theorems We will now show that the all isomorphism theorems hold in the category of coarse groups. As a preliminary remark, recall that—by deﬁnition—if f : G → H is a coarse homomorphism then f is a coarsely G-equivariant map, where G coarsely acts on itself by left multiplication and it coarsely acts on H multiplying on the left by its image under f. In symbols, the coarse action on H is the composition

∗H ◦ ( f × idH): G × H → H.

Let K ≤ G be a coarse subgroup such that f(K) = eH. Then f descends to a G-equivariant map of the coset space ¯f : G/K → H. Our study of coset spaces allows us to easily prove the following: Theorem 7.2.1 (Coarse First Isomorphism Theorem). Let f : G → H be a coarse homomorphism, K ≤ G a coarse subgroup such that f(K) = eH and let G/K be the coarse cosets space. The following are equivalent:

81 (1) the coarse map ¯f : G/K → H is proper;

(2) K = Ker( f);

(3) K is normal in G and ¯f : G/K → Im( f) is an isomorphism of coarse groups.

−1 Proof. Let G = (G, E), H = (H, F ), K = [K] and let ∗G, eG, (-) be adapted representatives for the operations of G. We can further assume that f (eG) = eH. We realise G/K as (G, D) where D = hE, ∆G ∗ (K × K)i. By Corollary 6.6.10, a subset B ⊆ G is D-bounded if and only if B 4E g ∗ K for every g ∈ B. (1) ⇒ (2): If the map f : (G, D) → (H, E) is proper, the preimage f −1(B) of every bounded neighborhood of the identity B ⊆ H is itself a D-bounded bounded neighborhood of eG. It follows that f −1(B) is E-coarsely contained in K, showing that [K] is the coarse kernel. (2) ⇒ (1): Assume that K = [K] is the coarse kernel. If B ⊆ H is bounded and f −1(B) , ∅, −1 pick some g ∈ f (B). The left multiplication maps [g−1 ∗G] and [ f (g)∗H] are coarse equivalences, and hence they always allow us to consider preimages (Lemma 3.5.9). Let B = [B]. We see that −1 −1 [ f (g)∗H] (B) = eH and hence ([ f (g)∗H] ◦ f) (B) = K because K is the coarse kernel of f. Since f is −1 a coarse homomorphism, f = [ f (g)∗H] ◦ f ◦ [g−1 ∗G]. Hence we deduce that f (B) also exists and it is −1 −1 equal to [g−1 ∗G] (K) = [g∗G](K). By deﬁnition, this implies that f (B) is E-coarsely contained in g ∗ K and it is hence D-bounded. (3) ⇒ (1): This is clear. (1) + (2) ⇒ (3): We already know that K = Ker( f) is a coarsely normal. It follows from Theo- rem 6.6.12 that G/K is indeed a group, hence it makes sense to ask whether ¯f is an isomorphism. Since we realise G/K as (G, D) and the maps f and ¯f (resp. ∗G and ∗¯ G) coincide setwise, it follows that ¯f : G/K → H is a coarse homomorphism. Since ¯f is proper, it is a coarse embedding (Corollary 5.1.9). Since ¯f is coarsely surjective on its image, it follows from Lemma 2.3.9 that ¯f : G/K → Im( f) is a coarse equivalence and hence an isomorphism of coarse groups. Given two coarse subspaces A = [A] and B = [B] of a coarse group G, their product A ∗ B is the coarse subspace [A ∗ B], where A ∗ B = {a ∗ b | a ∈ A, b ∈ B}. Equivalently, A ∗ B ⊆ G is the coarse image under ∗ of the coarse subspace A × B ⊆ G × G. The latter point of view makes it clear that if C ⊆ G is a third coarse subspace then (A ∗ B) ∗ C = A ∗ (B ∗ C). Theorem 7.2.2 (Coarse Second Isomorphism Theorem). Let H, N be coarse subgroups of G. If N is invariant under the H-coarse action by conjugation, then the coarse subspace (H ∗ N) ⊆ G is a coarse subgroup and N E (H ∗ N). Moreover, if the coarse intersection H ∩ N exists, then (1) H ∩ N is a coarsely normal coarse subgroup of H;

(2) there is a canonical isomorphism (H ∗ N)/N → H/(H ∩ N). Proof. Let G = (G, E) and H = [H], N = [N]. By deﬁnition, N is invariant under conjugation by H if and only if c(H, N) 4 N, where c: G × G → G denotes the conjugation c(h, g) = (h ∗ g) ∗ h−1. Since [H] = [H−1], we also have c(H−1, N) 4 N and hence there exists an E ∈ E such that for every n ∈ N and h ∈ H there exist n0, n00 ∈ N with

−1 0 −1 00 (h ∗ n1) ∗ h - E → n and (h ∗ n1) ∗ h - E → n .

In particular, if N is normalized by H then N ∗ H = H ∗ N. In fact N ∗ H 4 H ∗ N because

−1 0 n ∗ h - - → h ∗ (h ∗ n) ∗ h - ∆G∗E → h ∗ n (9) where - - → is shorthand for a ﬁnite composition of Eid, Einv, Eas and their multiplication with ∆G The coarse containment H ∗ N 4 N ∗ H is analogous.

82 Since ∗ is associative on coarse subsets, it follows that

(H ∗ N) ∗ (H ∗ N) = H ∗ (N ∗ H) ∗ N = H ∗ (H ∗ N) ∗ N = H ∗ N.

− − − − The coarse containment (H ∗ N) 1 4 (H ∗ N) follows from the fact that (h ∗ n) 1 - - → n 1 ∗ h 1. This proves that H ∗ N is a coarse subgroup. Both H and N are coarse subgroups of H ∗ N and it is immediate to verify that N is also coarsely normal. For the second part of the statement, assume that H ∩ N exists. We already know (Lemma 5.2.4) that H ∩ N is a coarse subgroup of H (and of N). Normality follows easily from the compatibility of the multiplication in H and G. Namely, if cH and cG denote the conjugation in H and G respectively, when embedding H in G we have cH(H, H ∩ N) = cG(H, H ∩ N). The right hand side is coarsely contained in N by assumption and it is obviously coarsely contained in H as well. It is hence coarsely contained in H ∩ N, as desired. It remains to prove that (H ∗G N)/N H/(H ∩ N). Let say let say that H ∩ N = [K] for some K ⊆ H. For every g ∈ H ∗ N choose h(g) ∈ H and n(g) ∈ N with g = h(g) ∗ n(g) and deﬁne a function φ: H ∗ N → H sending g to h(g). Equip H with the quotient coarse structure D = hE|H, ∆H ∗ (K × K)i. We claim that φ: (H ∗ N, E|H∗N) → (H, D) is controlled and that the coarse map [φ] does not depend on the choices made.

Claim. For every E ∈ E there exists a D ∈ D such that for every pair of elements g1, g2 such that gi = hi ∗ ni with hi ∈ H and ni ∈ N for i = 1, 2, if g1 ← E → g2 then h1 ← D → h2.

Proof of Claim. Fix gi, hi, ni as by the statement. Then

−1 −1 −1 −1 −1 −1 h2 ∗ h1 ← - → h2 ∗ (h1 ∗ n1) ∗ n1 ← ∆G∗E∗∆G → h2 ∗ (h2 ∗ n2) ∗ n1 ← - → n2 ∗ n1 where we have again been somewhat casual with the use of parentheses and used ← - → to denote some (ﬁxed) composition of Eid, Einv, Eas and multiplication with ∆G. The bottom line is that there exists an −1 −1 E1 ∈ E depending only on E so that h2 ∗ h1 ← E1 → n2 ∗ n1 . −1 −1 Let E2 ∈ E be a symmetric relation large enough so that H ∗ H ⊆ E2(H) and N ∗ N ⊆ E2(N). 0 0 0 −1 0 −1 Then we see that there exist h ∈ H and n ∈ N with h ← E2 → h2 ∗ h2 and n ← E2 → n2 ∗ n1 . In 0 particular, h ∈ H ∩ (E2 ◦ E1 ◦ E2(N)). However, it follows from the deﬁnition of coarse intersection that there exists some E3 ∈ E depending only on E such that H ∩ (E2 ◦ E1 ◦ E2(N)) ⊆ E3(K). At this point we are done, because

−1 0 h1 ← - → h2 ∗ (h2 ∗ h1) ← ∆G∗E3 → h2 ∗ h ← ∆G∗(E3(K)×{e}) → h2 ∗ e ← - → h2 showing that h1 and h2 are uniformly D-close.

Applying the claim to the diagonal E = ∆G we see that [φ] is independent of the choices made. Applying the claim to arbitrary E ∈ E shows that φ is controlled.By (9), the product (h1 ∗ n1) ∗ (h2 ∗ n2) is uniformly E-close to (h1 ∗ h2) ∗ n3 for some n3 ∈ N. It follows that [φ] is a coarse homomorphism. Now we only need to show that N E H is the coarse kernel of [φ]. Explicitly, we need to show that for every D-bounded neighbourhood of the idenity e ∈ B ⊆ H the preimage φ−1(B) is E-coarsely contained in N. By the deﬁnition of φ, the preimage φ−1(B) is contained in B ∗ N. On the other hand, it follows from Corollary 6.6.10 that the D-bounded set B is E-coarsely contained in K. In turn, since K is −1 E-coarsely contained in N, we see that B ∗ N 4E K ∗ N 4E N and hence φ (B) 4E N, as desired.

Theorem 7.2.3 (Coarse Third Isomorphism Theorem). Let N E G be a coarsely normal coarse subgroup. There exists a bijective correspondence:

{coarse subgroups of G/N} ←→ {K | N ≤ K ≤ G}.

83 This correspondence preserves normality. Furthermore, for every K E G coarsely containing N there is a canonical isomorphism (G/N)/(K/N) G/K. Proof. The projection G → G/N is a coarse homomorphism, hence the coarse image of every coarse subgroup of G is a coarse subgroup of G/N. We need to show that this map is a bijection when restricted to coarse subgroups of G that coarsely contain N. Let G = (G, E), N = [N]E and G/N = (G, D) with D = hE, ∆G ∗ (N × N)i. By Corollary 6.6.10, A 4D B if and only if A 4E N ∗ B. Let us prove injectivity. Let [K1]E and [K2]E be coarse subgroups of G coarsely containing [N]. Since the quotient map is the identity, they are sent to [K1]D and [K2]D. If K1 4D K2, it means that K1 4E N ∗ K2. Since N 4E K2, the latter is coarsely contained in K2 ∗ K2 4E K2. Therefore [K1]E ⊆ [K2]E. The converse coarse containment is analogous. For the surjectivity, let [K]D be a coarse subgroup of G/N. Note that N ∗ K D K, so that we can use N ∗ K instead of K as a representative of [K]D. This way we ensure that N 4E N ∗ K. It remains to see that [N ∗ K]E is a coarse subgroup of G. By assumption, (N ∗ K) ∗ (N ∗ K) 4D K. Therefore we have

(N ∗ K) ∗ (N ∗ K) 4E N ∗ K, showing that [N ∗ K]E is coarsely closed under multiplication in (G, E). Closure under inversion is analogous. In the coarse setting, the last statement is almost a tautology. Fix a representative of K = [K]E. We can also assume that N ⊆ K. We can realize the quotients as G/N = (G, D) and K/N = (K, D|K). Then (G/N)/(K/N) = (G, hD, ∆G ∗ (K × K)i). Since N ⊆ K, we see that hD, ∆G ∗ (K × K)i = hE, ∆G ∗ (K × K)i, and the latter is precisely the coarse structure that makes G into G/K.

7.3 A criterion for the existence of coarse kernels We now wish to provide a criterion to determine whether a coarse homomorphism admits a coarse kernel. To do so, we start by adapting the notion of ‘defect’ for R-quasimorphisms to the setting of coarse groups (we deﬁned R-quasimorphisms in Example 5.1.3). Recall that the defect of an R-quasimorphism −1 φ: G → R is deﬁned as Dφ B supx,y∈G|φ(xy) − φ(x) − φ(y)|. It is common to assume that φ(x ) = −φ(x) or, at the very least, that φ(eG) = 0. In this case, the defect has the obvious property that for every x, y ∈ G

−1 φ(xy) ∈ φ(x) + φ(y) + [−Dφ, Dφ] and φ(x ) ∈ −φ(x) + [−Dφ, Dφ].

The above is the property which we wish to reproduce for coarse homomorphisms of coarse groups. As usual, asking for pointwise containments of sets is not generally feasible (especially if the operations of H are not adapted, as in Example 4.3.8). It is better to introduce a notion of ‘defect relation’. Let (G, E) and (H, F ) be coarse groups. The defect relation of a controlled map f : G → H is the relation

−1 −1 F f B f (x1) ∗H f (x2) , f (x1 ∗G x2) | x1, x2 ∈ G ∪ f (x) , f (x ) | x ∈ G ⊆ H × H.

−1 −1 By deﬁnition, f (x1 ∗G x2) - F f → f (x1) ∗H f (x2) and f (x ) - F f → f (x) for every x1, x2, x ∈ G. In particular, f is a coarse homomorphism if and only if F f ∈ F . Note that for this characterisation of −1 −1 coarse homomorphism it is not really necessary to add to F f the points ( f (x) , f (x) ). However, doing so will simplify some algebraic manipulations later on. We learned the idea behind the following observation from Heuer–Kielak [KH21]. Some versions of it seem to go much further back in time and are implicit in the work of Meyer [Mey00] and some constructions of quasi-crystals (see also [BH18]). Informally, it says that if a subset B of the image of

84 a coarse homomorphism f has the property that a “suitably large” neighbourhood of B can be covered −1 with boundedly many translates f (a) ∗ B, then the preimage f (F f (B)) deﬁnes a coarse subgroup. The precise statement is as follows:

Lemma 7.3.1. Let f : G → H be a function between coarse groups, ﬁx some B ⊆ H and let K B −1 f (F f (B)). Assume that there exists a bounded neighbourhood of the identity eG A ⊆ G such that for every point y in op F f F f (B) ∗H F f (B) ∪ (F f (B) ⊆ H there is an a ∈ A with y ∗H f (a) ∈ B. Then [K] is coarse subgroup of (G, E).

−1 Proof. We begin by showing that K ∗G K 4 K ∗G A . Fix x1, x2 ∈ K, then we have f (x1 ∗G x2) ∈ F f f (x1) ∗H f (x2) ⊆ F f F f (B) ∗H F f (B) and therefore there is an a ∈ A such that f (x1 ∗G x2) ∗H f (a) ∈ B. Again, we have f (x1 ∗G x2) ∗G a ∈ F f f (x1 ∗G x2) ∗H f (a) ⊆ F f (B)

−1 and therefore (x1∗G x2)∗Ga ∈ K. Multiplying by a , we see that x1∗G x2 belongs to a controlled thickening −1 of K ∗G a (the control on the thickening is uniform in x1, x2, as it is obtained by one application of −1 −1 (Associativity), (Inverse) and (Identity)). The same argument also shows that K 4 K ∗G A . Since −1 A eG, this also shows that K ∗G K 4 K and K 4 K.

Remark 7.3.2. A few comments:

1. Recall that a subset X ⊆ G of a set-group G is an approximate subgroup if X = X−1 and there exists a finite A ⊂ G with X ∗ X ⊆ A ∗ X. If G is a set-group and f : G → H is a function so that −1 Lemma 7.3.1 applies for some finite set A ⊆ G, then the above proof shows that K B f (F f (B)) satisfies K ∗K ⊆ K ∗A−1. If one also knows that K = K−1 then this implies that K is an approximate subgroup of G (a priori, Lemma 7.3.1 only implies that K−1 ⊆ K ∗ A−1). This fact is used in [KH21] to show that (homogeneous) R-quasimorphisms have an approximate subgroup as “quasi-kernel”.

2. If the representatives of the operations of H are cancellative (Deﬁnition 4.4.5) it is possible to use a “defect set” instead of a defect relation. Namely, we can deﬁne the defect set of f : (G, E) → (H, F ) as the smallest set D f ⊆ H such that

op F f ∪ F f ⊆ ∆H ∗ (D f × {eH}).

The existence of such a set follows from Proposition 4.5.2. Now, for every B ⊆ H we have F f (B) ⊆ D f ∗ B. We can thus rephrase the statement of Lemma 7.3.1 as “if there exists A eG −1 −1 such that D f ∗ ((D f ∗ B) ∗ (D f ∗ B)) ⊆ B ∗ f (A) then [ f (D f ∗ B)] is a coarse subgroup ”.

Further extending the idea of Lemma 7.3.1, we prove a criterion for the existence of coarse kernels. This is based on the existence of appropriate coarse exhaustions by bounded sets:

Deﬁnition 7.3.3. A sequence of subsets An of a coarse space (X, E) is a coarse exhaustion of X if there is a controlled covering b ∈ C(E) such that for every bounded subset C ⊆ f (G) there exists n ∈ N with C ⊆ St(An, b).

Proposition 7.3.4. Let [ f ]: (G, E) → (H, F ) be a coarse homomorphism. If there exists a sequence of bounded neighbourhoods of the identity eG An ⊆ G whose image is a coarse exhaustion of f (G) ⊆ (H, F ), then [ f ] has a coarse kernel.

85 Proof. We will prove the claim by explicitly constructing a bounded subset of H whose preimage under −1 f is a representative of the coarse kernel of [ f ]. Let C B {y ∗H y | y ∈ H}, then C eH because (H, F ) is a coarse group. Let b ∈ C(F ) be a controlled covering such that that f (An) is a coarse exhaustion of −1 −1 f (G). We claim that if we are given x ∈ G and y ∈ St( f (x), b), then y ∗H f (x ) ∈ St C, b ∗H f (x ) . In −1 fact, y ∈ St( f (x), b) if and only if there is a B ∈ b with y, f (x) ∈ B. In particular, both y ∗H f (x ) and −1 −1 f (x) ∗H f (x ) are in B ∗H f (x ). The claim follows because

−1 −1 −1 −1 B ∗H f (x ) ⊆ St f (x) ∗H f (x ) , b ∗H f (x ) ⊆ St C , b ∗H f (x ) .

S −1 −1 Let A B n∈N An. The partial covering b ∗H f (iA ) = {B ∗H f (a ) | B ∈ b, a ∈ A} is controlled. We let −1 C B St C , b ∗H f (iA ) , and note that C C eH. Since [ f ] is a coarse homomorphism, F f (C) is also bounded. Let K B −1 −1 f (F f (C)), we claim that for every bounded D ⊆ H we have f (D) 4 K. This shows that [K] is the coarse preimage of eH and hence it is the coarse kernel of [ f ]. −1 Fix such a D, by hypothesis there is an n ∈ N such that D ⊆ St( f (An), b). Fix any x ∈ f (D) and let y B f (x). By construction, there is an a ∈ An such that y ∈ St( f (a), b) and by the previous argument we deduce that −1 −1 f (x) ∗H f (a ) = y ∗H f (a ) ∈ C. −1 We can now proceed as in the end of the proof of Lemma 7.3.1 to see that x ∗G a ∈ K and hence x −1 belongs to a controlled thickening of K ∗G An . The choice of this last thickening does not depend on −1 −1 −1 x ∈ f (D) and therefore f (D) 4 K ∗G An 4 K, as desired. Recall that if G is a set-group equipped with an equi-bi-invariant coarse structure then a function φ: (G, E) → (R, Ed) deﬁnes a coarse homomorphism if and only if it is a controlled R-quasimorphism (Example 5.1.3). Proposition 7.3.4 implies the following.

Corollary 7.3.5. Let G be a set-group and φ: G → R an R-quasimorphism. If E is a coarsely connected equi-bi-invariant coarse structure on G so that φ: (G, E) → (R, Ed) is controlled then [φ] has a coarse kernel.

Proof. Assume for convenience that φ is homogeneous, i.e. φ(xk) = kφ(x) for every k ∈ Z (every R-quasimorphism is close to a homogeneous one). If φ is trivial then Ker([φ]) = [G] and there is nothing to prove. k If φ is non-trivial, there exists an a ∈ G such that φ(a) > 0. Letting An B {a | −n ≤ k ≤ n}, we see S that n∈N φ(An) is a lattice in R and hence the sequence of sets φ(An) is a coarse exhaustion of R. Since (G, E) is coarsely connected and An is ﬁnite for every n ∈ N, the sets An are bounded neighbourhoods of the identity. It follows from Proposition 7.3.4 that φ has a coarse kernel. Explicitly, one can see that [φ−1((− f (a), f (a)])] is the coarse kernel of [φ].

More generally, it is convenient to spell out the following consequence of Proposition 7.3.4 for coarsely connected coarse groups admitting cancellative operations.

Corollary 7.3.6. Let (G, E) be coarsely connected and [ f ]: (G, E) → (H, F ) be a coarse homomorphism. Assume that the operations on (H, F ) have cancellative representatives and let D f ⊆ H be the defect set of f (Remak 7.3.2). If there exists a bounded neighbourhood of identity B ⊆ H so that for every bounded set C ⊆ H there exist g1,..., gn ∈ G with C ⊆ f (g1) ∗H B ∪ · · · ∪ f (gn) ∗H B

−1 then [ f (D f ∗H B)] is the coarse kernel of [ f ].

86 7.4 Some comments and questions In view of developing a structure theory for coarse groups, it is important to understand which coarse homomorphisms have a coarse kernel. Namely, we are interested in the following general question:

Question 7.4.1. When does a coarse homomorphism have a coarse kernel?

Recall that we already know the answer to the above for coarse homomorphism f : G → H where G = (G, Emin) is a trivially coarse group (Example 7.1.6). Moreover, if H is not coarsely connected then its coarse connected component containing the identity element is a coarse subgroup He ≤ H for which −1 it is easy to show that the coarse preimage f (He) ≤ G always exists. In particular, f : G → H has a −1 | −1 → coarse kernel if and only if f f (He) : f (He) He has a coarse kernel. We thus ﬁnd that Question 7.4.1 is particularly interesting for coarse homomorphisms of coarsely connected coarse groups.

Note that the same argument as in Corollary 7.3.5 shows that if G is a coarsely connected coarse group then every coarse homomorphism G → (R, Ed) has a coarse kernel. Examining the proof suggests that the main reason why this is the case is the fact that (R, Ed) is “one dimensional”. More generally, it seems plausible that every coarse homomorphism f : G → H where H is a coarse group of ﬁnite asymptotic dimension has a coarse kernel (see [Roe03, Chapter 9] for a deﬁnition of asymptotic dimension).

It is also interesting to restrict the scope of Question 7.4.1 to speciﬁc classes of coarse groups. For example, we will show in Subsection 12.2 that a coarse homomorphism between two Banach spaces has a coarse kernel if and only if the image of its homogenization is a closed vector subspace.

The above remark shows that Question 7.4.1 is non trivial also when restricted to controlled homomorphisms of set-groups. That is, the following seems interesting:

Question 7.4.2. Let G, H be set-groups equipped with equi-bi-invariant coarse structures E, F and let f : G → H be a homomorphism that is controlled as a function (G, E) → (H, F ), so that [ f ] is a coarse homomorphism. When does [ f ] have a coarse kernel?

87 Part II Selected topics

8 Coarse structures on set-groups

In this section we will start looking into coarsifications of set-groups. Definition 8.0.1. A coarsified set-group is a coarse group G = (G, E, [∗], [e], [-]−1) where (G, ∗, e) is a set-group. In Section9 we see Z admits many inequivalent coarsifications. The situation for other set-groups is of course more varied and in fact it is already an interesting question to understand which set-groups admit “interesting” coarsifications at all! More precisely, one obvious way to coarsify any set group G is by giving it the bounded coarse structure Emax. We also saw another uninteresting way to coarsify G, namely by giving it the trivial coarse E E structure min and viewing it as a trivially coarse group. In general, if is a coarse structure on G so that G = (G, E) is a coarse group (i.e. E is equi-bi-invariant), let G0 be the coarsely connected component of the identity. Since the left and right multiplications are controlled, it follows that G0 is a normal subgroup of G. Moreover, the set of coarsely connected components of G is naturally identified with the quotient set-group G/G0. In other words, the coarsified set-group G can be decomposed as a coarsely connected coarsified set-group G0 and a trivially coarse group. In view of this, we find that the most interesting equi-bi-invariant coarse structures on a set-group are the coarsely connected ones. The general question thus becomes: Question 8.0.2. Which set-groups admit unbounded, coarsely connected coarsifications? In the remainder of this section we will give examples of groups that have various non-trivial coarsifications and others that do not have any. The lack of unbounded coarsely connected coarsifications can be seen as rigidity phenomenon. Namely, it says that any attempt to ‘blur’ the algebraic structure of the set-group collapses to produce something bounded or coarsely disconnected.

8.1 Bi-invariant metrics

Recall that a coarse structure Ed induced by some metric d is always coarsely connected. We also know that if G is a set-group and d is a bi-invariant metric on it, then (G, Ed) is a coarse group. The only thing one need to check to know whether this is an “interesting” coarsification is whether the metric d is unbounded. Namely, the following is a special case of Question 8.0.2: Question 8.1.1. Which set-groups admit unbounded bi-invariant metrics? The above is a rather natural question: specific instances of it have appeared in various pieces of literature. In particular, we can readily draw from the literature various examples of set-groups with interesting coarsifications: much of the following material is taken from [BIP08] and [GK11]. To navigate the literature, it is useful to recall that a distance function on G is bi-invariant if and only if the induced length function |g| B d(g, e) is invariant under conjugation (Example 4.2.11). The problem of finding unbounded bi-invariant metric is thus often phrased in terms of the existence of unbounded length functions that are invariant under conjugation.12

One of the most important classes of bi-invariant metrics are the cancellation metrics that we encountered in Example 4.2.12. Recall that if a set S ⊆ G normally generates G, we deﬁned the

12In various pieces of literature length functions are called norms.

88 −1 cancellation metric d× induced by S as the word metric associated with the generating set S = {gsg | s ∈ S, g ∈ G}. This is a bi-invariant metric because S is invariant under conjugation. It is easy to see the reason why cancellation metrics are important for us. In fact, if S is a ﬁnite normally generating set of a set-group G and d is any bi-invariant metric on G, then it follows that

d(g1, g2) ≤ (max{d(s, e) | s ∈ S })d×(g1, g2) for all g1, g2 ∈ G. In particular, if d× happens to be bounded then so is every other bi-invariant metric. In Subsection 8.3 we will see that in many cases even more is true, namely if d× is bounded then G does not admit unbounded coarsely connected coarsification at all. Remark 8.1.2. The vast majority of bi-invariant metric that we will consider are special examples of cancellation metrics (see Section 10 for examples). Of course, this does not meant that there are no other interesting classes of (unbounded) bi-invariant metrics. To mention one, there is a rather different kind of bi-invariant which metrics plays an important role in the study of groups of Hamiltonian diffeomorphisms of symplectic manifolds [Hof90, LM95]. Two elementary lemmas that are often used to show that a set-group has unbounded bi-invariant metrics are as follows.

Lemma 8.1.3. Let (H, dH) be a set-group with a bi-invariant metric. If there is a homomorphism ψ: G → H such that ψ(G) has inﬁnite diameter then G admits an unbounded bi-invariant metric.

Sketch of proof. Let |-|H be the length function associated with the bi-invariant metric on H. Up to letting 0 |h|H B |h|H + 1 for every h ∈ H r {e}, we can assume that |ψ(g)|H ∈ {0} ∪ [1, ∞) for all g ∈ G. We can then let   0 g = e  |g|G B 1 g ∈ ker(ψ)   |ψ(g)|H otherwise and observe that |-|G is a conjugation invariant length function.

Corollary 8.1.4. If G surjects onto Z then it admits unbounded bi-invariant metrics.

Lemma 8.1.5. If G is normally generated by a subset S and φ: G → R is an unbounded R-quasimorphism so that {|φ(s)| | s ∈ S } is bounded, then G admits an unbounded bi-invariant metric.

Sketch of proof. It is easy to verify that the cancellation metric associated with S is unbounded.

Corollary 8.1.6. A non-elementary Gromov hyperbolic group admits unbounded bi-invariant metrics. More generally, the same is true for acylindrically hyperbolic groups.

Proof. It was proved by Epstein–Fujiwara that such a hyperbolic group G has unbounded R-quasimorphisms [EF97]. Since G is ﬁnitely generated, the boundedness condition is trivially satisﬁed. Similarly, it follows from [HO13] that acylindrically hyperbolic groups have unbounded R-quasimorphisms.

Remark 8.1.7. It is possible to merge Lemma 8.1.5 and Lemma 8.1.3 into a more general criterion. From the point of view of coarse structures, this is just a manifestation of the fact that if (H, F ) is a coarsely connected coarse group and [ f ]: (G, Emin) → (H, F ) is a coarse homomorphism with unbounded image, then (G, f ∗(F )) is a coarsely connected unbounded coarsiﬁcation of G by Lemma 5.1.8.

89 8.2 Metric vs arbitrary coarsiﬁcations

A coarse structure E on a set X is metrizeable if it coincides with the coarse structure Ed defined by a metric d on X. It is well-known that a connected coarse structure is metrizeable if and only if E is countably generated ([Roe03, Theorem 2.55], see also [Ros17, Section 4]). The same holds true for coarsifications of set-groups: Lemma 8.2.1. Let (G, E) be a coarsely connected coarsified set group. There exists a bi-invariant metric d so that E = Ed if and only if E is countably generated.

Proof. One implication is obvious: a metric coarse structure Ed is always generated by the countable family En B {(x1, x2) | d(x1, x2) ≤ n}. op For the other implication, let E = hEn | n ∈ Ni. Replacing En with ∆G ∗ (En ∪ En ) ∗ ∆G, we may assume that each En is symmetric and invariant under left and right translation. We deﬁne a function d0 : G × G → G by 0 d (x, y) B inf{n | (x, y) ∈ En} ∪ {+∞}. By construction, d0 is symmetric and invariant under both left and right multiplication. It need not be a metric, as it may fail to be transitive and can take inﬁnite values. Both issues are solved by letting

k X 0 d(x, y) B inf d (xi−1, xi), k,x ,...,x 0 k i=1 where the infimum is taken over all the finite sequences x0,..., xk ∈ G such that x0 = x and xk = y. In fact, transitivity is built into the definition of d, and d takes only finite values because we are assuming that E is connected. It is easy to verify that E = Ed. Recall that a coarse structure is coarsely geodesic if it is connected and generated by a single relation (Definition 6.4.2). We may then recognise that coarsely geodesic coarsifications of G are precisely the coarsifications arising from cancellation metrics: Proposition 8.2.2. Let (G, E) be a coarsely connected coarsified set group. Then E is coarsely geodesic if and only if there exists a cancellation metric d× so that E = Ed.

Proof. It is clear from the deﬁnition of cancellation metric that the relation E1 B {(x1, x2) | d×(x1, x2) ≤ 1} generates Ed× . This proves one implication. Vice versa, let E be generated by a single relation E. As before, we may assume that E is symmetric and invariant under left and right translation. Let S B eE = {x ∈ G | (e, x) ∈ E}. Note that S is invariant under conjugation: for every s ∈ S and g ∈ G the pair (e, g ∗ s ∗ g−1) belongs to E because

−1 −1 −1 (e, g ∗ s ∗ g ) = (g, g) ∗ (e, s) ∗ (g , g ) ∈ ∆G ∗ E ∗ ∆G.

It follows that the cancellation metric d× associated with S is in fact equal to the word metric dS . Also note that E = ∆G ∗ ({e} × S ). We then see that the formula for d in the proof of Lemma 8.2.1 deﬁnes precisely the word metric dS . As before, we now see that E = EdS = Ed× .

8.3 Intrinsically bounded set-groups Not every set-group admits interesting coarsifications. For example, the only way to make a finite set-group into a coarsely connected coarse group is by giving it the bounded coarse structure. More generally, we give the following definition.

Deﬁnition 8.3.1. A set-group G is intrinsically bounded if the bounded coarse structure Emax = P(G ×G) is the only connected equi-bi-invariant coarse structure on G.

90 By definition, (G, E) is coarsely connected if and only if each pair {g1, g2} is a E-bounded set. By Egrp composition, each finite subset of G must be bounded. Recall that we defined fin as the minimal coarse structure for which all the finite sets are bounded (Example 4.3.7). It then follows that G is intrinsically Egrp E bounded if and only if fin = max. In turn, we can prove the following: Proposition 8.3.2. A set-group G is intrinsically bounded if and only if there exist finitely many conjugacy classes [g1],..., [gn] ⊆ G and an m ∈ N such that every element of G can be written as the product of at most m elements in [g1],..., [gn].

Proof. For the sufficiency, assume that we are given conjugacy classes [g1],..., [gn] as by the statement. { } E ∈ Since e, g is fin-bounded for every g G it follows from condition (U5) of Subsection 4.6 that every Egrp conjugacy class of G is fin -bounded. We then see that G is bounded because it is contained in a finite union of finite products of bounded sets. Vice versa, we know from Subsection 4.6 that any family U of subsets of G that is closed under taking subsets, inverses, finite products and closure under conjugation is the family of bounded sets of some equi-bi-invariant coarse structure on G. Let U be the family of all the sets A ⊆ G such that there exist finitely many conjugacy classes [g1],..., [gn] ⊆ G and an m ∈ N such that every element of A can be written as the product of at most m elements in [g1],..., [gn]. It is easy to see that this U is closed under the above operations, and hence it is the family of bounded sets for some connected equi-bi-invariant coarse structure. On the other hand, it follows from the proof of the other implication that U is contained in the family of Egrp Egrp U Egrp fin -bounded sets. By the minimality of fin , it follows that is equal to the family of fin -bounded sets. Egrp In particular, we see that G is fin -bounded if and only if there exist [g1],..., [gn] as by the statement. As a sample application of the above criterion, we immediately see that the infinite dihedral group

2 −1 D∞ = Z o Z/2Z = hr, t | r = 1, rt = t ri is intrinsically bounded. Other interesting examples are the set-groups and SL(n, Z) with n ≥ 3: this follows from the fact that these groups are boundedly generated by the set of elementary matrices [CK83, Mor05]. More precisely, let Ei j be the matrix having coefficient 1 on the diagonal and on position (i, j) and 0 elsewhere. One can then show that every elementary matrix in SL(n, Z), n ≥ 3 can be written m as a commutator [Ei, j, Ek,l] (see [BIP08, Example 1.1]) and hence the finite set of matrices Ei j satisfies the hypotheses of Proposition 8.3.2.

Remark 8.3.3. Note that Z is an index 2 subgroup of D∞. This shows that intrinsic boundedness is a ﬁne algebraic notion that has little to do with the “geometry” of a group—at least as far as geometric group theory is concerned. Rather conveniently, we can now see that set-groups that are “small enough” are intrinsically bounded if and only if they admit no unbounded bi-invariant metrics:

Proposition 8.3.4. Let G be a set-group. If G is either

• ﬁnitely normally generated;

• countable; then it admits an unbounded coarsely connected coarsification if and only if it admits an unbounded bi-invariant metric. Egrp Proof. Recall if a set-group G is normally generated by a finite set S , then fin is equal to the coarse structure Ed× defined by the cancellation metric associated with S (Example 4.5.4). This means that such a group is intrinsically bounded if and only if d× is bounded.

91 Assume now that G is countable. Then the family of all finite subsets of G is countable. It follows Egrp from Lemma 4.3.5 that the coarse structure fin is countably generated. We may then apply Lemma 8.2.1 Egrp E to deduce that fin = d for some bi-invariant metric d. Hence G has an unbounded coarsification if and only if d is unbounded.

Knowing this, we can once refer to the literature to look for more examples of intrinsically bounded groups. Two results of special interest are the following:

Theorem 8.3.5 ([Dus12, MP11]). A Coxeter group is intrinsically bounded if and only if it is of spherical or aﬃne type.

Theorem 8.3.6 ([GK11]). Many Chevalley groups are intrinsically bounded.

Remark 8.3.7. It is not known whether a lattice Γ in higher rank semisimple real Lie groups with ﬁnite center must be intrinsically bounded ([GK11, Question 1.2]). If that was the case, it would follow from the Margulis Normal Subgroup Theorem that such a Γ is “strongly intrinsically bounded” in the sense that the coarsely connected components of any equi-bi-invariant coarse structure on Γ are bounded. This should be contrasted with the inﬁnite dihedral group D∞, which is intrinsically bounded but can be given a (disconnected) coarse structure making it coarsely equivalent to the disconnected union of two copies of (Z, E|-|). This being said, in general we still do not know the answer to the following:

Question 8.3.8. Do there exist set-groups that do not have unbounded bi-invariant metrics but are not intrinsically bounded?

By the previous discussion we know that a set-group as in Question 8.3.8 cannot be countable nor have a finite normally generating set. In particular, such a group cannot be finitely generated. A good place to start would have been by looking at groups of diffeomorphisms of manifolds e.g. because it it is n shown in [BIP08, Theorem 1.11] that Diff0(S ) does not have unbounded bi-invariant metrics. However, this does not work either because such groups are often simple and hence finitely normally generated (we refer to [BIP08] or the book [Ban13] and their references for an in-depth discussion of groups of geometric origin).

9 Coarse structures on Z

In this section we will provide a zoo of examples of coarse structures on the integers, Z, endowed with which Z becomes a coarse group. We show that there is at least a continuum of distinct coarsifications Z. It is interesting to contrast this with the fact that the infinite dihedral group D∞ is intrinsically bounded (see Section 8.3). Since Z < D∞ is a finite index subgroups, this indicates that determining what are the possible coarsifications of a given set-group requires some finesse in general. left There are two main approaches to generating a coarse group structure on Z: using Ecpt from Example 6.1.3, and by choosing a set of generators for the Cayley graph of Z. We explore both.

9.1 Coarse Structures Generated by Topologies One of the easiest way to deﬁne a coarsiﬁcation of set-group G is by specifying its bounded neighborhoods of the identity. Namely, we know from Subsection 4.6 that if U is a family of subsets of G satisfying

(U0) e ∈ U for every U ∈ U,

(U1) if U ∈ U and e ∈ V ⊂ U then V ∈ U,

92 (U3) if U ∈ U then U−1 ∈ U,

(U4) if U1, U2 ∈ U then U1 ∗ U2 ∈ U, (U5) if U ∈ U then S{g ∗ U ∗ g−1 | g ∈ G} ∈ U, grp then there exists a unique bi-invariant coarse structure EU so that U is the set of bounded neighborhoods of the identity (condition (U2) is trivially satisﬁed for set-groups). Note that if G is abelian, condition (U5) is always true. We already remarked in Example 6.1.3 that when G is a topological group, the family of relatively compact subsets of G containing e does satisfy (U0)–(U4). In particular, if G is also abelian we can use the family of compact sets to obtain a nice coarsiﬁcation of G. Explicitly, we have

grp −1 left Ecpt = {E ⊆ G × G | ∃K ⊆ G compact s.t. ∀(g1, g2) ∈ E, g2 g1 ∈ K} = Ecpt Remark 9.1.1. An alternative way of saying this is that every topological group has an equi-left-invariant left coarse structure Ecpt whose bounded neighbourhoods of the identity are the relatively compact subsets of G containing e (Example 4.6). If G is an abelian set-group, then every equi-left invariant coarse structure is actually equi-bi-invariant and hence defines a coarsification of G. grp If G is not abelian, one may still define a coarse structure Ecpt by taking the minimal equi-bi-invariant coarse structure so that compact sets are bounded. However, in general such a coarse structure will differ left from Ecpt and it will have many non relatively compact bounded sets. This makes it fairly complicated to understand. We can use topologies on the set of integers to coarsify Z. In the following, τ will denote a group left topology on Z (i.e. a topology such that addition and inversion are continuous) and Eτ B Ecpt will be the induced equi-bi-invariant coarse structure. In Eτ the bounded sets are the relatively compact sets. Thus two group topologies give rise to different coarse structures on Z if and only if they have different relatively compact sets. There are two obvious examples of coarse structures of topological origin. If τ is a topology so that Z is compact (e.g. the indiscrete topology) then Z is bounded and Eτ = Emax. If τ is the discrete topology E Egrp E then τ = fin = |-| coincides with the standard coarse structure defined by the Euclidean metric. Note that if a topology τ is contained in a finer topology τ0 then every τ0-compact set is also τ-compact. Hence 0 Eτ0 ⊆ Eτ (if the topologies are Hausdorff then Eτ0 = Eτ if and only if every τ-compact is also τ -compact).

Warning. If a topology τ is induced by a bi-invariant metric d then Eτ ⊆ Ed, but in general the containment may be strict (the equality holds if and only if d is proper).

Example 9.1.2. Another interesting group topologies on Z is the Furstenberg topology τF. This is the topology having the infinite arithmetic progressions {aZ + b | a, b ∈ Z} as basis for the open sets [Fur55] (one may also note that this topology is the subset topology defined by the inclusion of Z in the profinite completion bZ, more on this in Subsection 9.2). This topology is a nice, metrizable topology on Z and—importantly for us—Z is not compact in this topology (this can be seen e.g. by considering the infinite covering by open sets {2n+1Z − 2n | n ≥ 1}). It follows that Z is not EτF -bounded and hence EτF ( Emax. At the same time, one may easily verify that the sequence n! converges to 0 in τF. It follows that the infinite set {n! | n ∈ N} ∪ {0} is compact and hence

EτF -bounded. We then see that we have strict inclusions Egrp E E ﬁn ( τF ( max, and hence (Z, EτF ) is a genuinely new coarsiﬁcation of Z. At this point we do not know much else about the coarse structure EτF : to understand it better it would be useful to have some characterization of the τF-compact subsets of Z. We refer to [LM15] and references therein for more properties of the Furstenberg topology.

93 Example 9.1.2 is but the tip of the iceberg: infinite abelian groups tend to have a huge variety of group topologies. Of course, this information on its own is not enough to deduce that Z has many different coarsifications because we still need to understand the relatively compact sets in those topologies. To begin, it is certainly necessary to focus of non-compact topologies. Secondly, one also needs to make sure that the topology does have infinite compact sets. These two steps are necessary to ensure that the E Egrp E coarse structure τ induced by τ differs from fin and max. As shown in Example 9.1.2, one convenient way to make sure that a topological space has infinite compact sets is by showing that it admits some infinite converging sequence. Using group invariance, one may also focus on sequences converging to 0. This point of view helps navigate the literature. For example, in [ZP91] Zelenyuk and Protasov completely characterize those sequence (an)n∈N for which there exist a τ Hausdorff topology τ so that an −→ 0 (they call them T-sequences). By construction, all the topologies Egrp witnessing that some sequence is a T-sequence generate coarse structures that strictly contain fin .A very partial list of related references includes [BDMW03, DMT05, HRG12, Nie72, Heg05, Skr20] The above discussion leaves us with a wide variety of candidate topologies on Z. However, this also leaves us with the daunting task of understanding their compact sets. This seems to be a rather interesting but difficult question. In the next subsection we will content ourselves to study coarse structures induced by profinite topologies: this will be enough to deduce that Z has a continuum of different coarsely connected coarsfications.

9.2 Proﬁnite coarse structures

Let Q be a non-empty set of primes and consider the family Q∗ B {m ∈ N | (m, p) = 1 ∀p prime p < Q} of positive numbers whose factorization consists of powers of primes in Q. The family of quotients Z/mZ with m ∈ Q∗ forms an inverse system of ﬁnite groups. We deﬁne the pro-Q completion of Z as the inverse limit Z lim Z/mZ. Q B ←−− m∈Q∗

The limit topology is a metrizable, compact group topology on ZQ (one may realize ZQ as a subgroup of the inﬁnite product Q Z/mZ and the limit topology coincides with the subspace topology). Two examples of special interest are the proﬁnite completion bZ B Z{all primes} and the pro-p completion Zp B Z{p} (this is also known as the group of p-adic integers). Using the Chinese Reminder Theorem, one can show that every pro-Q completion is isomorphic (as a topological group) to a product of pro-p completions: Y ZQ Zp. p∈Q

For any choice of Q, the natural homomorphism Z → ZQ is an embedding with dense image. When Q identifying ZQ with the product of pro-p completions, the embedding Z ,→ p∈Q Zp is the diagonal embedding. Since Z is a dense proper subset of ZQ and the latter is Hausdorﬀ, Z is not compact in ZQ.

Deﬁnition 9.2.1. Let τQ denote the subspace topology on Z ⊂ ZQ. The pro-Q coarse structure on Z is the equi-left-invariant coarse structure EQ B EτQ induced by the pro-Q topology τQ.

τQ Concretely, a subset A ⊆ Z is EQ-bounded if and only if its closure A ⊆ Z is compact. Note that τQ ZQ this deﬁnition is not trivial, because closure A = A ∩ Z is generally not closed in ZQ and hence not compact. In particular, EQ is never equal to the bounded coarse structure Emax because Z is not compact in ZQ. Remark 9.2.2. It is not hard to see that the sets of the form mZ + k with m ∈ Q∗ and k ∈ N are a basis of open sets for τQ. In particular, this shows that the Furstenberg topology τF is indeed equal to the topology τ{all primes} induced by the proﬁnite completion of Z.

94 Lemma 9.2.3. For every non-empty set of primes Q the pro-Q coarse structure is the intersection \ EQ = Ep. p∈Q Proof. Since equi-left-invariant coarse structures are determined by their bounded sets (Proposition 4.5.2), it is enough to verify that a subset K ⊂ Z is τQ compact if and only if it is τp compact for every p ∈ Q. Q This is easily done identifying ZQ with the product p∈Q Zp and seeing τQ as the topology induced by the diagonal embedding. Q Since the projections p∈Q Zp → Zp are continuous, it is clear that any τQ compact set is τp compact for every p ∈ Q. Assume now that K ⊂ Z is τp compact for every p ∈ Q, we need to show that the Q diagonal embedding K ,→ p∈Q Zp has compact image. This embedding factors as a diagonal embedding into a product of copies of K (equipped with the various pro-p topologies) followed by the inclusion Y Y K ,→ K ⊂ Zp. p∈Q p∈Q Q Q By the assumption, p∈Q K is compact. Since the topologies τp are Hausdorﬀ, the diagonal K ⊆ p∈Q K is closed and hence compact.

0 0 Proposition 9.2.4. If Q an Q are two sets of primes, then EQ ⊆ EQ0 if and only if Q ⊇ Q . Proof. One implication follows immediately from Lemma 9.2.3. It remains to show that if Q0 is not 0 contained in Q then EQ * EQ0 . Let q be a prime in Q r Q. We need to find a τQ-compact set K ⊂ Z that 0 0 is not contained in any τQ -compact set. Since τq ⊆ τQ, it is enough to check that K is not contained in any τq-compact set. We will do so by constructing a sequence (kn)n∈N such that kn → 0 with respect to τQ but it converges to some point ξ ∈ Zq r Z with respect to the pro-q topology. Assuming that such a sequence exists, we S see that K = {kn | n ∈ N} ∪ {0} is τQ-compact. Since Zq is Hausdorff, ξ is the unique accumulation point of K in Zq. It follows that K has no τq-accumulation point in Z and hence it is not τq-compact. On the other hand, K ∪ {ξ} is compact in Zq, hence K is τq-closed. It follows that K is not contained in any τq-compact set, otherwise K would be compact. It remains to find an appropriate sequence. Note that for every l ∈ N any product of powers of primes in Q is coprime with ql and hence belongs to the group of units (Z/qlZ)∗. For any x ∈ (Z/qlZ)∗, let l ∗ m l ordql (x) be the order of x in (Z/q Z) (i.e. the smallest exponent so that x ≡ 1 mod(q )). We shall now construct the sequence iteratively. We will deal with the case where Q is infinite, the finite case is simpler and can be dealt with similarly. Choose an ordering p1, p2,... for the elements of Q a1 and let a1 B ordq(p1). We define k1 B p1 . l1 a2 Choose some l1 large enough so that k1 ≤ q and let a2 B ordql1 (p1 p2). Define k2 B k1(p1 p2) . With this choice, we see that

l1 k2 ≡ k1 ≡ 1 mod(q) and k2 ≡ k1 . 1 mod(q ).

Moreover, the power of p1 appearing in the factorization of k2 is strictly larger than that of k1. l Inductively, assume that kn−1 has already been defined. Choose ln so that kn−1 ≤ q n , let an B an ordqln (p1 p2 ··· pn) and define kn B kn−1(p1 p2 ··· pn) . We claim that the sequence kn does what we need. N For each i and each N, we see that pi divides kn for every n large enough. By definition of the pro-Q l topology, this shows that kn → 0 in ZQ. On the other hand, for each j ∈ N we see that kn ≡ k j mod(q j ) for every n ≥ j. This shows that the sequence Y l j ([k1], [k2], [k3],...) ∈ Z/q Z j∈N

95 l defines an element ξ ∈ Zq = lim Z/q j Z and that kn → ξ in Zq. Finally, this ξ does not belong to Z ←−− j∈N l because it is not eventually constant: for every j ∈ N and n < j we have kn . kn−1 mod(q j ). Corollary 9.2.5. The pro-Q coarse structures provide a continuum of different coarsely connected coarsifications of Z.

9.3 Coarse structures via Cayley graphs and other questions Another method to define interesting coarse structures on Z is to consider word lengths associated with infinite generating sets. In other words, for any choice of (possibly infinite) generating set S we can construct the Cayley graph Cay(Z, S ) and use the graph metric to define a coarse structure ECay(S) on Z. Of course, ECay(S) is equi-bi-invariant because the metric is left-invariant and Z is abelian. Recall the classical version of Milnor-Svarc tells us Cayley graphs are quasi-isometric only for finite generating sets. E Eleft E If S is finite we did not discover anything new: Cay(S) is nothing but fin = |-|. However, when S is infinite ECay(S) is a strictly larger coarse structure: the Cayley graph is not locally finite and there are bounded sets of infinite cardinality. If the set S is too large, ECay(S) may be equal to the maximal coarse structure Emax. This is obviously the case if we let G = S or any other ridiculously large set. On the other hand, if the set S is very sparse it becomes an interesting question to know whether ECay(S) = Emax. Explicitly, ECay(S) is equal to the maximal coarse structure if and only if there exists some fixed N so that every integer k ∈ Z can be written as a sum or difference of at most N elements of S . In particular, if

S = P is the set of all primes the equality ECay(P) = Emax becomes a very weak version of the Goldbach conjecture. It is a classical fact that every integer is equal to the sum of a bounded number of primes [Vin37]. In other words, it is already known that Cay(Z, P) does indeed have finite diameter (its value is also known as the Shnirelman constant). Helfgott’s proof of the ternary Goldbach conjecture [Hel15] implies that this diameter is at most 4 (the strong Goldbach conjecture would imply that it is 3). On the positive side, it is fairly simple to see that if S = {an | n ∈ N} is a geometric progression for some a > 1 then Z is not ECay(S)-bounded. It is also true that Z is unbounded if S is the set of all numbers whose factorization only contains primes in some fixed finite set [Nat11b, Theorem 3]. In particular, it follows that also that sets like S = {2n | n ∈ N} ∪ {3n | n ∈ N} give rise to a Cayley graph of infinite diameter. For a given set S , deciding whether Z is ECay(S)-bounded seems to be a hard number theoretic problem. It is also an interesting problem to understand when two different generating sets S 1, S 2 give rise to different coarse structures. This problem is solved in [Nat11a] in the case where both S 1 and S 2 are geometric progressions: for S = {an | n ∈ N} and S = {bn | n ∈ N}, E = E if and only if 1 2 Cay(S1) Cay(S2) an = bm for some n, m ∈ N. Little else is known otherwise (see also [BLPY17, Nat11c] for related work).

We end this section with a few open questions. We showed that there exists at least a continuum 2ℵ0 ℵ of different coarsifications, but the set of coarse structures on Z could in principle be as large as 22 0 . Question 9.3.1. What is the cardinality of the set of distinct coarsifications of Z? We provided two constructions of coarsifications for Z: a topological and a geometric one. It seems very unlikely that this should be an exhaustive list. Moreover, the relation between the two is is not clear. For example, we do not know the answer to the following:

Question 9.3.2. Does there exist an inﬁnite generating set S so that ECay(S) is a pro-Q coarse structure? How about other topological coarse structures? By construction, the Cayley graph is a geodesic metric space. Therefore, one possible approach to the above question would be by studying whether topological coarse structures are coarsely geodesic.

Question 9.3.3. When is (Z, Eτ) coarsely geodesic (Deﬁnition 6.4.2)?

96 Finally, if we are given two coarsiﬁcations (Z, E1) and (Z, E2) with E1 , E2 it may still be the case that the coarse group (Z, E1) and (Z, E2) are isomorphic via some function that is not the identity. This raises an obvious question:

0 Question 9.3.4. Given two diﬀerent sets of primes Q , Q , can (Z, EQ) and (Z, EQ0 ) be isomorphic coarse groups? Let S = {2n | n ∈ N} and S = {3n | n ∈ N}, are (Z, E ) and (Z, E ) isomorphic? 1 2 Cay(S1) Cay(S2) A version of the last question has already appeared in the literature. Namely, it is a question attributed n to Richard E. Schwarz whether the Cayley graphs associated with the sets S 1 = {2 | n ∈ N} and n S 2 = {3 | n ∈ N} are quasi-isometric. Note that in this case the Cayley graphs are quasi-isometric if and only if they are coarsely equivalent (this is because they are geodesic metric spaces). Our question diﬀers from Schwarz’s in that we ask for a coarse equivalence that is also a coarse homomorphism. Both questions are open.

10 On cancellation metrics

Cancellation metrics are rather fundamental objects which play a role in various diﬀerent subjects. Many special examples of cancellation metrics have been studied, generally under diﬀerent names. We collect below some of the instances that we are aware of, but we are sure that the list is far from being exhaustive.

1. If G is a perfect group (i.e. G = [G, G]) then the commutator length is the cancellation length associated with the set of commutators. By now, the commutator length is a rather classical object of study (especially in its “stabilized” form) which has beautiful connections with bounded cohomology, R-quasimorphisms and low-dimensional topology. We refer to [Cal09] for an introduction to the subject. If G is not perfect, the commutator length still is a cancellation metric on the commutator subgroup G0 B [G, G]. It is interesting to note that if G0 is unbounded with respect to this metric then it is also possible to deﬁne an unbounded bi-invariant metric on G [BIP08, Proposition 1.4].

2. If G is generated by torsion elements, one can consider the cancellation metric associated with the set of all the torsion elements. This is sometimes called torsion length ([Kot04]).

3. Let G = π1(Σ) is the fundamental group of a closed surface of genus at least 2. The set S ⊂ G of loops homotopic to simple closed curves is invariant under conjugation and generates G. Answering a question of Farb, Calegari showed in [Cal08] that the cancellation metric associated with S is unbounded on G (see also [BRH12]).

4. The reflection length on a Coxeter group W is defined as the cancellation length associated with the standard generating set [MP11]. It is known that the reflection length on W is bounded if and only if W is spherical or of affine type [Dus12]. See also [DP21, LMPS19] and references therein for some recent works on reflection lengths.

5. Let w be a word in some letters x1,..., xn. We can see w as a functions with variables x1,..., xn and, for any given a group G, we can then consider the set S of elements in G that can be obtained by replacing each xi with some gi ∈ G. The cancellation length associated with this set is also called w-length on G (technically, the w-length is only a length function if S generates G). Sometimes, −1 −1 this is also called verbal length. For example, when w = x1 x2 x1 x2 the w-length coincides with the commutator length. An initial study of some (stable) w-lengths is carried out in [CZ11]. See also [BBF19] and references therein for some recent results.

97 6. Let FS be the free group generated by S . Jiang deﬁned the width of a word w ∈ FS to be its cancellation length relative to the set S 0 B {sk | s ∈ S, k ∈ Z}. This quantity is relevant to count the number of ﬁxed points of a self-homeomorphism of a surface [Jia89] (see also [GK91]).

7. Cancellation metrics on a free group FS can be seen as combinatorial areas. Namely, a subset R ⊆ FS normally generates FS if and only if hS | Ri is a presentation of the trivial group. The cancellation length of an element w ∈ FS is then equal to the minimal number of conjugates of relations in R that is necessary to multiply in order to show the word w represents the unit element in hS | Ri. In other words, |w|× equals the minimal area of a van Kampen diagram of w. This is the point of view taken by Riley in [Ril17].

8. Let M be a smooth manifold and G B Diff0(M) be the group of diffeomorphisms isotopic to the identity. The Fragmentation Lemma states that G is generated by the subset of diffeomorphisms that are supported on embedded open balls [Ban13]. The associated cancellation length is known as the fragmentation norm on G. It is an interesting problem to understand which manifolds M give rise to diffeomorphism groups with unbounded fragmentation norm [BK18, BIP08, Tsu08].

9. Let G B Diff0(Σ, Area) be the group of area preserving diffeomorphisms of a closed surface that are isotopic to the identity. This G is generated by a special class of “autonomous” diffeomorphisms. The autonomous norm cancellation length associated with the set of autonomous diffeomorphisms. It can be shown that the autonomous norm is unbounded on G [BK13, Bra15, GG04].

10.1 Computing the cancellation metric on free groups In this subsection we will discuss in greater detail the cancellation metric on a free group. This is largely done in order to introduce some notation that will be used in Subsection 11.2. However, it is worth remarking that this metric is of special interest in surprisingly diverse topics ranging from biology to engeneering: it appears in the study of RNA-folding [NJ80] and the design of liquid crystals [MRZ10]. See also [Ril17] for a short survey of these application. −1 Let S be a finite set and w a word with letters in S ∪ S . The cancellation length |w|× is defined as the number of letters that is necessary to cancel from w in order to obtain a word whose reduced representative is the empty word. We already remarked (Example 4.2.12) that the cancellation length of an element g in an arbitrary group G = hhS ii is equal to the minimum of the cancellation lengths of the words w representing it. A priori, different words may reqire a different number of cancellations and one thus needs to take the minimum number over all the words representing g. The first observation to make −1 when specializing to the free group G = FS is that if w is any word in the alphabet S t S representing an element g ∈ FS , then |g|× = |w|×. That is, for the free group FS the cancellation length of an element is equal to the cancellation length of any word representing it.13 Remark 10.1.1. A similar phenomenon is also true for right angled Artin and Coxeter groups, and, more generally, for all groups with a “balanced” presentation [BGKM16, Theorem 1.C]. We set up some notation:

Notation. We will work with (possibly non reduced) words with letters in the alphabet S t S −1. We will denote red(w) the reduced word obtained by reducing the word w. We will not use notation [w] to diﬀerentiate between words and elements of FS . However, we will write v ≡ w to clarify that v and w are equal as words, while v = w means that they are equal as elements of FS (i.e. v = w if and only if

13This is not always the case. For one, this property depends on the choice of generating sets: if we take S 0 = {a, b, a2} as a 2 generating set for F2 then the two words aa and a have diﬀerent cancellation lengths (we are seeing the latter as a 1-letter word in S 0). For a more interesting example, notice that for the Baumslag–Solitar groups BS(2, 5) = ha, b | ba2b−1 = a5i the words ba2b−1 and a5 represent the same element but have diﬀerent cancellation length [BGKM16, Example 2.G].

98 red(v) ≡ red(w)). We may sometimes stress the diﬀerence between ≡ and = by saying that the latter holds in FS . The concatenation of two words is denoted vw. By v−1 we mean the word obtained by reversing the order of the letters v and inverting each letter. With our conventions, vv−1 = ∅ but vv−1 . ∅. A word w subword of v if v ≡ awb. It is a proper subword if at least one of a and b is non-empty (w itself may be empty). If a ≡ ∅, w is a starting subword. It is an ending subword if b ≡ ∅.

−1 By deﬁnition, the distance between two words w1, w2 is d×(w1, w2) = |w2 w1|×. However, it is convenient to have a more explicit characterization. Given a letter x in a word w, we denote by w r x the word obtained by omitting x. We say that a cancellation move M between reduced words is a −1 −1 transformation sending a reduced word v ≡ w1uxu w2 to w1w2, where x ∈ S t S . That is, M takes v to red(v r x). An addition move is the inverse of a cancellation move, i.e. it adds a conjugate of a letter in such a way that the end result is a reduced word. In either case, we say that the subwords w1 and w2 are preserved by the move.

Lemma 10.1.2. Given reduced words w, w0, let d˜(w, w0) B min{n | can take w to w0 with n moves}. Then d˜ = d×.

Proof. Both d˜ and d× are path metrics on the set of reduced words, so it is enough to verify that 0 0 0 0 −1 d˜(w, w ) = 1 if and only if d×(w, w ) = 1. Note that d×(w, w ) = 1 if and only if w = wuxu for some x ∈ S ∪ S −1. Up to reordering, if d˜(w, w0) = 1 we can assume that w0 is reached with a single addition move: 0 −1 0 w ≡ w1w2 and w ≡ w1uxu w2. Then we see that d×(w, w ) = 1 because

0 −1 −1 −1 −1 −1 w = ww2 uxu w2 = w(w2 u)x(w2 u) .

0 0 −1 −1 Conversely, if d×(w, w ) = 1 then w ≡ red(wvxv ). We can assume that the word vxv is reduced (otherwise it can be written as a shorter conjugate of x). If wvxv−1 is reduced there is nothing to prove. If it is not reduced, let h be the largest starting subword of vxv−1 that is cancelled when reducing wvxv−1. −1 We thus have w ≡ w1h . 0 −1 −1 There are now two cases. If h is a subword of v then v ≡ hv1 and w ≡ w1v1 xv1 h is reduced. Thus 0 −1 −1 w is obtained by adding v1 xv1 into w1h = w. If h is not a subword of v, then we decompose v ≡ v1v2 −1 0 −1 −1 −1 −1 in such a way that h ≡ v1v2 xv2 . We then see that w ≡ w1v1 is obtained from w = w1v2 x v2 v1 by cancelling the letter x−1 and reducing the word.

Remark 10.1.3. Since we are not differentiating between words and group elements, the notation d×(v, w) is also defined for non-reduced words and defines a pseudo-distance. We could have also extended the definition of d˜ by declaring that d˜(v, w) = 0 if red(v) = red(w). Of course, this d˜ is still equal to d×. −1 0 Remark 10.1.4. If w ≡ w1w2 and uxu are reduced and w is the—possibly non reduced—word −1 −1 w1uxu w2, then there can be some cancellation between w1 and u or between u and w2, but not both. −1 Assume that the cancellation happens between w1 and u, and let h be the initial substring of uxu w2 that is cancelled when doing the reduction. If h is contained in u, we then see as in the proof of Lemma 10.1.2 that w can be taken to red(w0) with an addition move. Otherwise, red(w0) is obtained from w via a cancellation move.

0 Lemma 10.1.5. The distance d×(w, w ) can always be realised by a sequence of moves M1,..., Mn such that all the cancellations are performed ﬁrst.

0 −1 00 0 Proof. It is enough to show that if w ≡ w1w2, w ≡ w1uxu w2 and w is obtained from w with one cancellation move M, then it is possible to take w to w00 by using two addition moves or one cancellation move followed by any other move. We can assume that the removed letter y comes after the letter x in w0.

99 If the subword uxu−1 is preserved by M, then M commutes with the addition move and there is nothing to prove. Assume ﬁrst that the letter y belongs to w2, i.e. w2 ≡ w3yw4. Cancelling the letter y, takes w ≡ 00 −1 −1 −1 −1 −1 w1w3yw4 to red(w1w3w4). Note that w = w1w3w4(w3 w4 u)x(w3 w4 u) is in FS . By Lemma 10.1.2, 00 we can take red(w1w3w4) to w in one move. −1 −1 The case where the letter y belongs to u is slightly more involved. We can write u = u1y u2, so that w00 is the reduction of −1 −1 −1 w1u1y u2 xu2 u1 w2. 000 −1 −1 With one move we can take the word w to the word w ≡ red(w1u1u2 xu2 u1 w2). This move may be either an addition or a cancellation (see Remark 10.1.4). The word w00 is the reduced representative of

−1 −1 −1 −1 −1 (w1u1y u1 w1 )(w1u1u2 xu2 u1 w2).

We can hence take w000 to w00 in one move, but we need to pay some extra care to see whether it is an −1 −1 −1 addition or a cancellation move. The above expression will ﬁrst reduce to w1u1y u2 xu2 u1 w2 and, by −1 −1 construction, any further reduction will originate from the product u2 u1 . By Remark 10.1.4, we know that the move taking w000 to w00 is an addition move unless the reduction −1 −1 −1 process starting at u2 u1 goes all the way past the letter y . However, when this is the case the reduction will also have gone past x.

We obtain as a corollary a formal proof of the fact that the coarse group (FS , Ed× ) is not coarsely abelian:

−1 Corollary 10.1.6. For every n < m ∈ N and x1 , x2 ∈ S t S , the cancellation norm of the commutator n m E [x1, x2 ] equals 2n. In particular, the coarse group (FS , d× ) is not coarsely abelian. Remark 10.1.7. The reader should be warned that the cancellation distance can be quite deceptive. For 2 example, it is generally not true that |w |× = 2|w|×. In fact, one can see that the word w = abAAB has cancellation norm 3 while |abAABabAAB|× = 4. However, there exists an easy-to-implement polynomial time algorithm that computes |w|× (see below). This can be very helpful to test conjectures involving the cancellation metric. Remark 10.1.8. A cubic time algorithm for computing the cancellation length of words was described in [NJ80] and was rediscovered in [BGKM16]. Faster algorithm are also known: an algorithm with less than cubic running time is described in [BGSW16]. See also the discussion in [Ril17].

10.2 Cancellation metrics on finitely generated set-groups and their subgroups The starting point of Geometric Group Theory is the observation that every finitely generated set-group can be given a left-invariant word metric and that this choice does not depend on the generating set up to coarse equivalence.14 We already remarked that the same is true for cancellation metrics: choosing a different generating set will give rise to a coarsely equivalent (in fact, bi-Lipschitz equivalent) cancellation metric. It is hence natural to indulge in a study of the coarse geometry of cancellation metrics of finitely generated set-groups. Given the many similarities between word metrics and cancellation metrics one may expect that it should be fairly simple to adapt techniques from Geometric Group Theory to the study of cancellation metrics. However, this is generally not the case: cancellation metrics tend to behave very differently and appear to be somewhat more complicated to control.

14One generally says that the metrics are quasi-isometric. However, for geodesic metric spaces coarse equivalences and quasi-isometries coincide.

100 One instance where the difference between word metrics and cancellation metrics becomes apparent is in the study of finite index subgroups. It is a basic fact that a finitely generated group equipped with its word metric is quasi-isometric to any of its finite index subgroups. This fact fails dramatically for the cancellation metric: to see this it is enough to recall that the infinite dihedral group D∞ is intrinsically bounded even though it has Z as an index-2 subgroup and (Z, Ed× ) = (Z, E|-|) is unbounded. Remark 10.2.1. On the other hand, it is still true that if N C G is a finite normal subgroup and Q = G/N is G Q the quotient then (G, E G ) and (Q, E Q ) are isomorphic coarse groups, where d and d are the relevant d× d× × × cancellation metrics. In fact, if S is a finite generating set of Q and q: G → Q is the quotient map, it is immediate to verify that the cancellation metric on G associated with the finite set q−1(S ) generates the ∗ pull-back coarse structure q (E Q ). d× One may hope that the example Z < D∞ is some sort of pathological case, maybe dependent on the fact that D∞ is not torsion free. However, the fact that the inclusion of finite a index subgroup is not a coarse equivalence seems to be more of a rule than an exception. One interesting example is the following:

Example 10.2.2. Let F2 B ha, bi be the 2-generated free group and let H C F2 be the index-2 subgroup consisting of all the elements represented by words of even length. The group H is a rank-3 free group 2 F2 H generated by the elements x = a , y = ab and z = ba. Let d× (resp. d× ) be the cancellation metric on F2 (resp. H) associated with the generating set {a, b} (resp. {x, y, z}). We claim that the inclusion

ι:(H, EdH ) ,→ (F2, E F2 ) is not an isomorphism of coarse groups. × d× It is immediate to verify that ι is a controlled map. Since H is a ﬁnite index subgroup and the coarse structure E F2 is connected, it is also clear that ι is coarsely surjective. However, ι is not a proper map. In d× other words, we claim that there is a strict containment EdH ⊂ (E F2 )|H. To see this, note that the set × d× B B {a(ab)na(ab)−n | n ∈ N} ⊂ H

F2 is a d× -bounded neighbourhood of the identity. On the other hand, writing the set B in terms of the generators x, y, z we see that B = {xzny−n | n ∈ N} and, using Lemma 10.1.5, it is easy to show that B is H not d× -bounded. This example can also be used to show the somewhat surprising fact that a self-commensuration of a coarsified set-group need not give rise to a coarse group automorphism. More precisely, it follows from the properties of the cancellation metric that whenever φ: H → H is an isomorphism of set-groups then [φ]: (H, E H ) → (H, E H ) is an isomorphism of coarse groups. Since H is coarsely dense in F , one d× d× 2 may expect that it should be possible to extend φ to a coarse homomorphism φˆ : (F2, E F2 ) → (F2, E F2 ). d× d× However, the fact that EdH , (E F2 )|H warns us that this is only a naïve hope. In fact, consider the × d× −1 F2 set-group isomorphism φ: H → H fixing x, z and sending y to y . The d× -bounded set B defined before is then sent to φ(B) = {xznyn | n ∈ N} = {a(ab)na(ab)n | n ∈ N}, which is clearly unbounded in

(F2, E F2 ). This shows that ι ◦ φ: (H, EdH ) → (F2, E F2 ) is not controlled and hence cannot be extended d× × d× to an isomorphism of the coarse group (F2, E F2 ). d× When dealing with normally finitely generated groups, it would be interesting to understand under what circumstances is the cancellation metric of a subgroup H < G compatible with the restriction of the coarse structure of G. As a preliminary step, it would be interesting to investigate the this for finitely generated normal subgroups: Question 10.2.3. Let G be a finitely generated set-group, N C G a finitely generated normal subgroup G N and d , d the associated cancellation metrics. Under what conditions is the containment E N ⊆ (E G )| × × d× d× N an equality?

In Example 10.2.2 we showed that the inclusion F3 ,→ F2 as the index-2 subgroup of words of even length is not an isomorphism of coarse groups. However, this does not rule out the possibility that

101 (F2, Ed× ) and (F3, Ed× ) be isomorphic coarse groups via some other map. More generally, we do not know the answer to the following:

Question 10.2.4. Let Fn and Fm be free groups with n, m ≥ 2. Is it true that if (Fn, Ed× ) and (Fm, Ed× ) are isomorphic coarse groups then n = m?

Remark 10.2.5. We now know that commensurable set-groups need not be isomorphic coarse groups when equipped with their cancellation metrics. A fortiori, we would then expect that the knowledge that two groups G1, G2 are quasi-isometric when equipped with their word lengths should give us remarkably little information over the coarse groups (G1, d×) and (G2, d×). However, there are instances where knowledge about the quasi-isometry type of the word length of a set-group give non-trivial information about the cancellation metric. One such example is given by hyperbolic groups: the property of being non-elementary hyperbolic is invariant under quasi-isometry and Corollary 8.1.6 shows that any such group has an unbounded cancellation metric. In hindsight, we ﬁnd that the above fact is very surprising. It would be very interesting to understand more precisely what are the circumstances that allow one to say something about the cancellation metric by looking at the quasi-isometry type of the word metric or vice versa.

11 A quest for coarse groups that are not coarsiﬁed set-groups

Many of the explicit examples of coarse groups that we have given so far were obtained by equipping a set-group with an equi-bi-invariant coarse structure. It is natural to ask whether every coarse group is a coarsiﬁed set-group. As stated, the answer to this question is clearly negative. For example, we already remarked that it is easy to construct a coarse group so that the operations do not even admit cancellative representatives (Example 4.4.4). However, that example is isomorphic (as a coarse group) to a coarsiﬁed set-group. The correct question to ask is:

Question 11.0.1. Is every coarse group isomorphic to a coarsiﬁed set-group?

We conjecture that the answer is negative (see Conjecture 11.2.1). In this section we will develop some tools that can be used to understand when a coarse group is isomorphic to a coarsiﬁcation of a set-group and provide some evidence towards our conjecture. Remark 11.0.2. Given a coarse group G, we believe that it is generally important to know whether G is isomorphic to a coarsiﬁcation of a set-group: when this is not the case we see that the operation is in some way “intrinsically not associative”. Such a feature should surely have important consequences for the properties of G and its coarse actions.

11.1 General observations In this subsection we provide a general criterion characterizing which coarse groups are isomorphic to coarsiﬁcations of set-groups. Given any set S , we denote by FS the free set-group generated by S . −1 Elements of FS can be identiﬁed with reduced words in the alphabet S ∪ S . For any x ∈ S we denote by x ∈ FS the one-letter word x. The other elements of FS are generally denoted by w.

−1 Proposition 11.1.1. Let G B (G, E, [∗G], [e], [-] ) be a coarse group. Then G is isomorphic to a coarsiﬁcation of a set-group if and only if there exists a coarse homomorphism [Ψ]: (FG, Emin) → (G, E) with a representative Ψ ∈ [Ψ] such that Ψ(g) = g for every g ∈ G.

Proof. One implication is clear: if there exists such a Ψ then the pull-back Ψ∗(E) is equi-bi-invariant on ∗ FG by Lemma 5.1.8 and G is isomorphic to (FG, Ψ (E)).

102 For the converse implication, let (H, ∗H) be a set group with an equi-bi-invariant coarse structure F such that there exists a isomorphism [ f ]: (G, E) → (H, F ). Fix representatives f ∈ [ f ] and f¯ ∈ [ f ]−1. Since H is a set-group, there exists a unique set-group homomorphism Φ: FG → H sending g ∈ G to f (g). A fortiori, such Φ deﬁnes a coarse-group homomorphism Φ: (FG, Emin) → (H, F ). The composition [ f ]−1 ◦ [Φ] = [ f¯ ◦ Φ] is therefore a coarse homomorphism. Deﬁne Ψ by ( g if w = g for some g ∈ G, Ψ(w) B ( f¯ ◦ Φ)(w) otherwise.

By construction, Ψ differs from f¯ ◦ Φ only on the 1-letter words in FG. However, since Φ(g) = f (g) and f¯ ◦ f is close to idG, we see that Ψ is still close to f¯ ◦ Φ. In particular, [Ψ] is a coarse homomorphism and it has the required properties. The interesting part of the above criterion is that it can make a rather abstract problem into a very elementary one. Here “elementary” means “easy to state”, but alas not easy to solve! In particular, Proposition 11.1.1 implies the following. −1 Corollary 11.1.2. Let G B (G, Ed, [∗], [e], [-] ) be a coarse group where d is a metric on G. Then G is isomorphic to a coarsification of a set-group if and only if there exists a constant D > 0 and a function Ψ: FG → G such that • Ψ(g) = g for every g ∈ G, • d Ψ(w1w2) , Ψ(w1) ∗ Ψ(w2) ≤ D for every w1, w2 ∈ FG. We will later use this criterion to give evidence to support our conjecture that some coarse subgroups of the coarse group (F2, Ed× ) are not isomorphic to coarsifications of set-groups.

One can apply Proposition 11.1.1 to prove that some coarse group is isomorphic to a coarsified set-group if one can find “special” representatives for the multiplication function: Definition 11.1.3. Let S = (S, E) be a coarse space. A controlled function ∗: (S, E) × (S, E) → (S, E) is coarsely multi-associative if there exists a bounded set B ⊆ S such that

{x1 ∗ · · · ∗ xn | associated in any order} ⊆ B for every n ∈ N and every choice of xi ∈ S . Corollary 11.1.4. If G B (G, E, [∗], [e], [-]−1) is a coarse group and ∗ is coarsely multi-associative then G is isomorphic to a coarsiﬁed set-group.

Proof. Deﬁne the map Ψ: FG → G by sending a reduced word w = g1 ··· gn to g1 ∗ · · · ∗ gn (with an arbitrarily chosen order of association).

It is easy to produce examples G B (G, E, [∗]) where we can choose a representative ∗ that is not coarsely multi-associative and yet G is isomorphic to the coarsification of a coarse group. For example, equip (G, E) B (Z, Ed) × (Z/Z2, Emin) with the following operation: (a, 0) ∗ (b, 0) B (a + b, 0) (a, 0) ∗ (b, 1) B (a + b + 1, 1) (a, 1) ∗ (b, 1) B (a + b, 0) (a, 1) ∗ (b, 0) B (a + b + 1, 1). Obviously, ∗ is close to the natural group operation +, therefore (G, E, [∗]) = (G, E, [+]) is a coarsified set-group. However by multiplying (1, 1) with n copies of (1, 0) one immediately sees that ∗ is not coarsely multi-associative. On the other hand, we are not sure whether the converse of Corollary 11.1.4 holds true. That is, it is unclear whether the existence of a coarsely multi-associative representative for [∗] is a necessary requirement for (G, E, [∗]) to be isomorphic to a coarsified set-group.

103 11.2 A conjecture on coarse groups that are not coarsiﬁed set groups

Recall that an R-quasimorphism of a set-group G is a function φ: G → R such that |φ(g1g2) − φ(g1) − k φ(g2)| is uniformly bounded, and that φ is homogeneous if φ(g ) = kφ(g) for every k ∈ Z (every R-quasimorphism is close to a unique homogeneous one [Cal09, Section 2.2]). Let FS be the free group freely generated by the set S and let d× be the associated cancellation metric 10.1). It is then easy to see that [φ]: (FS , Ed× ) → (R, Ed) is a coarse homomorphism if and only if φ is a controlled R-quasimorphism. It is also easy to check that a homogeneous R-quasimorphism is controlled if and only if there is some D ≥ 0 such that |φ(x)| ≤ D for every x ∈ S . In particular, if S is finite then every R-quasimorphism is controlled. Further recall (Definition 7.1.3) that the coarse kernel of a coarse homomorphism f : G → H is −1 defined as f (eH)—if it exists. We already showed (Corollary 7.3.5) that controlled R-quasimorphisms always have a coarse kernel. In particular, this applies to controlled homomorphisms FS → R. For concretness, for the rest of the section we will work with the rank-2 free group F2 generated by the set S = {a, b}. The goal of this subsection is to justify the following.

Conjecture 11.2.1. The coarse kernel of a homomorphism φ: (F2, Ed× ) → (R, Ed) is isomorphic to a coarsiﬁcation of a set-group if and only if its image is discrete.

One implication is easy: assume that f has discrete image, Im( f ) = {ka | k ∈ Z}. It follows from Corollary 7.3.6 that Ker([ f ]) = [ f −1((−a, a))] = [ker( f )]. In particular, Ker([ f ]) is isomorphic to

(ker( f ), Ed× |ker( f )), which is a coarsiﬁed set-group. Let φ be a homomorphism with dense image. Up to rescaling or replacing a, b with a−1, b−1, we may −1 assume that φ sends a to −1 and b to β ∈ R>1 r Q. Let K B φ ([0, 1)), then [K] = Ker([φ]). Spelling out the deﬁnition of coarse subgroup, the set K can be equipped with operations ∗, [-]−1 that make

(K, Ed× ) into a coarse group so that the inclusion K ,→ F2 is a coarse homomorphism (here d× = d×|K is the restriction of the metric to K). The conjecture states that K B (K, Ed× ) is not isomorphic to a coarsiﬁcation of a set-group. We rephrase it by using Corollary 11.1.2:

Conjecture 11.2.2. There do not exist a function Ψ: FK → K and a constant D ≥ 0 such that

(1) Ψ(k) = k for every k ∈ K;

(2)d × Ψ(w1w2) , Ψ(w1)Ψ(w2) ≤ D for every w1, w2 ∈ FS .

Remark 11.2.3. Note that the second condition in the above statement only involves the actual set-group multiplications in FK and FS , and the operation ∗ on K does not play any role. This is because we know a priori that ∗ is close to the usual multiplication in FS and we can hence use the latter (possibly at the cost of choosing a larger constant D). We now use the remainder of this section to outline a strategy of proof (for which we are indebted to Nicolaus Heuer). The overall strategy is to obtain a contradiction by showing a certain word is “almost periodic”. We were unfortunatly unable to carry out this plan because we lack appropriate tools to compute the cancellation distance d× on F2. For every n ∈ N let

bnβc n un B a b and note that un belongs to K (recall that bnβc denotes the ﬂoor). To prove Conjecture 11.2.2 it would be enough to prove the following statement and show that it leads to a contradiction.

104 If Ψ and D satisfy conditions (1) and (2), then there exist some constants C D and n ∈ N large enough so that N N d× Ψ(un ) , un ≤ C (♣) N for every N ∈ N (in the above inequality, the un that appears on the left denotes the word consisting N of N repetitions of the letter un ∈ K. The un that appears on the right is the element of F2 obtained by multiplying N copies of un).

To see why (♣) leads to a contradiction, it is enough to note that for every g1, g2 ∈ F2 the diﬀerence |φ(g1) − φ(g2)| is bounded by βd×(g1, g2). Therefore, (♣) implies that

N N |φ Ψ(un ) − φ(un )| ≤ βC

N for every N ∈ N. Since φ(un) = Nφ(un) goes to inﬁnity with N, we deduce that for N large enough N Ψ(un ) does not belong to K, against the hypotheses. N We now wish to explain why we think that (♣) is a plausible claim. By hypothesis, both Ψ(un)Ψ(un ) N N+1 and Ψ(un )Ψ(un) are at most at distance D from Ψ(un ). Since Ψ(un) = un, this means that the word N Vn,N B Ψ(un ) ∈ F2 almost commutes with un: d× unVn,N , Vn,Nun ≤ 2D.

N This poses strong restrictions on the possible choices for Vn,N. If D = 0 it is clear that Vn,N ≡ un . Intuitively, it seems very plausible that—so long as D n—something similar should hold also for D > 0. As a warning, we point out that it is deﬁnitely not the case that a word commuting with un must be close k to being a power of un. For example, letting W B (aun) for any k ∈ N, we see that d× unW, Wun = 2 k even though d×(un, W) = k is unbounded. However the following might be true: So long as n D, there exists a constant C0 depending only on D so that every reduced word W ♠ ≤ k1 ··· kC0 ∈ ≤ 0 ( ) satisfying d× unW, Wun D is of the form: W = v1 vC0 , where ki Z and d×(un, vi) C for every i = 1,..., C0. Importantly, C0 does not depend on n.

Remark 11.2.4. A more audacious claim could be as follows. Say that a reduced word u has D-stable almost centralizer if (♠) holds true (namely, there exists a constant C0 such that every word satisfying 0 0 d×(uW, Wu) ≤ D can be decomposed as a product of at most C powers of words that are C -close to u). Further, say that a word u is R-cyclically reduced if it is cyclically reduced and every word obtained bnβc n by cancelling up to R letters from u is still cyclically reduced. Note that the words un = a b are R-cyclically reduced for every R. It might be the case that for every D there is a R D such that every R-cyclically reduced word has D-stable almost centralizer. Claim (♠) does not directly imply (♣). However, it might be possible to use some trick to prove ♣ ∈ m km,1 ··· km,C0 0 ( ). Specifically, fix some N n. For every m N, we know that Ψ(un ) = vm,1 vm,C0 . Since C is m fixed while |ψ(un )|× goes to infinity with m, we deduce that for some M N there must exist an index 0 1 ≤ i ≤ C so that kM,i N. Let M = M1 + N + M2 for some M1, M2 ≥ 0. Then

k k 0 M1 N M2 ≈ M M,1 ··· M,C Ψ(un )Ψ(un )Ψ(un ) 2D Ψ(un ) = vM,1 vM,C0 .

The idea would then be to find appropriate M1 and M2 so that the most efficient way to take the word k 0 M1 N M2 kM,1 ··· M,C N kM,i Ψ(un )Ψ(un )Ψ(un ) to vM,1 vM,C0 takes Ψ(un ) to a subword of vM,i . At this point it should not be hard to prove (♣) or some modification of it which is still sufficient to prove Conjecture 11.2.2. The problem with the last part of the plan is that it is difficult to control the amount of cancellation N M1 M2 that can happen when composing Ψ(un ) with Ψ(un ) and Ψ(un ). More generally, we find that we

105 currently lack some technical tools allowing us to study the cancellation distance. This seems to be a rather interesting combinatorial problem.

Remark 11.2.5. If Conjecture 11.2.1 holds true, it most likely holds also when F2 is replaced with any free group FS (equipped with the relevant cancellation norm) and φ: FS → X is a homomorphism into any Banach space X.

11.3 A few more questions As already remarked, we believe that it is an important problem to understand precisely which coarse groups are isomorphic to coarsified set-groups. To be a little more specific, let G be a fixed coarse group (possibly a coarsified set-group). A general question that would be very interesting to answer is the following:

Question 11.3.1. When is the coarse kernel of a coarse homomorphism f : G → H isomorphic to the coarsiﬁcation of a set-group?

Any kind of “if and only if” characterization would most likely be useful for any attempt of classiﬁ- cation of coarse groups. It would already be interesting to answer the above question for some special classes of coarse homomorphism. For example:

Question 11.3.2. Let φ: F2 → R be an R-quasimorphism. When is Ker([φ]) < (F2, Ed× ) isomorphic to the coarsiﬁcation of a coarse group?

Note that Conjecture 11.2.1 is our attempt to solve Question 11.3.2 for R-quasimorphisms that are close to set-group homomorphisms. Another case of special interest is the following:

Question 11.3.3. Let φ: F2 → R be a Brooks quasimorphism. Is Ker([φ]) isomorphic to the coarsiﬁca- tion of a coarse group?

One reason why the above question is interesting is that the homogenization of a Brooks quasimorphism has discrete image in R (hence Ker([φ]) = [φ˜−1(0)], where φ˜ is the homogenization of φ). If it turns out that these kernels are not isomorphic to the coarsiﬁcation of a coarse group it would mean that the natural extension of Conjecture 11.2.1 to (homogeneous) R-quasimorphisms is not true.

We would also like to know the answer to the following:

Question 11.3.4. Is every coarsely abelian coarse group isomorphic to a coarsiﬁcation of a set-group?

12 On coarse homomorphisms and coarse automorphisms

In this section we study the group of coarse automorphisms of (coarsely connected) coarse groups. If G = (G, E) is a coarse group, we deﬁne AutCrs(G) = AutCrs(G, E) as the group of self-isomorphisms of G. That is, AutCrs(G) < MorCrs(G, G) is the subset of coarse-homomorphisms that have a coarse inverse.

12.1 Elementary constructions of coarse automorphisms The most elementary examples of coarse automorphisms for a coarse group G = (G, E) are the conju- −1 gations gc(h) B (g ∗ h) ∗ g . However, we already noted that, whenever g lies in the same coarsely connected component of e, the map gc is close to the identity (Example 5.1.5). In particular, in the case of special interest where G is coarsely connected, the conjugation induces a trivial map G → AutCrs(G) sending every g ∈ G to idG.

106 Remark 12.1.1. We emphasize that [gc] = idG for each g ∈ G does not imply that the coarse action by conjugation G y G is trivial (see Subsection 6.2). In fact, for a given coarse space X the set-group AutCrs(X) is very ill-suited to describe coarse actions G y X. In a sense, this is because AutCrs(X) is the set-group of “coarsely connected components of the space of transformations of X”. From this point of view, it is clear that AutCrs(X) cannot detect any coarse action of a coarsely connected coarse group G. This idea is made precise and further developed in Subsections 14.3 and 14.4, where we construct a suitable “space of transformations” of a coarse space. If G = (G, E) where G is a set-group, every controlled homomorphism f : G → G determines a coarse homomorphism [ f ]: G → G. In particular, if an automorphism f ∈ Aut(G) such that both f and −1 f are controlled then [ f ] ∈ AutCrs(G). Notice that set-group automorphisms need not be controlled in 2 2 general. For example, the linear function f : ` (N) → ` (N) sending the sequence (an)n∈N to (nan)n∈N is a set-group automorphism of `2(N), but it is not controlled with respect to the metric coarse structure. In some cases one can say a priori that every set-group automorphism is automatically controlled. When this happens, we obtain a natural homomorphism Aut(G) → AutCrs(G).When G is coarsely connected, conjugations are close to the identity. Thus, it follows that the above homomorphisms quotient through the space of outer automorphisms Out(G) → AutCrs(G). Example 12.1.2. One instance where set-group homomorphisms are automatically controlled is the Egrp Egrp case G = (G, fin ), where fin is the minimal equi-bi-invariant coarse structure whose bounded sets ∗ Egrp → are finite. To see this, recall that the pull-back f ( fin ) for a set-group homomorphism f : G G is Egrp ⊆ ∗ Egrp equi-bi-invariant (Lemma 5.1.8). Since f sends finite sets to finite sets, we see that fin f ( fin ) by minimality. An example of special interest is the set-group Zn. Since the group Zn is abelian and finitely generated, Egrp E the coarse structure fin is equal to the coarse structure defined by the Euclidean metric k-k. We thus n n n n obtain a set-group homomorphism Out(Z ) → AutCrs(Z , Ed). Moreover, Z is coarsely dense in (R , Ek-k) n n n n and therefore the inclusion Z ,→ R gives a natural isomorphism of coarse groups (Z , Ek-k) (R , Ek-k). Noting that the set-group of outer automorphisms of Zn is Out(Zn) = Aut(Zn) GL(n, Z), we obtain a homomorphism n n GL(n, Z) → AutCrs(Z , Ek-k) = AutCrs(R , Ek-k). We will later see that the right hand side is in fact isomorphic to GL(n, R) and the above map is nothing but the inclusion GL(n, Z) < GL(n, R). Egrp Recall that for normally finitely generated set-groups G the coarse structure fin coincides with the coarse structure defined by the cancellation metric d× (Example 4.5.4). Therefore, in this case we always have a homomorphism Out(G) ,→ AutCrs(G, Ed× ).

Question 12.1.3. Let G be a (normally) ﬁnitely generated group. When is the homomorphism Out(G) ,→

AutCrs(G, Ed× ) injective?

We will later see that this homomorphism is injective for the free group (Corollary 12.4.7). On the other hand, if K is a ﬁnite set-group with non-trivial Out(K) then Out(K) → AutCrs(K, Ed× ) = {id} is not injective. Similarly, if G = H × K for some K as above then Out(K) ≤ Out(G) while (G, Ed× ) (H, Ed× ) and again we see that Out(G) → AutCrs(G, Ed× ) is not injective. This suggests that Question 12.1.3 should be particularly interesting for torsion-free ﬁnitely generated set-groups.

Other examples of set-groups G so that Out(G) does not embed into AutCrs(G, Ed× ) can be obtained by looking at intrinsically bounded set-groups (Subsection 8.3). Remark 12.1.4. In geometric group theory there are various results that are proved for (torsion free) finitely generated groups G with finite or trivial Out(G) (e.g. [Hua18]). It would not be surprising if some of those arguments can be rephrased by saying that some finiteness condition on AutCrs(G, Ed× ) implies some degree of rigidity for G.

107 12.2 Coarse homomorphisms into Banach spaces

Let (V1, k-k1) and (V2, k-k2) be normed vector spaces (we will only deal with vector spaces over R).

We can then see their additive groups as the coarse groups (V1, Ek-k1 ) and (V2, Ek-k2 ). It follows by the deﬁnition that a linear operator T : (V1, k-k1) → (V2, k-k2) is controlled if and only if it is bounded (i.e. continuous). We therefore have a natural homomorphism B (V1, k-k1), (V2, k-k2) → MorCrs (V1, Ek-k1 ), (V2, Ek-k2 ) , where B is the space of bounded linear operators. Note that the above homomorphism is always injective. Furthermore, if (V2, k-k2) is complete (i.e. a Banach space) then it is also surjective:

Proposition 12.2.1. Let (V1, k-k1) be a normed vector space and (V2, k-k2) a Banach space. If [ f ]: (V1, Ek-k1 ) → ¯ (V2, Ek-k2 ) is a coarse homomorphism of additive groups, then there exists a representative f ∈ [ f ] that is linear and continuous.

Proof. Define f¯ by f (nv) f¯(v) B lim n→∞ n for every v ∈ V1. We know from Lemma 5.1.15 and Remark 5.1.16 that f¯ is well-defined, it is close to f 1 1 and satisfies f (kv) = k f (v) for every k ∈ Z. This also implies that f ( k v) = k f (v). D ≥ f D k f v v − f v f v k Let 0 be the defect of . That is, = supv,w∈V1 ( 1 + 2) ( 1) + ( 2) 2. Then 1 D k f¯(v + w) − f¯(v) − f¯(w)k2 = lim k f (n(v + w)) − f (nv) − f (nw)k2 ≤ lim = 0. n→∞ n n→∞ n This shows that f¯ is additive (and hence linear over Q). Since f is controlled, there is some R > 0 such that f sends the ball of radius 1 centred at the origin of V1 into the ball of radius R centred at the origin of V2. The same holds for f¯ as well. It follows that ¯ 1 ⊆ R ¯ ¯ f (BV1 (0; k )) BV2 (0; k ) and hence f is continuous at 0. Together with Q-linearity, this implies that f is R-linear and continuous.

Corollary 12.2.2. If (V2, k-k2) is a Banach space then HomCrs (V1, Ek-k1 ), (V2, Ek-k2 ) B (V1, k-k1), (V2, k-k2) .

Corollary 12.2.3. If k-k is a norm on Rn then

n AutCrs(R , Ek-k) GL(n, R).

Using Corollary 12.2.2 and the Open Mapping Theorem, we can characterize those coarse homomorphisms between Banach spaces which admit a coarse kernel (Deﬁnition 7.1.3):

Proposition 12.2.4. Let [T]: (V1, Ek-k1 ) → (V2, Ek-k2 ) be a coarse homomorphism between Banach spaces and let T be its linear representative. Then [T] has a coarse kernel if and only if T(V1) is a closed subspace in V2. When this is the case, Ker([T]) = [ker(T)]. Proof. We start by showing that if the coarse kernel exists then it must be [ker(T)]. This is done as in

Example 5.1.18: let [H] be any coarse subgroup of (V1, Ek-k1 ). The inclusion ι: H ,→ (V1, Ek-k1 ) is a coarse homomorphism. We can hence use Lemma 5.1.15 to construct ¯ι: H → (V1, Ek-k1 ) that is close to ι and such that ¯ι(h ∗ h) = 2¯ι(h). In order for T ◦ ¯ι to have bounded image it is necessary that ¯ι(H) ⊂ ker(T) and hence H 4 ker(T). This proves our claim.

108 It follows by the deﬁnition that [ker(T)] is the coarse kernel of [T] if and only if for every r > 0 there is an R > 0 so that −1 T (BV2 (0; r)) ⊆ BV1 (0; R) ∗ ker(T) . (10)

Assume now that T(V1) is closed in V2. Then T(V1) is a Banach space and we can apply the Open Mapping Theorem to T : V1 → T(V1). It follows that for every r > 0 there is an R > 0 so that −1 −1 BT(V1)(0; r) ⊆ T(BV1 (0; R)). In turn, this implies that T (BV2 (0; r)) = T (BT(V1)(0; r)) ⊆ BV1 (0; R). This proves that [ker(T)] is the coarse kernel of [T]. For the converse implication, (10) implies that T : V1 → T(V1) is an open map and hence descends to a homeomorphism of the quotient T : V1/ ker(T) → T(V1). Since ker(T) is a closed subspace, V1/ ker(T) is a Banach space. Then T(V1) is also complete and therefore closed in V2.

12.3 Hartnick-Schweitzer quasimorphisms Hartnick and Schweitzer [HS16] introduced a notion of “quasimorphism” between set-groups which will allow us to construct a zoo of coarse automorphisms. These quasimorphisms (which we will call HS-quasimorphisms) are deﬁned to be those functions that are well-behaved with respect to composition with R-quasimorphisms. Recall that an R-quasimorphism on a set-group G is a function φ: G → R which has the property that |φ(gh) − φ(g) − φ(h)| is uniformly bounded. We already encountered R-quasimorphisms in Example 5.1.3, as they provide examples of coarse homomorphisms (G, E) → (R, E|-|) (an R-quasimorphism is a coarse homomorphism as soon as it is controlled as a function).

Deﬁnition 12.3.1 ([HS16]). A HS-quasimorphism between set-groups G, H is a function f : G → H such that if φ is a R-quasimorphism on H then f ◦ φ is a R-quasimorphism on G. Two HS-quasimorphisms f, g: G → H are equivalent if |φ ◦ f − φ ◦ g| is a bounded function for every such φ (equivalently, (φ ◦ f ) ≈ (φ ◦ g) as functions into (R, E|-|)).

It is immediate from the definition that compositions of HS-quasimorphisms are HS-quasimorphisms, and that equivalence of HS-quasimorphisms is preserved under composition. Hartnick and Schweitzer designed their definition to be “categorical”. Thus their definition fits very naturally in the theory of coarse groups and coarse homomorphism. More precisely, the link between HS-quasimorphism and coarse homomorphisms is provided by the pull-back coarses structures defined in Example 5.1.10. Namely, any set-group G can be equipped with the coarse structure \ n ∗ o E(G,Emin)→(R,E|-|) B φ (E|-|) φ:(G, Emin) → (R, E|-|) coarse homomorphism \ n ∗ o = φ (E|-|) φ: G → R, R-quasimorphism

(this can be described as the maximal coarse structure for which quasimorphisms are controlled). For brevity, we will denote this coarse structure by EG→R. The following is easy to prove.

Proposition 12.3.2. A function between set-groups f : G → H is a HS-quasimorphism if and only if [ f ]:(G, EG→R) → (H, EH→R) is a coarse homomorphism.

Proof. Let φ: H → R be an R-quasimorphism. Then, by deﬁnition, [φ]: (H, EH→R) is controlled and hence a coarse homomorphism. If [ f ] is a coarse homomorphism then the composition [φ ◦ f ] is a coarse homomorphism and hence an R-quasimorphism. This shows that f is indeed an HS-quasimorphism. Assume now that f is an HS-quasimorphism. Then \ n ∗ o ∗ EG→R ⊆ (φ ◦ f ) (E|-|) φ: G → R, R-quasimorphism = f (EH→R),

109 and hence f : (G, EG→R) → (H, EH→R) is controlled. It remains to check that f (g1) ∗H f (g2) and f (g1 ∗G g2) are uniformly close (with respect to EH→R). For every R-quasimorphism, both φ ◦ f and φ are coarse homomorphisms. Therefore, there are F1, F2 ∈ E|-| such that (φ ◦ f )(g1 ∗G g2)- F1 → (φ ◦ f )(g1) + (φ ◦ f )(g2) - F2 → φ( f (g1) ∗H f (g2))

−1 and hence f (g1 ∗G g2)- (φ×φ) (F1◦F2) → ( f (g1) ∗H f (g2). This shows that

∗ f (g1) ∗H f (g2) , f (g1 ∗G g2) | g1, g2 ∈ G ∈ φ (E|-|) for every such φ.

It is an interesting fact that amenable set-groups tend to manifest some sort of rigidity, which allowed Hartnick and Schweitzer to classify HS-quasimorphisms in this case (see below for more on this). On the contrary, non-abelian free groups seem to be very flexible and admit a huge variety of HS-quasimorphisms. In what follows, let S be a finite set and FS be the free group freely generated by S . Denote by xi the elements of S ∪ S −1. The following examples are taken from [HS16, Section 5]: Example 12.3.3. Fix some k ∈ N and a function r : (S ∪ S −1)k → G. The function r plays the role of a substitution rule and allows us to define a function fr : FS → G that sends a reduced word w = x1 ··· xn ∈ FS to e ∈ G if n < k and to the product

r(x1,..., rk)r(x2,..., rk+1) ··· r(xn−k,..., xn) ∈ G if n ≥ k. That is, fr(w) is computed by substituting each letter xi of the word w with some element of G according to a rule that only looks at the k letters following xi. For any substitution rule r, the function fr : FS → G is an HS-quasimorphism. We call these local HS-quasimorphism.

Example 12.3.4. Two non-empty reduced words w1, w2 ∈ FS are overlapping if they are subwords of one another or if there is a non-empty subword that is initial in one of them and terminal in the other. If w1,..., wk is a set of words that are pairwise non-overlapping and such that each wi is not self-overlapping, then every reduced word wi ∈ FS can be uniquely decomposed as a concatenation w = u1 ··· un such that each ui is equal to one of the w j or it does not have any w j as a subword. −1 −1 Let w1, w2 be two reduced words with the same ﬁrst and last letter and such that {w1, w1 , w2, w2 } is a set of pairwise non-overlapping and non-self-overlapping words. We may then use the unique decomposition property of above to deﬁne a function fw1,w2 : FS → FS that “swaps w1 and w2”. Namely, if w = u1 ··· un is the unique decomposition of a reduced word w, we let fw1,w2 (w) B fw1,w2 (u1) ··· fw1,w2 (un) where   w2 if ui = w1   w1 if ui = w2  −1 −1 fw1,w2 (ui) B  w2 if ui = w1  −1 −1  w1 if ui = w2   ui otherwise.

The function fw1,w2 is an HS-quasimorphism, called replacement quasimorphism. It is worthwhile 2 noting that ( fw1,w2 ) = idFS and that the composition with fw1,w2 swaps the Brooks counting R-quasimorphisms φw1 , φw2 (the Brooks R-quasimorphisms are deﬁned in Example 5.1.3). The latter fact can be used to show that the group of invertible HS-quasimorphisms acts almost transitively on the set of counting R-quasimorphisms associated with non-self-overlapping words ([HS16, Corollary 5.18], see also [HS16, Theorem 5.21]). One may also observe that every replacement HS-quasimorphism is equivalent to a local HS-quasimorphism.

110 Example 12.3.5. Deﬁne the wobbling group W(N>0) as the set-group of permutations on N>0 = {1, 2,...} with bounded displacement (i.e. those σ so that |σ(k) − k| is uniformly bounded as k ranges in N>0). As above, given a non self-overlapping cyclically reduced word v we can uniquely decompose any reduced k k k word w as a concatenation u0v 1 u1v 2 u2 ··· v n un where no ui has v as a subword, ki , 0 for every i, and ui , ∅ for 0 < i < n. For every σ ∈ W(N>0) we may deﬁne a function fv,σ : FS → FS by acting with σ on the exponents ki appearing in the above decomposition. Namely, we let

σ±(k1) σ±(k2) σ±(kn) fv,σ(w) B u0v u1v u2 ··· v un where σ±(k) = σ(k) if k is positive and σ±(k) = −σ(−k) otherwise. −1 The functions fv,σ are invertible HS-quasimorphisms, as fv,σ−1 = fv,σ. One may also check that fv,σ−1 is equivalent to a local HS-quasimorphism if and only if σ has ﬁnite support. Note that the group W(N>0) is uncountable, and it has elements of arbitrary torsion. Remark 12.3.6. We may generalize the above example as follows. Let v be a non self-overlapping k k cyclically reduced word, so that any reduced word w has a unique decomposition w = u0v 1 u1 ··· v n un as before. Take a Z-quasimorphism f : Z → Z that is symmetric (i.e. so that f (−k) = − f (k) for every k ∈ Z). Then one may check that applying f to the exponents of v in the decompositions of reduced words deﬁnes an HS-homomorphism FS → FS

k1 k2 kn f (k1) f (k2) f (kn) u0v u1v u2 ··· v un 7→ u0v u1v u2 ··· v un.

The reason why it is relatively simple to exhibit examples of HS-quasimorphisms of the free group is that elements in FS can be uniquely identﬁed with reduced words. To prove that the above examples are HS-quasimorphisms, Hartnick and Schweitzer use the following criterion:

Proposition 12.3.7 ([HS16, Proposition 5.3]). Let f : FS → G be a function from the free group to an arbitrary set-group G. If there exists a ﬁnite set B ⊂ G such that

(i)f (w1w2) ∈ B ∗ f (w1) ∗ B ∗ f (w2) ∗ B whenever w1, w2 and their concatenation w1w2 are reduced words;

(ii)f (w−1) ∈ B ∗ f (w)−1 ∗ B; then f is an HS-quasimorphism.

Taking the point of view of coarse homomorphisms, it is not hard to state and prove a more general version of the above:

Lemma 12.3.8. Let f : FS → G be a function from the free group to an arbitrary set-group G. If there exists a controlled set E ∈ EG→R such that

(i)f (w1w2)- E → f (w1) ∗ f (w2) whenever w1, w2 and their concatenation w1w2 are reduced words;

−1 −1 (ii)f (w )- E → f (w) for every w ∈ FS ; then f is an HS-quasimorphism.

Proof. We can directly show that whenever φ: G → R is an R-quasimorphism the composition f ◦ φ is an R-quasimorphism as well. Given two elements g1, g2 ∈ FS , their reduced representative can be −1 uniquely decomposed as g1 = w1u and g2 = u w2 so that w1w2 is the reduced representative of the product g1 ∗ g2.

111 By the deﬁnition of the coarse structure EG→R, condition (i) implies that there is some constant D independent of g1, g2 so that

|φ( f (w1w2)) − φ( f (w1) ∗ f (w2))| ≤ D

|φ( f (w1u)) − φ( f (w1) ∗ f (u))| ≤ D −1 −1 |φ( f (u w2)) − φ( f (u ) ∗ f (w2))| ≤ D.

Similarly, it follows from condition (ii) that |φ( f (u−1)) − φ( f (u)−1)| ≤ D as well. We then see that

−1 |φ ◦ f (g1 ∗ g2) − φ ◦ f (g1) − φ ◦ f (g2)| = |φ( f (w1w2)) − φ( f (w1u)) − φ( f (u w2))| ≤ |−φ( f (u)) − φ( f (u−1))| + 3D ≤ 4D.

This proves that φ ◦ f is an R-quasimorphism and hence f is an HS-quasimorphism.

Proposition 12.3.7 follows from Lemma 12.3.8 because any finite set B ⊂ G is of course EG→R-bounded. More precisely, for any fixed B we know that the family {B ∗ f (w1) ∗ B ∗ f (w2) ∗ B | w1, w2 ∈ FS } is a controlled partial cover of G. Assume that e ∈ B. If condition (i) of Proposition 12.3.7 is satisfied, we see that the bounded set B ∗ f (w1) ∗ B ∗ f (w2) ∗ B contains both f (w1w2) and f (w1) ∗ f (w2). From this we deduce that hypothesis (i) of Lemma 12.3.8 is satisfied as well. A similar argument of course works for condition (ii).

Equivalence of HS-quasimorphisms is preserved under composition. Therefore, the set of equivalence classes of HS-quasimorphisms f : G → G is a monoid, denoted Q(G, G). Hartnick and Schweitzer deﬁne the set-group of quasioutomorphisms QOut(G) of a set-group G as the set-group of invertible elements of Q(G, G). It follows from Proposition 12.3.2 that quasioutomorphisms can be identiﬁed with coarse automorphisms of the coarse group (G, EG→R):

Corollary 12.3.9. For any set-group G there is a canonical isomorphism QOut(G) AutCrs(G, EG→R).

Proof. It is enough to show that two HS-quasimorphisms f1, f2 are equivalent if and only if [ f1] = [ f2]. The ‘if’ implication is clear by the deﬁnition. The proof that equivalent HS-quasimorphisms must be close with respect to EH→R is analogous to the ﬁnal argument in Proposition 12.3.2.

As mentioned above, one interesting result proved in [HS16] is that amenable groups are somewhat rigid from the point of view of HS-quasimorphisms. Namely, they prove the following classiﬁcation of quasioutomorphisms:

Theorem 12.3.10 ([HS16]). If a set-group G is amenable and its abelianization has ﬁnite rank k then there is a natural isomorphism QOut(G) GL(k, R).

On the other hand, the situation for non-abelian free groups is much more ﬂexible. The examples of HS-quasimorphisms constructed in Example 12.3.4 and Example 12.3.5 are all invertible and hence deﬁne elements in QOut(FS ). Using appropriate R-quasimorphisms, it is not hard to show that these examples are not equivalent. In particular, this shows that that the wobbling set-group W(N>0) embeds in QOut(FS ). Of course, every automorphism of a set-group G gives rise to an invertible HS-quasimorphism, so there is a natural homomorphism Aut(G) → QOut(G). This homomorphism factors through the set-group of outer automorphisms Out(G). With some work, it is possible to show that this homomorphism is injective:

112 Proposition 12.3.11 ([HS16, Proposition 5.1]). The canonical homomorphism

Out(FS ) ,→ QOut(FS ) AutCrs(FS , EFS →R) is injective.

We conclude this subsection with some ideas for future research: Remark 12.3.12. At first sight, it would be natural to guess that the existence of “exotic” quasioutomorphisms could be yet another instance of the amenable vs. non-amenable divide. However, this is not the case. In fact, we already noted in Subsection 8.3 that the set group SL(n, Z) is intrinsically bounded for every n ≥ 3. Since QOut(SL(n, Z)) AutCrs(SL(n, Z), ESL(n,Z)→R) and ESL(n,Z)→R = Emax by intrinsic boundedness, it follows that QOut(SL(n, Z)) = {e}. It is therefore very unclear which set-groups G admit exotic examples of quasioutomorphisms. One issue that we are facing is the lack of examples for non-free groups: the examples illustrated in this subsection make heavy use of the identification between group elements and reduced words. In view of this, it seems fairly natural to start investigating HS-quasimorphisms for finitely presented groups that have particularly nice presentations.

Remark 12.3.13. It would be interesting to understand more precisely the coarse structures EG→R for specific examples of set-groups G. We can of course make a few simple observations. For any set-group G, EG→R is a connected coarse structure because (R, E|-|) is itself coarsely connected (in other words, Egrp ⊆ E E E fin G→R). In particular, if G is an intrinsically bounded set-group (Subsection 8.3) then G→R = max is the bounded coarse structure. Similarly, it follows from the abelianity of R that (G, EG→R) is necessarily coarsely abelian. In Egrp → E particular, the identity map [id]: (G, fin ) (G, G→R) factors through the coarse abelianization of Egrp E (G, fin ) (Remark 6.2.4). It would be interesting to characterize the set groups G for which (G, G→R) is the coarse abelianization. By means of Bavard’s Duality [Bav91] (see also [Cal09, Theorem 2.70]), this problem relates to the question of bounding the stable commutator length in terms of commutator length.

12.4 Coarse homomorphisms of groups with cancellation metrics

As already remarked, every ﬁnitely normally generated set-group has a coarsiﬁcation of special importance.

Namely, the coarse group (G, Ed× ) where G = hhS ii, S ⊆ G is a ﬁnite normally generating set and d× is the associated cancellation metric (Example 4.2.12). This coarse structure is particularly important E Egrp because d× = ﬁn is the minimal connected coarse structure that makes G into a coarse connected coarse group (in particular, it is canonical and does not depend on the choice of S ). The aim of this subsection is to provide examples of coarse homomorphisms for such a (G, Ed× ) and explore how they relate to other notions of quasimorphisms.

It follows from the minimality of Ed× that in order to check whether a function deﬁnes a coarse homomorphism it is enough to verify that it is well behaved under multiplication. Namely, the following holds true:

Lemma 12.4.1. Let G be any set-group and H = (H, F ) a coarsely connected coarse group. If f : G → H ◦ ∗ ≈ ∗ Egrp → is a function so that f ( - G - ) f ( - ) H f ( - ), then f : (G, ﬁn ) H is controlled and [ f ] is a coarse homomorphism.

Egrp E Egrp → Proof. Since G is a set-group, min = min is the minimal coarse structure. It follows that f : (G, min) H is controlled and hence a coarse homomorphism. By Lemma 5.1.8, the pull-back f ∗(F ) is equi-bi-invariant. Since F is a connected coarse structure, every finite set U ⊆ G is f ∗(F )-bounded. By minimality, Egrp ⊆ ∗ F Egrp → it follows that fin f ( ) and hence f :(G, fin ) H is controlled.

113 For ﬁnitely generated groups, we can prove a variation on the above. When we ﬁx a generating set S for G, every element g ∈ G can be represented (non uniquely) as a reduced word w with letters in S ∪ S −1. We then have a version of Lemma 12.3.8:

Lemma 12.4.2. Let G = hS i be a ﬁnitely generated group, H = (H, F ) a coarsely connected coarse group and f : G → H be a function such that there exists F ∈ F with

(i)f (w1w2)- F → f (w1) ∗H f (w2) whenever w1, w2 and their concatenation w1w2 are reduced words;

− − (ii)f (w 1)- F → f (w) 1 for every reduced word w; then f :(G, Ed×) → H is controlled and [ f ] is a coarse homomorphism.

Proof. By Lemma 12.4.1, it is enough to show that f (g1 ∗G g2) ≈ f (g1) ∗H f (g2). Choose reduced words −1 v1, v2 representing the elements g1, g2. We can decompose them as v1 = w1u and v2 = u w2, so that the concatenation w1w2 is reduced. Now we have:

f (g1 ∗ g2) = f (w1w2)- F → f (w1) ∗H f (w2)

−1 f (g1) ∗H f (g2) = f (w1u) ∗H f (u g2) −1 - F → f (w1) ∗H f (u) ∗H f (u ) ∗H f (w2) −1 - ∆H∗F∗∆H → f (w1) ∗H f (u) ∗H f (u) ∗H f (w2)

- F → f (w1) ∗H f (w2).

We can hence apply Lemma 12.4.1.

Unsurprisingly, the above is especially useful for free groups:

Corollary 12.4.3. All the examples of HS-quasimorphisms f : FS → H constructed in Examples 12.3.3,

12.3.4 and 12.3.5 deﬁne coarse homomorphisms [ f ]: (FS , Ed× ) → (H, F ) whenever F is an equi-bi-invariant coarse structure on H.

Proof. The proofs that those examples are indeed HS-quasimorphisms rely on Proposition 12.3.7. Since every ﬁnite subset of H is F -bounded, Lemma 12.4.2 applies to any function satisfying the hypotheses of Proposition 12.3.7.

Corollary 12.4.4. Any replacement quasioutomorphism f : FS → FS deﬁnes a coarse automorphisms

[ f ] ∈ AutCrs(FS , Ed× ) and Example 12.3.5 gives a homomorphism of the wobbling group W(N>0) →

AutCrs(FS , Ed× ).

The reader should not expect that every HS-quasimorphism f : FS → FS is controlled as a function f : (FS , Ed× ) → (FS , Ed× ). In fact, the coarse structure EFS →R is strictly larger than Ed× (for instance, notice that (FS , EFS →R) is coarsely abelian while (FS , Ed× ) is not). One may then easily ﬁnd a representative f for a coarse homomorphism [ f ]: (FS , EFS →R) → (FS , EFS →R) that is not controlled as a function

(FS , Ed× ) → (FS , Ed× ). On the contrary, it is often true that coarse homomorphisms of G give rise to HS-quasimorphisms:

E E → Egrp Lemma 12.4.5. Let G, H be set-groups and (G, ) any coarsiﬁcation of G. If [ f ]: (G, ) (H, ﬁn ) is a coarse homomorphism, then f is an HS-quasimorphism. Any two representatives f1, f2 ∈ [ f ] are equivalent HS-quasimorphisms.

114 → Egrp → E Proof. Every R-quasimorphism φ: H R deﬁnes a coarse homomorphism [φ]: (H, ﬁn ) (R, | - |). Therefore, the composition [ f ◦ φ]: (G, E) → (R, E| - |) is itself a coarse homomorphism and hence an R-quasimorphism. The statement about equivalence is clear, because [φ ◦ f1] = [φ] ◦ [ f ] = [φ ◦ f2] and hence |φ ◦ f1 − φ ◦ f2| is bounded.

Corollary 12.4.6. If G and H are ﬁnitely normally generated set-groups, there is a natural map

HomCrs((G, Ed× ), (H, Ed× )) → Q(G, H) sending coarse homomorphism to equivalence classes of HS-quasimorphisms.

Corollary 12.4.7. The natural homomorphism Out(FS ) → AutCrs(FS , Ed× ) is an embedding.

Proof. The composition Out(FS ) → AutCrs(FS , Ed× ) → Q(FS ) is injective by Proposition 12.3.11.

We already remarked that not every HS-quasimorphism f gives rise to a coarse homomorphism. However, our examples are rather unsatisfactory as such f may still be equivalent to an HS-quasimorphism that is a coarse homomorphism. In other words, we ﬁnd that the following is a more natural question.

Question 12.4.8. Let G, H be ﬁnitely generated groups. Under what circumstances is the map

HomCrs((G, Ed× ), (H, Ed× )) → Q(G, H) surjective? When is it injective?

Thus far we compared coarse homomorphisms with HS-quasimorphism. However one may also consider other notions of R-quasimorphism. Following Hartnick–Schweitzer and Fujiwara–Kapovich, we use the following:

Deﬁnition 12.4.9 ([HS16, FK16]). Let G, H be set groups. A function f : G → H is

• an Ulam quasimorphism if

f (g1g2) ∈ f (g1) f (g2)B ∀g1, g2 ∈ G,

• a middle quasimorphism if

f (g1g2) ∈ f (g1)B f (g2) ∀g1, g2 ∈ G,

• a geometric quasimorphism if

f (g1g2) ∈ f (g1)B f (g2)B ∀g1, g2 ∈ G,

• an algebraic quasimorphism if

f (g1g2) ∈ B f (g1)B f (g2)B ∀g1, g2 ∈ G, where B ⊆ H is some ﬁxed ﬁnite set.

115 If G and H are ﬁnitely normally generated, we have the following chain of containments:

{ f : G → H | Ulam q.morph.} { f : G → H | middle q.morph.} ⊂ ⊃ { f : G → H | geometric q.morph.} ⊂

{ f : G → H | algebraic q.morph.} ⊂

{ f :(G, Ed× ) → (H, Ed× ) | controlled} ⊂

{ f : G → H | HS-q.morph.}.

Remark 12.4.10. In [FK16] Fujiwara and Kapovich use more general definition of quasimorphisms where they equip H with some left-invariant metric d and ask for the set B to be bounded. Their definition coincides with the one we gave if d is proper and discrete. Considering more general metrics complicates the above diagram of inclusions. As before, it is easy to see that these containments are usually strict by adding some perturbation. However, we are interested in equivalence classes of quasimorphisms. More general quasimorphisms which are more tolerant to error should be considered only up to a more general equivalence relation. For example, two geometric quasimorphisms f1, f2 : G → H should be considered equivalent if there is 0 0 some finite B ⊆ H so that f1(g) ∈ f2(g)B for every g ∈ G. In contrast, algebraic quasimorphisms should 0 0 0 be equivalent if there is some finite B ⊆ H so that f1(g) ∈ B f2(g)B for every g ∈ G. It is now natural to extend Question 12.4.8 to this setting and ask when it is the case that the maps induced by the above inclusions are injective/surjective when quotienting by the relevant equivalence relation. A special instance of this question already asked in [HS16], where ask they whether there exist algebraic quasimorphisms that are not equivalent to any geometric quasimorphism. It is a result of Fujiwara–Kapovich [FK16] that there are very few Ulam quasimorphisms f : FS → FS . At the opposite end, the examples of Hartnick–Schweitzer show that Q(FS ) is very rich. This shows that the composition of the maps

{Ulam q.morph.} → {geom. q.morph.}/∼ → {alg. q.morph.}/∼ → AutCrs(FS , Ed× ) → Q(FS ) is far from being surjective for the free group. However, it is still not clear whether surjectivity fails for all of the above. In general, we find that this area of research still presents a number of open problems. Remark 12.4.11. We are not currently aware of any set-group G that has an unbounded, coarsely connected coarsification (G, E) so that AutCrs(G, E) is countable or finite.

13 Proper coarse actions

The aim of this section is to study the notion of properness for coarse actions of coarse groups. We already gave a temporary definition in Subsection 6.4 when discussing the coarse geometric approach to the Milnor–Švarc Lemma. Here we will give a more conceptual definition by analogy with the definition of properness for continuous actions of topological groups.

116 13.1 Properness via bornologies Informally, properness is the property of preserving boundedness under taking pre-images. The most appropriate language to describe boundedness is that of bornologies.

Definition 13.1.1 ([Bou87, Chapter 3]). A bornology B on X is a collection of subsets of X (called bounded sets) that covers X and is closed under taking subsets and finite unions. A function between 15 bornological spaces f : (X1, B1) → (X2, B2) is bornological if f (B1) ∈ B2 for every B1 ∈ B1. We also −1 say that f is proper if f (B2) ∈ B1 for every B2 ∈ B2 . If E is a coarse structure on X, the set of E-bounded sets is a bornology. That is, to every coarse space is naturally associated a unique bornology (the converse is not true: a given bornology can be induced by distinct coarse structures). A controlled map between coarse spaces f : (X1, E1) → (X2, E2) is bornological with respect to the induced bornologies. Moreover, f is proper if and only if it is proper in the sense of bornologies. For these reasons we can mix coarse structures and bornologies, so that it makes sense to say that a function f : (X, B) → (Y, E) from a bornological space to a coarse space is bornological. The product of two bornological spaces (X1, B1) × (X2, B2) is the cartesian product X1 × X2 equipped with the bornology B1 ⊗ B2 generated by the products B1 × B2 where B1 ∈ B1 and B2 ∈ B2. Equivalently, B ∈ B1 ⊗ B2 if and only if π1(B) ∈ B1 and π2(B) ∈ B2. Definition 13.1.2. Let (G, E) be a coarse group and let B be a bornology on G. A coarse action [α]: (G, E) y (Y, F ) is B-proper if (α, πY ): (G, B) × (Y, F ) → (Y, F ) × (Y, F ) is bornological and proper, where πY denotes the projection onto Y. Since F is a coarse structure, this definition does not depend on the choice of representative for [α].

Explicitly, [α] is a B-proper coarse action if and only if it satisﬁes

(P1) α(B × C) is F -bounded for every B-bounded B ⊆ G and F -bounded C ⊆ Y;

−1 (P2) α (C1) ∩ (G × C2) is B ⊗ F -bounded for every F -bounded C1, C2 ⊆ Y

(condition (P1) implies that (α, πY ) is bornological, (P2) implies that it is proper). Since α is a coarse action, we are assuming that α: (G, E) × (Y, F ) → (Y, F ) is controlled. In particular, every E-bounded set must be contained in B in order for α to be proper with respect to B (i.e. B contains the bornology associated with E). In other words, the deﬁnition of B-proper coarse actions is designed to describe those coarse actions whose orbit maps are not proper, and B ought to be thought of as a way to keep track of those (possibly E-unbounded) subsets of G that are sent to bounded sets in Y. By the deﬁnition of the product bornology B ⊗ E, we can rephrase condition (P2) as:

−1 (P2’) for every F -bounded C ⊆ Y the set πG α (C) ∩ (G ×C) = {g ∈ G | (g ·C) ∩C , ∅} is B-bounded. E Recall that fin is the coarse structure on X of consisting of relations with finitely many points off the diagonal. Further recall that in Subsection 6.4 we gave a temporary definition of properness by declaring Egrp F { ∈ | · ∩ ∅} that an action of a trivially coarse group (G, min) y (Y, ) is proper if the set g G (g B) B , is F ⊆ E finite for every -bounded B Y. By (P2’) we then see that that definition is equivalent to fin-properness.

13.2 Proper actions are determined locally We already saw that equi-left-invariant coarse structures on a coarse group (G, E) are determined by their bounded sets. We are now going to leverage this fact to show that B-properness of coarse actions completely determines the equivariant coarse geometry of the orbits.

15Such a map is often called ‘bounded’, but in our setting we ﬁnd this terminology confusing.

117 Let (G, E) be a coarse group and B be a bornology on G containing all the E-bounded sets. Denote left left by RB {B × B | B ∈ B}. Let E E be the smallest equi-left-invariant coarse structure containing B B B RB RB. Since equi-left-invariant coarse structures are determined by their bounded sets (Proposition 4.5.2) left and every E-bounded set is in B, it follows that E ⊆ EB . We may then see that Eleft h ∗ R Egrp i h ∗ R Ei B = ∆G B , min = ∆G B , (see Remark 4.3.6). Proposition 13.2.1. If there exists a B-proper coarse action of (G, E) then B must coincide with the set left of EB -bounded sets. Given such a B, a coarse action [α]: (G, E) y (Y, F ) is B-proper if and only if ∗ left αy(F ) = EB for every ﬁxed y ∈ Y. ∗ Proof. We begin by showing that if [α]: (G, E) y (Y, F ) is B-proper then αy(F ) = h∆G ∗ RB , Ei. This proves the ﬁrst statement and one of the “if and only if” implications. Fix y ∈ Y. For every B ∈ B, ∗ αy(B) = α(B × {y}) is F -bounded because (α, πY ) is bornological. Hence B × B ∈ αy(F ). Since [α] is a ∗ ∗ coarse action, E ⊆ αy(F ) and the latter is equi-left invariant. It follows that also ∆G ∗ (B × B) ∈ αy(F ) ∗ for every B ∈ B and hence h∆G ∗ RB , Ei ⊆ αy(F ). The coarse structure h∆G ∗ RB , Ei is equi-left-invariant. Since equi-left-invariant coarse structures are determined by their bounded sets (Proposition 4.5.2), to prove the other containment it is enough to ∗ ∗ −1 show that every αy(F )-bounded set is B-bounded. Let B be αy(F )-bounded, we can assume B = αy (C) for some C ∈ F . On the other hand, −1 −1 αy (C) × {y} = α (C) ∩ (G × {y}) is B ⊗ F -bounded by hypothesis and hence B is B-bounded.

left Assume now that [α]: (G, E) y (Y, F ) is a coarse action such that B is the bornology of EB -bounded ∗ left sets and αy(F ) = EB for every y ∈ Y. Given a B-bounded B and a F -bounded C, choose y ∈ C. Then B × C = St(B × {y}, ig × C) where ig × C is the controlled partial covering {{g} × C | g ∈ G}. The image αy(B) is F -bounded by hypothesis, therefore

α(B × C) ⊆ St(α(B × {y}), α(ig × C)) = St(αy(B), α(ig × C)) is F -bounded. This shows that (α, πY ) is bornological. It only remains to check that (α, πY ) is proper. Let C1, C2 ⊆ Y be F -bounded sets. Then [ −1 −1 n −1 o α (C1) ∩ πY (C2) = αz (C1) × {z} z ∈ C2 , S −1 B ∈ we thus need to show that the set B B z∈C2 αz (C1) is -bounded. Fix y C2. Note that for every −1 g ∈ αz (C1) g · y ∈ g · C2 ⊆ St(g · z, iG · C2) ⊆ St(C1, iG · C2), where iG · C2 is the controlled partial covering {g · C1 | g ∈ G}. It follows that B is contained in −1 αy (St(C1, iG · C2)) and is hence B-bounded. Corollary 13.2.2. If it exists, a B-proper cobounded coarse action of (G, E) must be isomorphic to left [∗]:(G, E) y (G, EB ). In particular, such an action is unique up to isomorphism. −1 Assume now that the operations ∗, e, (-) on G are cancellative (Definition 4.4.5), and let Be be the set of B-bounded neighbourhoods of the identity. If B is the bornology of bounded elements of an equi-left-invariant coarse structure, then Be must certainly satisfy conditions (U0)–(U4) of Subsection 4.6. Using the cancellation hypothesis, it is also immediate to verify that B must be equal to the union of the translates of the identity neighbourhoods B = iG ∗ B. It follows easily from Proposition 4.6.1 that these two conditions are also sufficient: Corollary 13.2.3. If the operations on G are cancellative, then there exists an B-proper action if and only if B = iG ∗ Be and Be satisfies (U0)–(U4).

118 13.3 Topological coarse actions The deﬁnition of properness for coarse actions is modelled on the deﬁnition of properness for continuous actions of topological groups. A continuous action α: G y Y of a topological group G is topologically proper if (α × πY ): G × Y → Y × Y is a proper map (preimage of compact sets is compact). Unsurprisingly, the two notions of properness are tightly connected. Let α: G y Y be a continuous action of a topological group G. We have already seen in Exam- left left ple 6.1.4 that this induces a coarse action of the trivially coarse group (G, Emin) y (Y, Fcpt ). Here Fcpt is the unique (G, Emin)-equivariant coarse structure on Y whose bounded sets are the relatively compact subsets of Y. Explicitly,

left Fcpt = h∆G ·Fcpti = F ⊆ Y × Y | ∃K ⊆ Y compact s.t. ∀(y1, y2) ∈ F, ∃g ∈ G with y1, y2 ∈ g(K) .

In the above, Fcpt = {F ⊆ Y × Y | ∃K ⊆ Y compact, F ⊆ (K × K) ∪ ∆Y } is the minimal coarse structure on Y whose bounded sets are the relatively compact subsets of Y. Let K be bornology of relatively compact subsets of G. Then we have:

Proposition 13.3.1. Let G y Y be a continuous action of a topological group. If the action is topologi- left cally proper, the coarse action (G, Emin) y (Y, Fcpt ) is K-proper. If Y is Hausdorﬀ the converse is also true.

left Proof. It follows by the definitions that the K ⊗ Fcpt -bounded sets in G × Y are precisely the relatively compact subsets. It follows from continuity that the coarse action satisfies (P1), because the continuous image of a relatively compact set is relatively compact. left −1 Since Fcpt -bounded sets are relatively compact, condition (P2) amount to saying that (α × πY ) (C1 × C2) is relatively compact whenever C1 × C2 is. If the action is topologically proper, this condition is clearly satisfied. Vice versa, if Y is Hausdorff and (P2) is true then α × πY is a topologically proper map because a set is relatively compact if and only if its closure is compact.

left left Remark 13.3.2. If α is a proper action, the pull-back of Fcpt under the orbit map αy is equal to Ecpt . This left can be checked by hand, or by remarking that the Ecpt is the unique equi-left-invariant coarse structure on G whose bounded sets are the relatively compact subsets of G. left The coarse action [α]: (G, Emin) y (Y, Fcpt ) induced by a topological action is cobounded if and only if there exists a compact set K ⊆ Y such that iG · K = {g(K) | g ∈ G} covers Y. That is, if and only if α is cocompact. Combining Proposition 13.3.1 and Corollary 13.2.2 we obtain:

Corollary 13.3.3. If G y Y is a proper cocompact action of a topological group, then the orbit map left left induces an equivariant coarse equivalence between (G, Ecpt ) and (Y, Fcpt ).

Let (Y, d) be a metric space and denote by Ed the induced coarse structure. Recall that a set-group action G y Y induces a coarse action on (Y, Ed) if and only if it is an action by “uniform metric coarse equivalences” (see Example 6.1.5). If α: G y Y is a continuous action of a topological group which left does induce a coarse action of (G, Emin) on (Y, Ed), then Fcpt ⊆ Ed because compact sets are bounded and the latter is (G, Emin)-equivariant. If Y is a proper metric space (i.e. closed balls are compact) then left Ed ⊆ Fcpt as well. Corollary 13.3.4. If G is a topological group, (Y, d) a proper metric space and α: G y Y is a continuous action by uniform metric coarse equivalences that is proper and cocompact, then the orbit map induces a left (G, Emin)-equivariant coarse equivalence between (G, Ecpt ) and (Y, Ed).

119 Recall that a coarse space is coarsely geodesic if it is coarsely connected and the coarse structure is generated by a single relation. Thus a metric space Yd is coarsely geodesic if the coarse space (Y, Ed) is (recall that metric spaces are always coarsely connected). Remember coarsely geodesic metric spaces are coarsely equivalent if and only if they are quasi-isometric. In particular, we may rephrase “action by uniform metric coarse equivalences” as “action by uniform quasi-isometries” where “uniform” means that the quasi-isometry constants do not depend on the group element. Any generating set S of a set-group G can be used to deﬁne a word metric dS on G. Note that, a 0 priori, two inﬁnite generating sets S, S need not give rise to quasi-isometric word metrics dS and dS 0 . We may now prove a topological generalization of the Milnor–Švarc Lemma:

Proposition 13.3.5. Let G be a Hausdorﬀ topological group and (Y, Ed) a coarsely geodesic proper metric space. If there exists a continuous proper and cocompact action α: G y Y by uniform quasi-isometries, then G is compactly generated. Moreover, for any compact generating set K and y ∈ Y the orbit map αy :(G, dK) → (Y, d) is a quasi-isometry.

Proof. Let α: G y Y be an action satisfying the hypothesis of the theorem. The assumptions that (Y, d) be a proper metric space and that α is a proper action imply that G must be locally compact. Since G is Hausdorff, we may then apply the Baire Theorem to show that if we are 0 given two compact generating sets K and K then the induced word metrics dK, dK0 are quasi-isometric. S −1 n −1 n In fact, since G = n∈N(K ∪ K ) we see that for some n large enough (K ∪ K ) must have non-empty interior. It then follows by compactness of K0 that K0 ⊆ (K ∪ K−1)nm for some m ∈ N. left Note that the assumptions of Corollary 13.3.4 are satisfied. We deduce that (G, Ecpt ) is coarsely equivalent to (Y, Ed) and hence is coarsely geodesic. Therefore, there exists a compact set K¯ ⊆ G such that ¯ left EK¯ B ∆G ∗ (K × {e}) generates Ecpt . Since G is coarsely connected, for every g ∈ G the pair (g, e) belongs ¯ n ¯ to the n-fold composition EK¯ ◦ · · · ◦ EK¯ = ∆G ∗ (K × {e}). This shows that K is a compact generating E set for G and hence defines a word metric. Note that the induced coarse structure dK¯ coincides with left left Ecpt . We also know that for every y ∈ Y the orbit map αy : (G, Ecpt ) → (Y, Ed) is a coarse equivalence. By coarse geodesicity we deduce that αy : (G, dK¯ ) → (Y, d) is a quasi-isometry. By the previous paragraph, the same is true for every choice of compact generating set K.

Remark 13.3.6. Some comments.

1. The above proposition truly is a generalization of the Milnor–Švarc Lemma to general locally compact groups: the original lemma is obtained as the special case where G is a discrete group.

2. The Milnor–Švarc Lemma for discrete groups is sometimes stated without the requirement that Y be proper but requires that the action be metrically-proper instead ([BH13]). However, this is not a real weakening of the hypotheses, as it is easy to show that a metric space admitting a cocompact metrically proper action must indeed be proper.

3. If the metric space is coarsely geodesic but not proper, it is still possible to construct a word metric on G with which G is coarsely equivalent to (Y, Ed). This word metric will be determined by ∗ the choice of a αy(F )-bounded subset of G generating the coarse structure. However, the coarse left structure thus obtained will not coincide with Ecpt in general. Remark 13.3.7. Rosendal makes a more advanced study of coarse geometry of topological groups in the monograph [Ros17]. The focus there is somewhat different from ours. In our language, Rosendal is mostly concerned with the study of certain natural equi-left-invariant coarse structures that are “compatible” with the group topology. This different focus makes it somewhat complicated to directly compare our results with those of Rosendal. For example, his versions of topological Milnor–Švarc Lemma [Ros17, Theorems 2.56 and 2.57] are different (although related) from the one we just presented.

120 14 Spaces of controlled maps

The aim of this subsection is to study the space of controlled maps (X, E) → (Y, E). This is not to be confused with MorCrs(X, Y), as we are not identifying close functions. Ideally, we would like to deﬁne a natural coarse structure on this set. This is not possible, however there is a natural “fragmentary” coarse structure which will be useful to deal with families of controlled maps.

14.1 Fragmentary coarse structures A fragmentary coarse structure on a set X is a collection of relations satisfying all the requirements of a coarse structure with the exception that it need not contain the diagonal:

Deﬁnition 14.1.1. A collection E of relations on X is a fragmentary coarse structure if it is closed under taking subsets, ﬁnite unions and

(i) {(x, x)} ∈ E for every x ∈ X;

(ii) if E ∈ E then Eop ∈ E;

(iii) if E1, E2 ∈ E then E1 ◦ E2 ∈ E.

The relations E ∈ E are still called controlled sets. A set with a fragmentary coarse structure is a fragmentary coarse space (shortened to frag-coarse space). A subset Y ⊆ X is a fragment of the fragmentary coarse space (X, E) if ∆Y ∈ E. The completely fragmented fragmentary coarse structure on X generated by the set of diagonal singletons

n o EFrag B {(x1, x1),..., (xn, xn)} x1,... xn ∈ X, n ∈ N

(this is the minimal fragmentary coarse structure on X).

Remark 14.1.2. If (X, E) is a fragmentary coarse space and Y ⊆ X is a fragment, then the restriction of E to Y is a genuine coarse structure and hence (Y, E|Y ) is a coarse space. With the obvious modiﬁcations, basic properties and deﬁnitions for coarse structures hold true for fragmentary coarse structures as well. For example, Lemma 2.2.9 becomes:

Lemma 14.1.3. A relation F belongs to the fragmentary coarse structure generated by a family of relations {Ei | i ∈ I} if and only if it is contained in a finite composition F1 ◦ · · · ◦ Fn where each F j is a op finite union of Ei or Ei . Controlled maps are defined in the usual way. Note that if (X, E) is a coarse space and f : (X, E) → (Y, F ) is a controlled map to a fragmentary coarse space then the image ( f (X), F | f (Y)) is still a coarse space. Bounded sets and controlled partial coverings are also defined analogously. Notice that for any subset of a fragmentary coarse space Y ⊆ (X, E) the partial covering iY is E-controlled if and only if Y is a fragment of (X, E). The definition of closeness requires a modification, as it relies on the diagonal relation.

Deﬁnition 14.1.4. Two functions f, g: (X, E) → (Y, F ) between fragmentary coarse spaces are fragmentary close (shortened to frag-close) if f × g(∆Z) ∈ F for every fragment Z ⊆ X.

Note that Definition 14.1.4 depends on the fragmentary coarse structure on the domain (more precisely, it depends on the set of fragments of (X, E)). This was not the case for the usual definition of closeness (Definition 2.3.4), as it was assumed that ∆X always belong to the coarse structure. If ∆X ∈ E (i.e. (X, E) is a coarse space) the two notions coincide.

121 Since the difference between closeness and frag-closeness might create some confusion, we will be zealous and add the adjective “fragmentary” to those notions that rely on closeness. For example, a fragmentary coarse map (frag-coarse) is an equivalence class of a controlled maps up to frag-closeness. Fragmentary coarse equivalence is defined analogously. However, we will keep using bold symbols to denote fragmentary coarse spaces X = (X, E) and frag-coarse maps f : X → Y. Note that if X = (X, E) and Y = (Y, F ) are two coarse spaces, then two functions f, g: X → Y are frag-close if and only they are close in the usual sense. In particular, the bold symbol f represents the same set of equivalent functions regardless of whether we view X and Y as objects in Coarse or FragCrs. We denote the set of frag-coarse maps f : X → Y by MorFrCrs(X, Y). Remark 14.1.5. In the sequel we will not use fragmentary analogues of most of the notions of Section3, so we do not discuss them in detail. However, we do wish to point out that one should be a little careful when developing a theory of frag-coarse subspaces, images, intersections, etc.. The definition of coarse containment and asymptoticity of subsets (Definition 3.3.2) does not work well as it is, because it is not a reflexive property. Instead, one would have to adjust it as we did for closeness, i.e. by only checking coarse containment on fragments of the subsets. A different point of view is that of preserving the correspondence between “set-wise” and “categorical” frag-coarse subspaces. Since the categorical definition in terms of equivalence classes of monics relies on frag-closeness, it is certainly necessary to adapt accordingly the definition of frag-coarse containment. Remark 14.1.6. We can of course define the category FragCrs of fragmentary coarse spaces, and this category naturally includes the category Coarse of coarse spaces as a subcategory. Note that if X and Y are two coarse spaces then MorCrs(X, Y) = MorFrCrs(X, y). The main reason for introducing frag-coarse spaces is to have a language for spaces of controlled maps. In categorical terms, the key extra features that FragCrs has over Coarse is that it is a Cartesian closed category and that Coarse is enriched over FragCrs. We relegate this discussion to Appendix A.2.

Warning. The next example shows that a coarse space can be frag-coarsely equivalent to a frag-coarse space that is not a coarse space.

Example 14.1.7. It is easy to show that a frag-coarse space (X, E) is frag-coarsely equivalent to the one-point coarse space (pt, {pt}) if and only if there exists a point x¯ ∈ X such that for every fragment Y ⊆ X the product relation {x¯} × Y is in E (we say that such a space is frag-bounded). Concretely, the set R equipped with the fragmentary coarse structure E = {E | ∃t > 0, E ⊆ [−t, t] × [−t, t]} is frag-coarsely equivalent to {pt}, but ∆R < E. There are instances when the diagonal relation is very useful. This is the case for the fundamental equality E ⊗ F = (E ⊗ ∆Y ) ◦ (∆X ⊗ F) on (X, E) × (Y, F ). Fortunately, this equality can be reﬁned by writing

E ⊗ F = (E ⊗ ∆π1(F)) ◦ (∆π2(E) ⊗ F), (11) where π1 and π2 are the projection to the ﬁrst and the second coordinate. Importantly, if E is a relation in a fragmentary coarse structure E on X, then the projections π1(E) and π2(E) are fragments of X, so that both terms on the RHS of Equation (11) belong to E ⊗ F . We can actually prove a slightly more reﬁned result (which will be useful later):

Lemma 14.1.8. Let E be a fragmentary coarse structure. Then for every E ∈ E there exists a E˜ ∈ E such that E ⊆ E˜ and ⊆ E˜. In particular, for every E ∈ E the union π E ∪ π E is a fragment of ∆π1(E˜)∪π2(E˜) 1( ) 2( ) X.

Proof. Note that π1(E1 ◦ E2) ⊆ π1(E1) and π2(E1 ◦ E2) ⊆ π2(E2). Also note that

op op ∆π1(E) ⊆ E ◦ E and ∆π2(E) ⊆ E ◦ E.

122 It follows that E ⊆ E ◦ Eop ◦ E. Moreover,

op op π1(E ◦ E ◦ E) = π1(E ◦ (E ◦ E)) = π1(E),

op op op ˜ hence ∆π1(E◦E ◦E) ⊆ E ◦ E ◦ E. The same argument works for π2 as well, hence E B E ◦ E ◦ E satisﬁes our requirements. Corollary 14.1.9. A function between fragmentary coarse spaces f : (X, E) → (Y, F ) is controlled if and only if its restriction to every fragment of X is controlled.

We can deﬁne (X, E)-equicontrolled families of maps as in Subsection 2.6. That is, an X-indexed family of controlled functions fx : (Y, F ) → (Z, D) is (X, E)-equicontrolled if and only if the induced map f :(X, E) × (Y, F ) → (Z, D) is controlled.

Warning. Recall that a I-indexed family of functions fi : (Y, F ) → (Z, D) is equicontrolled if they are (I, Emin)-equicontrolled, where Emin = {∆I} as usual. We will use the same nomenclature even if Y and Z are frag-coarse spaces. In particular, note that if (X, E) is a frag-coarse space then a family of (X, E)-equicontrolled functions need not be equicontrolled.

The important observation characterizing controlled functions (X, E) × (Y, F ) → (Z, D) as those functions whose left and right sections are equicontrolled (Lemma 2.6.3) cannot hold for frag-coarse spaces. However, since Equation (11) holds true, we can recover the following fact:

Corollary 14.1.10. Let (X, E), (Y, F ) and (Z, D) be fragmentary coarse spaces. A function f : (X, E) × (Y, F ) → (Z, D) is controlled if and only if for every pair of fragments A ⊆ X and B ⊆ Y both families {x f :(Y, F ) → (Z, D) | x ∈ A} and { fy :(X, E) → (Z, D) | y ∈ B} are equicontrolled.

14.2 The fragmentary coarse space of controlled maps As announced, we introduced fragmentary coarse spaces because they are well-suited for studying spaces of controlled maps. We begin with lemmas and deﬁnitions, as we need them to introduce the notation. The heuristics behind these deﬁnitions are explained in Remark 14.2.8. Let X = (X, E) and Y = (Y, F ) be two fragmentary coarse spaces. With a slight abuse of notation, we denote the set of controlled maps X (Y,F ) (X, E) → (Y, F ) by Yctr (it would be more correct to write (X, E)ctr ). We see this as a subset of the set Y X of all the functions from X to Y. Given E ∈ E and F ∈ F , we denote

E X F B ( f1, f2) f1, f2 ∈ Yctr, f1 × f2(E) ⊆ F .

E The set F is a relation on the set of controlled functions from (X, E) to (Y, F ). Clearly, F1 ⊆ F2 implies E E E1 E2 E F1 ⊆ F2 and E1 ⊆ E2 implies F ⊇ F . It is also easy to verify that F satisﬁes the following properties:

E E E (i) F1 ∪ F2 ⊆ (F1 ∪ F2) ; op (ii)( FE)op = (Fop)(E ).

E X X E For any ﬁxed E ∈ E, we let F denote the set of relations C ⊂ Yctr × Yctr such that C ⊆ F for some E S E F ∈ F . In other words, F is the collection of relations obtained by making F∈F F closed under taking subsets. The following observation is important.

E Lemma 14.2.1. Let E ∈ E be a relation such that ∆π1(E)∪π2(E) ⊆ E. Then, for every relation D ∈ F there exists a F˜ ∈ F such that f × f (E) ⊆ F˜ for every f ∈ π1(D) ∪ π2(D). In other words, there exists F˜ ˜ E such that ∆π1(D)∪π2(D) ⊆ F .

123 Proof. Fix D ∈ F E. By hypothesis, there is F ∈ F such that f × g(E) ⊆ F for every pair ( f, g) ∈ D. Let X f ∈ π1(D), by deﬁnition there is g ∈ Yctr such that ( f, g) ∈ D. Note that

op f × f (E) ⊆ ( f × g(E)) ◦ (g × f (∆π2(E))) = ( f × g(E)) ◦ ( f × g(∆π2(E))) . (12)

∆π2(E) op Since ∆π2(E) ⊆ E, the pair ( f, g) is in F . That is, f × g(∆π2(E)) ⊆ F. It follows that f × f (E) ⊆ F ◦ F op An analogous argument shows that g × g(E) ⊆ F ◦ F whenever g ∈ π2(D). To conclude the proof it is enough to let F˜ B (Fop ◦ F) ∪ (F ◦ Fop). E Lemma 14.2.2. Let E ∈ E be a relation such that ∆π1(E)∪π2(E) ⊆ E. Then the collection F is a fragmentary coarse structure. Proof. The collection F E is closed under taking subsets by construction. Properties (i) and (ii) show that it is also closed under finite unions and symmetries. To prove that F E is a fragmentary coarse structure, all it remains to verify is that it is closed under composition. E Fix D1, D2 ∈ F , then there exist F1, F2 ∈ F such that f1 × f2(E) ⊆ F1 and g1 × g2(E) ⊆ F2 for every ( f1, f2) ∈ D1 and (g1, g2) ∈ D2. By definition, ( f, g) ∈ D1 ◦ D2 if and only if there exists h: X → Y such that ( f, h) ∈ D1 and (h, g) ∈ D2. It follows that for every ( f, g) ∈ D1 ◦ D2 we have × ⊆ × ◦ × op ◦ × ⊆ ◦ op ◦ op ◦ f g(E) f h(E) h h(E) h g(E) F1 (F1 F1) F2, ◦ ◦ op op ◦ ◦ ⊆ E where we used (12) for the last containment. Letting F B F1 (F1 F1 ) F2 shows that D1 D2 F E and hence D1 ◦ D2 is in F . It follows from Lemma 14.1.8 that the intersection T{F E | E ∈ E} is equal to the intersection T E {F | E ∈ E, ∆π1(E)∪π2(E) ⊆ E}. By Lemma 14.2.2, the latter is an intersection of fragmentary coarse structures and hence it is itself a fragmentary coarse structure. X Definition 14.2.3. The exponential fragmentary coarse structure on Yctr is the intersection \ F E B {F E | E ∈ E}. Given X = (X, E) and Y = (Y, F ), the fragmentary coarse space of controlled functions from X to Y (or, X X E exponential frag-coarse space) is the fragmentary coarse space Y B (Yctr, F ). E X Note that if D ∈ F and f ∈ Yctr, then all the functions in D( f ) are frag-close to f (they are in fact “equi-frag-close” to it). Furthermore, Lemma 14.2.1 has an immediate consequence on the exponential fragmentary coarse structure: E Corollary 14.2.4. For every relation D ∈ F the functions in π1(D) ∪ π2(D) are equicontrolled. Proof. E ∈ E E ⊆ E˜ ∈ E ⊆ E˜ D ∈ F E˜ Given , choose a with ∆π1(E˜)∪π2(E˜) . Since , by Lemma 14.2.1 there is an F ∈ F such that f × f (E) ⊆ F for every f ∈ π1(D) ∪ π2(D).

Example 14.2.5. Let (X, dX), (Y, dY ) be metric spaces. Then f : X → Y is controlled if and only if there is 0 0 0 a controlled function ρ+ : [0, ∞) → [0, ∞) with dY ( f (x), f (x )) ≤ ρ+(dX(x, x )) for every x, x ∈ X. A set X E X of controlled functions {gi ∈ Y | i ∈ I} is contained in a F -bounded neighbourhood of f ∈ Y if and only if for every r > 0 there is an R large enough so that gi B(X,dX)(x; r) ⊆ B(Y,dY )( f (x); R) 0 0 for every x ∈ X. Notice that this also implies that there is a ρ+ : [0, ∞) → [0, ∞) such that dY (gi(x), gi(x )) ≤ 0 0 0 ρ+(dX(x, x )) for every i ∈ I and x, x ∈ X (i.e. the gi are equicontrolled). One can also check that a set E D = {(gi, fi) | i ∈ I} is in F if and only if there exists a uniformly controlled function ρ+ : [0, ∞) → [0, ∞) with 0 0 0 max dY (gi(x), gi(x )) , dY ( fi(x), fi(x )) ≤ ρ+(dX(x, x )) 0 for every i ∈ I, x, x ∈ X, and a constant R ≥ 0 such that dY (gi(x), fi(x)) ≤ R for every i ∈ I, x ∈ X.

124 X Example 14.2.6. Given two controlled functions f, g ∈ Yctr, in order for the singleton {( f, g)} to be in F E it is necessary that f and g be frag-close. The converse is also true: for every E ∈ E the projection

π2(E) is a fragment of X and hence f × g(∆π2(E)) ∈ F . Since f × f (E) ∈ F by assumption, it follows that f × g(E) ⊆ ( f × f (E)) ◦ ( f × g(∆π2(E))) is in F . This show that there is a natural bijection between X MorFrCrs(X, Y) and the set of coarsely connected components of Y . Lemma 14.2.7. Given three coarse spaces X, Y, Z, the composition of functions induces a coarse map ZY × YX → ZX. Proof. Let X = (X, E), Y = (Y, F ) and Z = (Z, D). We need to check that the composition is a controlled Y X X F E map Zctr × Yctr → Zctr, i.e. that for every pair of controlled sets C1 ∈ D and C2 ∈ F the product E C1 ⊗ C2 is sent to a D -controlled set. E F Fix any E ∈ E. Then there exists F ∈ F with C2 ⊆ F , and there is a D ∈ D with C1 ⊆ D . The composition sends {( f1, f2), (g1, g2)} ∈ C1 ⊗ C2 to ( f1 ◦ f2, g1 ◦ g2) and we see that ( f1 ◦ f2) × (g1 ◦ g2) (E) = ( f1 × g1) ( f2 × g2)(E) ⊆ ( f1 × g1)(F) ⊆ D.

E That is, (◦ × ◦)(C1 ⊗ C2) ⊆ D . Since E is arbitrary, this concludes the proof. Remark 14.2.8. The following is a long heuristic remark providing motivation for the definition of F E. Informally, the space YX is designed to describe coarse properties of families of controlled functions. X The following example motivates why we define Y instead of simply using MorCrs(X, Y). For every t ∈ R let ft : R → R be the translation by t. For each fixed t, the function ft is close to the identity, and hence [ ft] = [id] ∈ MorCrs(R, R). That is, the set {[ ft] | t ∈ R} seen as a subset of MorCrs(R, R) is nothing but the singleton {idR}. This is rather unsatisfactory because the functions ft have larger and larger displacement as |t| grows: this non-trivial piece of information about the coarse geometry of the family of maps ft which is completely lost in MorCrs(R, R). The key point here is that each function ft is coarsely trivial, but the family of the ft’s is not. In order to describe ‘coarse properties’ of families of controlled functions it is ill advised to immediately quotient out close functions. Rather, it would be X 16 more appropriate to introduce some coarse structure on the set Yctr. The main difficulty is that coarse structures are not flexible enough to describe the current situation, hence we need to pass to fragmentary coarse spaces. The choice of F E might appear arbitrary at first sight, but it is actually natural. To begin with, note that for every F ∈ F the pairs of controlled functions f, g: X → Y such that ( f, g) ∈ F∆X are—by definition— X pairwise “uniformly close”. In other words, two families of controlled functions ( fi)i∈I, (gi)i∈I ⊂ Yctr ∆ such that ( fi, gi) ∈ F X for every i ∈ I should be “indistinguishable” from the coarse point of view. Thus ∆X X the relation F should belong to the (fragmentary) coarse structure that we wish to define on Yctr. A first X ∆X attempt would be to equip Yctr with the coarse structure generated by the sets F with F ∈ F . However, we will see that this is not quite the correct thing to do. One reason why F ∆X does not capture the essence of the “coarse geometry of functions” is that it behaves poorly under composition. Namely, say that we are given two “coarsely indistinguishable” families of controlled functions fi, gi : X → Y. If we are given another pair of “coarsely indistinguishable” 0 0 0 0 families of controlled functions fi , gi : Z → X, we would expect the compositions fi ◦ fi and gi ◦ gi to be “coarsely indistinguishable”. However, we see that the assumptions

∆X 0 0 ∆Z {( fi, gi) | i ∈ I} ∈ F and {( fi , gi ) | i ∈ I} ∈ E

0 0 ∆X are not enough to imply that {( fi ◦ fi , gi ◦ gi ) | i ∈ I} belongs F . For the latter to hold we need to E 0 0 ∆X assume that {( fi, gi) | i ∈ I} ∈ F where E ∈ E is a controlled set so that {( fi , gi ) | i ∈ I} ∈ E . So to 16By now it should be clear that coarse spaces can be seen as non-destructive quotients. Namely, when we equip X with a coarse structure E we are saying that close points should be thought of as being equal. However, we do not take the actual quotient by this equivalence relation, because it is useful to keep track of how unbounded families of points behave.

125 obtain a coarse geometric theory of controlled functions that is compatible with compositions we are forced to use the fragmentary coarse structures F E. The above also explains why we consider fragmentary coarse structures. The requirement that the theory be well behaved under composition implies that all the families of functions that we work with must be equicontrolled (Corollary 14.2.4). If the coarse space Y is unbounded, we may find a family of controlled functions fi : X → Y that are not equicontrolled and hence the set {( fi, fi) | i ∈ I} does E X not belong to F . That is, { fi | i ∈ I} is not a fragment of Y . Concretely, if X = [−1, 1] and Y = R are equipped with the metric coarse structure then the set of maps mt(x) B tx with t ∈ [0, ∞) is not a fragment of R[−1,1] because they are not uniformly Lipschitz. This shows that even well-behaved coarse spaces give rise to fragmentary coarse spaces. X There is a natural evaluation function ev: Yctr × X → Y defined by ev( f, x) B f (x). Furthermore, every function f : Z × X → Y admits a transposed function λ f : Z → Y X defined by λ f (z) B f (z, - ). Note that f = ev ◦ (λ f × idX). The following is the main result we prove about fragmentary coarse spaces of controlled functions. Once the formalism is set, the proof is actually rather simple: most of the ideas already appear in the examples above.

Theorem 14.2.9. Let X = (X, E), Y = (Y, F ), Z = (Z, D) be fragmentary coarse spaces. Then,

X E (1) the evaluation map ev: (Yctr, F ) × (X, E) → (Y, F ) is controlled; X E (2) a function h: (Z, D)×(X, E) → (Y, F ) is controlled if and only if its transpose λh: (Z, D) → (Yctr, F ) is controlled; ¯ X E ¯ (3) if h is controlled and h: (Z, D) → (Yctr, F ) is a function such that h is frag-close to ev ◦ (h × idX), then h¯ must be frag-close to λh (and it is hence controlled);

X (4) the transposition deﬁnes a canonical bijection MorFrCrs(Z × X, Y) −→ MorFrCrs(Z, Y ).

X Proof. (1) The product fragmentary coarse structure on Yctr × X is generated by controlled sets of the form C ⊗ E with C ∈ F E, E ∈ E. It is thus enough to verify that ev × ev(C ⊗ E) ∈ F . By the deﬁnition of F E, there exists F ∈ F such that f × g(E) ⊆ F for every ( f, g) ∈ C. On the other hand, we have [ ev × ev(C ⊗ E) = { f × g(E) | ( f, g) ∈ C} ⊆ F and so we are done.

(2) By construction, h = ev ◦ (λh × idX). If λh is controlled, then h is composition of controlled maps and hence controlled. Vice versa, assume that h is controlled, then for every E ∈ E we see: λh × λh(D) = (h(z1, -), h(z2, - )) | (z1, z2) ∈ D X ⊆ ( f, g) ∈ Yctr | f × g(E) ⊆ h × h(D ⊗ E) = (h × h(D ⊗ E))E.

E Since h is controlled, F B h × h(D ⊗ E) is in F and hence λh × λh(D) ∈ Fb . (3) Assume that ev ◦ (h¯ × idX) is frag-close to h, and ﬁx any E ∈ E. We can assume that π1(E) is a fragment of X. Let C be a fragment of Z, so that ∆C×π1(E) = ∆C ⊗ ∆π1(E) is in D ⊗ E. By hypothesis, there ¯ is an F ∈ F such that (ev ◦ (h × idX)) × h (∆C ⊗ ∆π1(E)) ⊆ F. Then, for every z ∈ C we have: (h¯(z) × λh(z))(E) = (h¯(z)(x1), h(z, x2)) | (x1, x2) ∈ E

⊆ F ◦ {(h(z, x1), h(z, x2)) | (x1, x2) ∈ E}

⊆ F ◦ (h × h(∆C ⊗ E))

126 E and hence h¯ × λh(∆C) ∈ F because h is controlled. (4) It follows from (3) that if h, h0 : (Z × X, D ⊗ E) → (Y, ⊗F ) are frag-close then λh and λh0 are frag-close. This implies that the transposition h 7→ λh deﬁnes a function

X λ: MorFrCrs(Z × X, Y) → MorFrCrs(Z, Y ). Since ev is controlled by (1), composition with ev preserves frag-closeness. We thus obtain a function in the other direction

X ev ◦ (- × idX): MorFrCrs(Z, Y ) → MorFrCrs(Z × X, Y).

By construction, λ and ev ◦ (- × idX) are inverse of one another (this is even true point-wise). Together with Lemma 2.6.3, item (2) of Theorem 14.2.9 has the following corollary.

Corollary 14.2.10. Fix a Z-indexed family of controlled functions z f :(X, E) → (Y, F ). Then: X E X 1. it is equicontrolled if and only if f : (Z, Emin) → (Yctr, F ) is controlled (i.e. the subset f (Z) ⊆ Yctr is a fragment);

X E 2. it is (Z, D)-equicontrolled if and only if f :(Z, D) → (Yctr, F ) is controlled. Convention 14.2.11. To keep true to our convention of using bold letters for frag-coarse maps, we will denote [λ f ] by λ f. A more pendantic approach would be to always see λ as a function λ: MorFrCrs(Z × X X, Y) → MorFrCrs(Z, Y ) and hence write λ(α) instead of λα. We ﬁnd that the latter approach would unneccessarily complicate the notation.

14.3 Coarse actions revisited Depending on the context, it may be more convenient to deﬁne set-group actions as functions G × Y → Y satisfying some commutative diagrams, or as homomorphism of G into the group of bijections G → Aut(Y). The frag-exponentials provide us with a language to make sense of the latter approach in the context of coarse actions. For later use, we record the following properties of λ. Lemma 14.3.1. The transposition satisﬁes the following identities. 1. Given frag-coarse maps f : A → B and g: B × X → Y, consider the composition

f×id g A × X X B × X Y. The following commutes: A λ(g◦( f×idX)) f B YX. λg

2. Given frag-coarse maps f : A × Y → Z and g: B × X → Y, consider the composition

id ×g f A × B × X A A × Y Z. The following commutes: λ f×λg A × B ZY × YX [◦] λ( f◦(idA×g)) ZX.

127 Proof. Both statements are true pointwise (i.e. the relevant equalities are alredsy true in Set).

Let X be any coarse space. By Lemma 14.2.7, the composition of functions deﬁnes a frag-coarse Y Y Y Y Y map [◦]: Y × Y → Y . Since ◦ is associative and idY ∈ Y is a unit, it follows that Y is a frag-coarse monoid (here we are using the obvious deﬁnition of monoid object in the category FragCrs). Recall that a homeomorphism between monoids is a map that sends the unit to the unit and commutes with the operation. It is then simple to prove the following.

Proposition 14.3.2. Let G be a coarse group. A coarse map α: G × Y → Y is a coarse action if and only if its transpose λα: G → YY is a coarse homomorphism of frag-coarse monoids.

Proof. By Theorem 14.2.9 we know that the transposition λ gives a bijection between MorCrs(G×G×Y, Y) Y and MorCrs(G × G, Y ). Therefore, the diagram

id ×α G × G × Y G G × Y

∗×idY α G × Y α Y commutes if and only if λ(α ◦ (idG × α)) = λ(α ◦ (∗ × idY)). By Lemma 14.3.1, this happens if and only if the diagram G × G λα×λα YY × YY ∗ [◦] G λα YY commutes. Similarly, we see that the diagram

(e,id ) Y Y G × Y α idY Y commutes if and only if λα(eG) = idY.

The above characterization of coarse actions is often useful. For example, to deﬁne coarse faithfulness:

Deﬁnition 14.3.3. A coarse action α: G y Y is coarsely faithful if λα: G → YY is proper (i.e. Ker(λα) = eG).

Example 14.3.4. The coarse action by left multiplication ∗: G y G of a coarse group G = (G, E) on itself is always faithful. In fact, let B ⊆ G be a subset so that (λ∗)(B) whose image is a bounded G E E neighbourhood of idG in (Gctr, E ). We may assume that idG ∈ B. By the definition of E it follows that there is a controlled set E such that (λ∗)(g) × idy({(e, e)}) ∈ E for every g ∈ B. In other words, g ∗ e ∈ E(e) for every g ∈ B. Since e is the coarse unit in G, it follows that B is a E-bounded neighbourhood of e. It is of course possible to find an equivalent formulation of Definition 14.3.3 without using λα. Namely, [α]: (G, E) y (Y, F ) is coarsely faithful if and only if every set B ⊆ G such that {α(B × {y}) | y ∈ Y} is a controlled partial covering of Y must be E-bounded. We find Definition 14.3.3 cleaner.

128 14.4 Frag-coarse groups of controlled transformations Proposition 14.3.2 falls short of the characterization of set-group actions as homomorphisms G → Aut(Y), in that we are not considering a coarse analog of the group Aut(Y) but only of the monoid YY = {functions from Y to itself}. More precisely, in Set it is trivially true that the subset of invertible functions Y → Y is itself a set-group Aut(Y) < YY and that for any set-group action α: G y Y the image of the homomorphism λα: G → YY is contained in Aut(G). In the context of (frag-)coarse spaces we lack a good analogue of Aut(Y) (as already explained, the set-group of coarse automorphisms AutCrs(Y) is ill-suited to describe coarse actions). In general, for any set-monoid M the subset of invertible elements M∗ ⊆ M is a set-group. Let Y = (Y, F ) be a frag-coarse space. Since YY is a frag-coarse monoid, it is tempting to look at the subset Y ∗ Y Y ∗ of frag-coarsely invertible functions (Yctr) ⊆ Yctr (i.e., (Yctr) , i.e., the set of frag-coarse equivalences Y ∗ (Y, F ) → (Y, F )). Clearly, (Yctr) is closed under composition and therefore it is a frag-coarse submonoid (YY)∗ ≤ YY. However, it is generally not a frag-coarse group because the operations need not be well-behaved with respect to the exponential frag-coarse structure. More precisely, there are two fundamental issues. The ﬁrst issue is that even if we assume that every X ∗ element f ∈ (Yctr) is coarsely invertible, we do not know whether we can choose an inversion function −1 Y ∗ Y ∗ (-) :(Yctr) → (Yctr) so that the (Inverse) diagram commutes up to frag-closeness:

−1 ([-] ,id Y ∗ ) (YY)∗ (Y ) (YY)∗ × (YY)∗

−1 idY (id(YY )∗ ,[-] ) [◦] (YY)∗ × (YY)∗ (YY)∗. [◦]

For instance, let Y = (N, Ed) and for each k ∈ N let fk(n) B n + k be a translation by k. The set of X ∗ −1 functions { fk | k ∈ N} is a fragment of (Y ) . However, there is no way to find coarse inverses fk such −1 −1 that the compositions fk ◦ fk and fk ◦ fk are uniformly close to idN. −1 Y ∗ Y ∗ The second issue is that even if there exists an inverse function (-) : (Yctr) → (Yctr) so that the diagram commutes, the function would not need to be controlled. To see this, let Y = (R, Ed) and let ft(x) B tx denote the multiplication by t. The set { ft | 0 < t ≤ 1} is a set of equicontrolled coarse Y ∗ −1 equivalences and it is hence a fragement of (Y ) . Still, the set of inverses { ft | 0 < t ≤ 1} = { ft−1 | 0 < t ≤ 1} does not consist of equicontrolled functions and it is hence not a fragment.17 We can go around these difficulties as follows. Let X = (X, E) and Y = (Y, F ) be two (frag-coarsely X Y equivalent) frag-coarse spaces, and define CE(X, Y) ⊆ Yctr × Xctr as the set X Y CE(X, Y) B ( f, g) | f ∈ Yctr, g ∈ Xctr frag-coarse inverses of one another . One advantage of considering such a set is that we have a natural candidate for the inversion function (-)−1: namely swapping the order of each pair. The composition ◦: CE(Y, Z) × CE(X, Y) → CE(X, Z) is defined componentwise. It now remains to define a frag-coarse structure such that these operations are controlled and make CE(Y, Y) into a frag-coarse group. Here we see a second advantage of working with pairs of functions. Since CE(X, Y) is a subset of the X Y E F product Yctr × Xctr, we can consider the restriction of the product frag-coarse structure F ⊗ E . When doing so, we automatically obtain that our candidate (-)−1 : CE(X, Y) → CE(Y, X) is a controlled function. We are still not done though. The product coarse structure does not allow us to control “how close to actual inverses” are the coarse inverse functions and hence (-)−1 need not make the (Inverse) diagram commute. Rather, we will use the following:

Deﬁnition 14.4.1. A relation C ⊆ CE(X, Y) × CE(X, Y) belongs to the frag-coarse structure CE(E, F ) if

17This example shows that Theorem 4.2.6 is not true for frag-coarse spaces.

129 (Ce1) C ∈ F E ⊗ EF ;

(Ce2) for every E ∈ E and F ∈ F there exist E0 ∈ E and F0 ∈ F such that

0 0 idX ×(g1 ◦ f1)(E) ⊆ E and idY ×( f2 ◦ g2)(F) ⊆ F .

The fragmentary coarse space of coarse equivalences is CE(X, Y) = (CE(X, Y), CE(E, F )). We shall denote CE(Y) B CE(Y, Y) and CE(Y) B CE(Y, Y)

We leave it to reader to verify that the above is a fragmentary coarse structure. Remark 14.4.2. Eﬀectively, to deﬁne CE(E, F ) we selected a subset of F E ⊗ EF in such a way that the fragments of CE(X, Y) are composed of pairs of functions ( f, g) such that f ◦ g and g ◦ f are uniformly close to idY and idX. The reader may verify that—thanks to condition (1)—condition (2) is equivalent to:

(Ce2’). Given fragments X0 ⊆ X and Y0 ⊆ Y there exist E ∈ E and F ∈ F such that

idX ×(g ◦ f )(∆X0 ) ⊆ E and idY ×( f ◦ g)(∆Y0 ) ⊆ F

for every ( f, g) ∈ π1(C) ∪ π2(C).

Now that we deﬁned the frag-coarse structure on CE(X, Y), it is routine to prove the following:

Lemma 14.4.3. The composition ◦: CE(Y, Z) × CE(X, Y) → CE(X, Z) and inversion (-)−1 : CE(X, Y) → CE(Y, X) are controlled maps. Conside also the identity element (idY , idY ) ∈ CE(Y), −1 then (CE(Y), [◦], [(idY , idY )], [-] ) is a frag-coarse group.

Since we defined the coarse structure CE(E, F ) as a refinement of F E ⊗ EF , it is clear that the projection on the first coordinate ( f, g) 7→ f defines a controlled map CE(X, Y) → YX. Notice that for X = Y this map is in fact a homomorphism of frag-coarse monoids. From here, it is easy to obtain the required characterization of coarse actions.

Proposition 14.4.4. Let G be a coarse group and Y a coarse space. A coarse map α: G × Y → Y is a coarse action if and only if λα: G → YY factors through a coarse homomorphism G → CE(Y).

Sketch of proof. Unpacking the notation, we have λα(g) = gα for every g ∈ G. Consider the map -α: G → CE(Y) sending g 7→ (gα, g−1 α). If α is a coarse action, it follows from Theorem 14.2.9 that F F -α: (G, E) → (CE(Y), F ⊗ F ) is controlled. The relation (-α × -α)(E) also satisfies condition (Ce2) of Definition 14.4.1 for every E ∈ E. Hence -α: (G, E) → (CE(Y), CE(F , F ) is controlled. It is also immediate to verify that -α defines a coarse homomorphism. The converse implication follows directly from Proposition 14.3.2.

Remark 14.4.5. Extending the deﬁntion of coarse action to frag-coarse groups, we see that Proposi- tion 14.4.4 holds true for frag-coarse actions of frag-coarse groups on frag-coarse spaces. As in Section 11, we say that a set-group equipped with a fragmentary coarse structure that makes it into a frag-coarse group18 is a frag-coarsiﬁed set-group. An interesting consequence of Proposition 14.4.4 is the following:

Theorem 14.4.6. Every coarse group is isomorphic to a frag-coarse subgroup of a frag-coarsiﬁed set-group.

18The failure of Theorem 4.2.6 for frag coarse spaces shows that condition is not equivalent to saying that the frag-coarse structure is equi-bi-invariant. That is, one needs to check it is equi-bi-invariant and that the inverse function is controlled.

130 Proof. Let G be a coarse group. By Theorem 4.4.6, G is isomorphic to a coarse group whose operations admit adapted representatives. Up to isomorphism, we may thus ﬁx a choice of such representatives ∗, e and (-)−1. Note that the set of bijective coarse equivalences

BijCE(G) B { f :(G, E) → (G, E) coarse equivalence | f bijective} is closed under inversion and composition. In particular, it is a set-group and the retriction of CE(E, E) makes it into a frag-coarse group. The action of G on itself by left multiplication is a coarse action and hence the map g 7→ g∗ defines a coarse homomorphism [-∗]: G → CE(G). Since we chose adapted representatives, we see that the image of G under -∗ is actually contained in BijCE(G). Since the action by left multiplication of a coarse group on itself is always faithful, [-∗] is a isomorphism between G and its coarse image. As we already pointed out, the definition of the exponential frag-coarse structure requires us to leave the category of coarse spaces. As a consequence, we see no obvious way to adapt the proof of the above theorem to upgrade the frag-coarsified set-group to a genuine coarsified set-group. More precisely, we do not know the answer to the following:

Question 14.4.7. Is every coarse group isomorphic to a coarse subgroup of a coarsiﬁed set-group?

The above is related to, but independent from, our conjecture that there exist coarse groups that are not isomorphic to any coarsified set-group. We do not yet have any intuition for what the answer to Question 14.4.7 should be (note that our candidate example for a coarse group that is not the coarsification of a group is defined as a coarse subgroup of (F2, E×)).

131 Part III Appendices

A Categorical aspects of Coarse

In this appendix we review some basic properties of the categories Coarse and FragCrs. We will recall most of the relevant categorical deﬁnitions, for more details and basic facts we refer to [ML13]. For the most part, the material concerning Coarse is not novel, we decided to include it to provide a somewhat complete treatment with uniﬁed conventions and notation. More details can be found e.g. [DZ17, DZ20, Luu07]. As far as we are aware, all the results concerning fragmentary coarse structures have not been considered anywhere else in the literature.

A.1 Limits and colimits

Taking the disconnected union shows that the category Coarse has arbitrary coproducts. Namely, if (Xi, Ei) is a family of coarse spaces over an index set I, then we obtain a coproduct by considering the disjoint union of the Xi with the coarse structure generated by the Ei: a G (Xi, Ei) B Xi , hEi | i ∈ Ii . i∈I i∈I F It is obvious that the inclusion Xi ,→ ∈ Xi is controlled for every i. Given controlled functions Fi I F fi : (Xi, Ei) → (Z, F ), their disjoint union fi : Xi → Z is controlled because it sends a generating family of controlled sets to controlled sets of Z. Note also that the coproduct of an empty family of coarse spaces is the empty coarse space, which is indeed an initial object in Coarse.

Warning. According to our conventions, in the next few paragraphs there will be two sets of bold symbols. Namely, we use upright bold characters for categories and italic bold characters for coarse spaces.

Given two morphisms f, g: X → Y in a category C, a coequalizer is an object Q together with a morphism coeq: Y → Q such that coeq ◦ f = coeq ◦ g and is universal with this property, i.e. every fZ : Y → Z such that fZ ◦ f = fZ ◦ g factors through Q:

f coeq X Y Q g . fZ Z

Lemma A.1.1. The category Coarse has coequalizers.

Sketch of proof. Given f, g: (X, E) → (Y, F ) controlled functions, let Q B Y equipped with the coarse structure D generated by F ∪ {( f × g)(∆X)}. Then the set-theoretic identity function id: Y → Q is such id that [id ◦ f ] = [id ◦g] and Y −→ Q is universal with this property.

Let J be a ﬁxed category. A diagram of shape J in a category C is a functor D: J → C. Given a diagram D: J → C, a cocone is an object A in C together with morphisms fJ : D(J) → A for every object

132 J of J and such that for every morphism j: J → J0 the following commutes:

D(J) fJ D( j) A.

f 0 D(J0) J

A colimit for a diagram D: J → C is an initial object in the category of cocones. That is, a colimit is a cocone given by an object lim D and morphisms p : D(J) → lim D such that for every other cocone A −−→ J −−→ there is an morphism f¯: lim D → A such that −−→

p D(J) J lim D −−→ f¯ fJ A commutes. Note that coproducts are colimits where the category J has no morphisms, and coequalizers are colimits where the category J consists of two objects and two parallel morphisms. A category is small if the collections of its objects and morphisms are sets. A category is small-cocomplete if every small diagram (i.e. a diagram of shape J for a small category J) has a colimit. In particular, a small-cocomplete category admits small-coproducts (i.e. coproducts of families of objects indexed over a set) and coequalizers. Vice versa, it is a standard fact that a category with small-coproducts and coequalizers is small-complete [ML13, Chapter V]. It is convenient to show this explicitly for the coarse category:

Lemma A.1.2. The category Coarse is small-cocomplete.

Sketch of proof. Let D be a small diagram in Coarse of shape J. For notational coherence, let (XJ, EJ) = 0 ` XJ B D(J) and for every morphism j: J → J let [ j] = j B D( j). Considering the coproduct XJ ` J we obtain natural inclusions [iJ]: XJ ,→ J XJ. To make the coproduct into a cocone it is necessary to 0 change the coarse structure so that [iJ] = [iJ0 ] ◦ [ j] for every j: J → J . This is readily done by equipping ` J XJ with the coarse structure D E lim E E | J object in J ∪ i × (i 0 ◦ j) | j morphism in J . −−→ J B J J J By definition, lim E is the minimal coarse structure making F X into a cocone and hence lim X −−→ J J J −−→ J B (F X , lim E ) is a colimit. J J −−→ J We now turn our attention to the dual notions of products, equalizers and limits. A positive result is that Coarse has arbitrary (small) products. Q Lemma A.1.3. Every set of coarse spaces Xi, i ∈ I has a product i∈I Xi in Coarse. Q Sketch of proof. Let Xi = (Xi, Ei) and consider the cartesian product ∈ Xi. Define the coarse structure N Q Q i I i∈I Ei containing a set D ⊆ ( i∈I Xi) × ( i∈I Xi) if and only if π j × π j(D) ∈ E j for every j ∈ I, where π j is the projection to X j. N Q It is immediate to check that i∈I Ei is indeed a coarse structure. The projections π j : i∈I Xi → X j are controlled by definition and it is also immediate to check that, given controlled functions fi : Z → Xi, Q Q the product i∈I fi : Z → i∈I Xi is controlled.

133 The situation is diﬀerent for equalizers. An equalizer of two morphisms f, g: X → Y in a category C, is an object S together with a morphism eq: S → X such that f ◦ eq = g ◦ eq and is universal with this property: eq f S X Y g

fZ Z

Lemma A.1.4. In Coarse, equalizers do not necessarily exist.

E → E Sketch of proof. Consider idN, 0: (N, min) (N, d), where idN is the (set theoretic) identity function, E ¯ { } E 0 is the constant function 0, min = ∆N is the minimal coarse structure and d is the coarse structure ¯ → E coming from the natural metric. For every coarse space Z and morphism fZ : Z (N, min) we have that [idN ◦ fZ] = [0 ◦ fZ] if and only if fZ(Z) is ﬁnite. Yet, since | fZ(Z)| can be arbitrarily large, there ¯ cannot be a eq: X → (N, E ) such that [idN ◦eq] = [0 ◦ eq] and is universal i.e. so that every [ fZ] with min ¯ [idN ◦ fZ] = [0 ◦ fZ] factors through [eq]. ¯ Since equalizers are limits (where limits are the dual notion of colimits), we have in particular that Coarse does not have limits and is hence not (small)-complete. Further, equalizers are also special examples of ‘pushouts’ (see any category book for a deﬁnition). We record these results in the following:

Corollary A.1.5. The category Coarse does not generally have (ﬁnite) limits nor pushouts. In particular, it is not complete.

Remark A.1.6. The absence of equalizers in Coarse is tied to the fact that it is not possible to deﬁne a notion of ‘intersection of coarse subspaces’ that is both general and well behaved. We preferred to use a deﬁnition that is very well-behaved but need not exist in general (Subsection 3.5).

A.2 The category of fragmentary coarse spaces The study of spaces of controlled maps naturally led us to introduce a weakening of the notion of coarse structure and to define fragmentary coarse spaces (Section 14). Here we explain more in detail some of the properties of the category of fragmentary coarse spaces. Recall that a fragmentary coarse structure on X is a set of relations E satisfying all the requirements of a coarse structure, except that it may not contain the diagonal. A fragment of X is a subset Z such that ∆Z ∈ E (by assumption, every singleton of X is a fragment). Controlled functions are defined as usual. On the other hand, the notion closeness is replaced by that of frag-closeness (two functions are frag-close if their restrictions to every fragment are close).19 The category FragCrs has frag-coarse spaces as objects and frag-closeness equivalence classes of controlled maps as morphisms. As we already remarked, various of the concepts and definitions of Section2 translate without difficulty to frag-coarse spaces. One exception is the relation between controlled maps of binary products and equicontrolledness (see the discussion following Corollary 14.1.9). Other exceptions are coarse embeddings and coarsely surjective maps: this is because we wish to preserve their relation with monomorphisms and epimorphisms. Namely, it makes sense to say that f : (X, E) → (Y, F ) is a coarse embedding if ( f × f )−1(F) ∈ E for every F ∈ F , even if X and Y are frag-coarse spaces. However, in this case Lemma 2.4.8 is no longer true (the same happens for coarse surjectivity). In view of this, it is advisable to give the following:

19More generally, we say that any property P whose definition requires considering the diagonal relation on X has a fragmentary analogue frag-P which is obtained by requiring that every fragment of X satisfies P. We already provided a few examples with precise definitions in Section 14.

134 Deﬁnition A.2.1. A controlled map between frag-coarse spaces f : (X, E) → (Y, F ) is a frag-coarse embedding if its restriction to every fragment of X is a coarse embedding. It is frag-coarsely surjective if for every fragment Y0 ⊆ Y there exists a fragment X0 ⊆ X and F ∈ F so that Y0 ⊆ F( f (X0)).

Of course, these notions are invariant under taking frag-close maps and they are hence properties of frag-coarse maps. Using this deﬁnition, it is not hard to adapt the proofs of Lemma 2.4.8, Lemma 2.3.9 and Lemma 3.1.1 to show that their frag-coarse analogs hold true. We thus deduce the following:

Proposition A.2.2. Let f : X → Y be a frag-coarse map between frag-coarse spaces. Then

(i) f is a monomorphism if and only if it is a frag-coarse embedding,

(ii) f is an epimorphism if and only if it is frag-coarsely surjective,

(iii) f is a frag-coarse equivalence if and only if it is a frag-coarsely surjective coarse embedding.

In particular, FragCrs is a balanced category (i.e. every morphism that is both monic and epic is an isomorphism).

With regards to limits and colimits, much of the material in Subsection A.1 generalizes to FragCrs. If (Xi, Ei) is a family of frag-coarse spaces indexed over some set I their coproduct is the fragmented union a G (Xi, Ei) B Xi , hEi | i ∈ Iifrag , i∈I i∈I where h - ifrag denotes the generated frag-coarse structure (it diﬀers from the generated coarse structure in that we do not add the diagonal. In particular, each fragment of the coproduct is completely contained in a unique Xi). In FragCrs, the coequalizer of two controlled functions f, g: (X, E) → (Y, F ) between frag-coarse spaces can be deﬁned equipping Q B Y with the fragmentary coarse structure D generated by F ∪ {( f × g)(∆Z) | Z fragment of X} and considering the set-theoretic identity function idY : Y → Q. Thus FragCrs is small-cocomplete. With regard to limits, the same construction of Coarse shows that FragCrs has products. In particular, if Xi is a family of coarse spaces then their product in Coarse is also a product in FragCrs (as we saw, this is not the case for coproducts). Unlike Coarse, the category FragCrs also has equalizers:

Lemma A.2.3. The category FragCrs has equalizers

Sketch of proof. Let [ f ], [g]: (X, E) → (Y, F ) be frag-coarse maps of frag-coarse spaces. For every controlled set F ∈ F let CF B {x ∈ X | ( f (x), g(x)) ∈ F} ⊆ X. Consider the union [ S B CF ⊆ X F∈F and equip it with the fragmentary coarse structure C B hE|CF | F ∈ F ifrag. We claim that the inclusion ι: (S, C) ,→ (X, E) is an equalizer. It is clear that [ f ◦ ι] = [g ◦ ι]. If fZ : (Z, D) → (X, E) is a controlled map with [ f ◦ fZ] = [g ◦ fZ], we must have that fZ(Z) ⊆ S . This is because each fragment of Z must be sent into CF for some large enough F ∈ F , and Z is equal to the union of its fragments. We now see that fZ : (Z, D) → (S, C) is controlled because for every D ∈ D the 0 union of the projections Z B π1(D) ∪ π2(D) is a fragment of Z and hence fZ × fZ(D) ∈ E| f (Z0) ⊆ C. Corollary A.2.4. The category FragCrs is small-complete.

The main result of Subsection 14.2 is a theorem on the properties of exponential frag-coarse spaces (Theorem 14.2.9). The language and conventions used there are motivated by the categorical deﬁnition of exponential object:

135 Deﬁnition A.2.5. Let X and Y be objects of a category C in which all binary products with Y exist. Then an exponential object is an object XY in C equipped with an evaluation morphism ev: XY × Y → X which is universal in the sense that, given any object Z and morphism f : Z × Y → X, there exists a unique morphism u: Z → XY such that the composition

u×id ev Z × Y −−−−→Y XY × Y −−→ X equals f . The morphism u is denoted by λ f and is called the transpose of f .

If it exists, an exponential object is of course unique up to isomorphism. The main content of Theorem 14.2.9 can be rephrased as follows:

Theorem A.2.6. In FragCrs every pair X and Y has an exponential object YX.

This implies that FragCrs is a particularly ‘nice’ category. More precisely, recall the following:

Deﬁnition A.2.7. A category is Cartesian closed if it contains a terminal object, it has all binary products, and the exponential of any two objects.

Since FragCrs has all ﬁnite products, Theorem A.2.6 implies that it is Cartesian closed:

Corollary A.2.8. The category FragCrs is Cartesian closed.

A.3 Enriched coarse categories Informally, an enirched category is a category where the sets of morphisms have some additional structure. A prototypical example is that of the category Vect of vector spaces over some fixed field, as the set of linear maps between two vector spaces is itself a vector space. In other words, Vect is enriched over itself. It is a general fact that a Cartesian closed category is enriched over itself (see [Kel82, Section 1.6]), therefore FragCrs is enriched just as Vect is. Further, it is easy to see Coarse can be enriched over FragCrs. Rather than giving a precise definition of enriched category, we will content ourselves with a hands-on description of the extra structure on the sets of morphisms of Coarse. The reader interested in the abstract formalism may refer [Kel82]. As mentioned above, the ‘enriched’ structure of Vect is easy to describe: if V and W are two vector spaces it is easy to define operations on the set of linear maps L(V, W) = MorVect(V, W) that make it into a vector space. The enriched structure of Coarse is more subtle, because we cannot define a good X (frag-)coarse structure on the set MorCrs(X, Y). Instead, we consider the frag-coarse space Y . More precisely, we define an enriched version of the coarse category Coarseen as the category having as X objects coarse spaces and so that the “morphisms” MorCoarseen (X, Y) are the frag-coarse space Y . The composition in Coarseen is defined by the composition morphism [◦]: ZY × YX → ZX. This structure makes Coarseen into a category enriched over FragCrs (or FragCrs-category). We cannot talk about morphisms in Coarseen, because an enriched category is not really a category: the morphisms are not a ‘set’, but an object in a different category (in this case FragCrs). In this setting, the correct terminology is that YX is the hom-object from X to Y. In particular, one should not think of an X element f ∈ Y as a morphism in MorCoarseen (X, Y). The correct way to recover the original category Coarse from its enriched version Coarseen is by noting that the sets of morphism in Coarse can be identified with frag-coarse maps from the terminal object to the hom-object:

X MorCrs(X, Y) MorFrCrs({pt}, Y ).

Of course, the same construction can be performed to deﬁne the category FragCrsen enriched over FragCrs (this is the standard way of enriching a cartesian closed category over itself).

136 A.4 The pre-coarse categories At times it may be useful to refrain from identifying close functions. We conclude this categorical appendix by noting that the categories whose morphisms are controlled maps (as opposed to equivalence classes thereof) have a natural enriched structure.

Deﬁnition A.4.1. The category of precoarse spaces PreCoarse (resp. of prefrag-coarse spaces Pre- FragCrs) has coarse spaces (resp. frag-coarse spaces) as objects and controlled maps as morphisms.

Remark A.4.2. By definition, we have a functor PreCoarse → Coarse preserving each coarse space and sending a controlled function f to its equivalence class f = [ f ]. The same is of course true for PreFragCrs and FragCrs. As in Subsection A.3, also PreCoarse and PreFragCrs can be enriched over PreFragCrs. In this case the situation is even simpler because we are not identifying close maps. Namely, given coarse X spaces X = (X, E) and Y = (Y, F ), the set of morphisms MorPreCoarse(X, Y) equal to Yctr by definition. We can hence direcly equip it with the frag-coarse structure F E. Namely, we are entitled to identify X MorPreCoarse(X, Y) with Y . Unlike for Coarse, here it is legitimate to say that an element f ∈ YX is a morphism. If one wanted to X be formal, this is the case because f can be uniquely identified with the morphism MorPreFragCrs({pt}, Y ) sending pt to f .

B Metric Groups

As already remarked, the prototypical example of a coarse space is obtained from a metric space (X, d) by considering the coarse structure Ed induced from the metric. It is hence natural to try to construct coarse groups by considering ‘metric groups’ with their induced coarse structure, where a metric group is a group object in the category of metric spaces. Interestingly, there is more than one choice for the ‘category of metric spaces’. In fact, even if it is clear what the objects should be, the choice of which morphisms to consider can vary accordingly to one’s necessity. In this section we will consider two such choices and characterise the group objects for both.

B.1 Groups in the 1-LipMet category The category 1-LipMet is the category whose objects are metric spaces and whose morphisms are 20 1-Lipschitz maps. Given two metric spaces (X, dX) and (Y, dY ), the p-product metric on the cartesian product X × Y is the metric given by

0 0 0 p 0 p 1 dp((x, y), (x , y )) B (dX(x, x ) + dY (y, y ) ) p .

0 0 0 0 The ∞-product metric is the maximum d∞((x, y), (x , y )) B max(dX(x, x ), dY (y, y )). It follows imediately from the triangle inequality that (X × Y, d∞) is a product of (X, dX) and (Y, dY ) in 1-LipMet. On the other hand, if neither X nor Y consist of a single point, then (X × Y, dp) with p , ∞ is not a product because the projections from (X × Y, d∞) to X and Y do not lift to a 1-Lipschitz map (X × Y, d∞) → (X × Y, dp). The metric space consisting of a single point is a terminal object in 1-LipMet. Since 1-LipMet has binary products and terminal object, we can study its group objects (i.e. metric spaces with morphisms satisfying the diagrams (♥) of Deﬁnition 4.1.2).

20From a categorical point of view it is very natural to restrict ones attention to 1-Lipschitz maps. In fact, these are the morphisms that arise naturally by considering metric spaces as enriched categories [Cite Lawvere].

137 Recall that an ultrametric on a set X is a metric d that satisfies the following strengthened triangle inequality: d(x, z) ≤ max{d(x, y), d(y, z)} for every x, y, z ∈ X. Proposition B.1.1. A group object in 1-LipMet is a set-group equipped with a bi-invariant ultrametric. Proof. Functions between metric spaces define the same morphism if and only if they coincide pointwise. In particular, functions ∗, (-)−1, e making (G, d) into a metric group must make it into a set-group as well. That is, the problem of characterising group objects in 1-LipMet is equivalent to the problem of characterising the set of metrics making a set-group G into a group object. Let (G, ∗, e, (-)−1) be a set-group. A metric d on G makes it into a group object if and only if it makes ∗ and (-)−1 into 1-Lipschitz maps. For ∗ to be 1-Lipschitz it is certainly necessary that both the left multiplication g∗: G → G and the right multiplication ∗g : G → G be 1-Lipschitz. Since their inverses −1 −1 ∗g = ∗g−1 and g∗ = g−1 ∗ are also 1-Lipschitz, we deduce that the metric d must be bi-invariant. Note that bi-invariance of d immediately implies that (-)−1 is an isometry. We claim that d must also be an ultrametric. Let |g| B d(g, e) = d(e, g) and note that |g| = |g−1| for every g ∈ G because d is bi-invariant. By definition we have d∞ (e, e), (g, h) = max{d(e, g), d(e, h)} = max{|g|, |h|}. If ∗ is 1-Lipschitz it follows that for every g, h, k ∈ G we have

−1 −1 −1 −1 −1 d(g, h) = |g h| ≤ d∞ (e, e), (g k, k h) = max |g k|, |k h| = max{d(g, k), d(k, h)}. Vice versa, it is immediate to check that any bi-invariant ultrametric makes ∗ 1-Lipschitz and hence (G, d, ∗, e, (-)−1) is a group object.

B.2 Groups in the LipMet category The category LipMet is the category whose objects are metric spaces and whose morphisms are all the Lipschitz maps. Again, the one point metric spaces is a terminal object and the ∞-product metric makes X × Y into a product. In particular, the inclusion of 1-LipMet into LipMet preserves group objects. That is, every group with a bi-invariant ultrametric is a group object in LipMet. Yet, there are many more group objects that are not of this form. As for Proposition B.1.1, group objects in LipMet must be set-groups. To characterise the metrics making an abstract group into a group object we need to introduce a piece of terminology. A family of maps fi :(X, dX) → (Y, dY ) is equi-Lipschitz if there is a constant L such that all the fi are L-Lipschitz. Proposition B.2.1. A group object in 1-LipMet is a group equipped with a metric so that both the left and the right multiplication are families of equi-Lipschitz maps. −1 Proof. Let (G, ∗, e, (-) ) be a group and d a metric on G. If ∗ is L-Lipschitz, then so are g∗ and ∗g for all g ∈ G. Vice versa, if g∗ and ∗g are equi-Lipschitz, say with Lipschitz constant L, then we have 0 0 0 0 0 0 0 0 0 0 0 0 d(gh, g h ) ≤ d(gh, gh ) + d(gh , g h ) ≤ L(d(h, h ) + d(g, g )) = Ld1 (g, h), (g h ) ≤ 2Ld∞ (g, h), (g h ) . The fact that also (-)−1 is Lipschitz is immediate. Corollary B.2.2. Every group with a bi-invariant metric is a group object in LipMet. There is a clear analogy between Proposition B.2.1 (and its proof) and Lemma 2.6.3. It is also easy to see that Proposition B.2.1 provides an alternative proof for Corollary 4.2.9. In fact, there is a natural functor sending a metric space (X, d) to the coarse space (X, Ed). This functor preserves products, and hence group objects.

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143 Index

(X, E)-equicontrolled, 123 coarse orbit, 68 coarse structures, 66 I-indexed partial covering, 31 coarse preimage, 36 control function, 24 R-quasimorphism, 53 coarse quotient, 34 controlled ∞-metric, 22 coarse quotient group, 61 indexed partial covering, ∞-product metric, 137 coarse retraction, 26 31 p-product metric, 137 coarse space, 21 controlled map, 24 coarsely connected, 21 controlled set, 21 action by conjugation, 66 coarsely connected controlled thickening, 31 adapted functions, 45 component, 21 controlledly proper, 25 asymptotic, 31 coarse structure, 21 covering, 20 generated, 22 refinement, 20 bornological compact, 22 controlled, 23 function, 117 compact fibre, 22 minimal, 20 bornologically proper action, connected, 21 partial, 20 117 finite off-diagonal, 21 star, 20 bornology, 117 maximal, 21 bounded minimal, 21 defect of quasimorphism, 53 subset, 22 coarse subgroup, 59 diagram of shape J, 132 bounded neighbourhoods of coarse subspace, 33 disconnected union, 23 the identity, 48 coarsely G-invariant, 66 cancellation length, 42 coarsely connected entourage, 21 cancellation metric, 42 components, 38 epic, 30 cancellative, 46 coarsely dense, 25 epimorphism, 30 Cartesian closed, 136 coarsely equivalent, 25 equalizer, 134 category of coarse groups, 54 coarsely equivariant map, 65 equi-asymptotic, 32 category of coarse spaces, 26 coarsely faithful, 128 equi-bi-invariant, 39 close function, 24 coarsely geodesic, 72 equi-coarsely contained, 32 Coarse abelianization, 67 coarsely invariant under equi-left-invariant, 39 coarse action, 63 conjugation, 67 equi-Lipschitz, 138 coarse centralizer, 67 coarsely Lipschitz, 24 equi-right-invariant, 39 coarse containment, 31 coarsely multi-associative, equicontrolled map, 29 subspaces, 35 103 equivalent monics, 33 coarse coset space, 75 coarsely normal, 67 evaluation function, 126 coarse embedding, 25 coarsely surjective, 25 exponential fragmentary coarse equivalence, 25 coarsified set-group, 88 coarse structure, coarse equivariant relations, cobounded 124 73 coarse action, 119 exponential object, 136 coarse exhaustion, 85 cobounded coarse action, 70 coarse group, 37 cocomplete category, 133 factors through, monics, 33 coarse homomorphism, 52 cocone, 132 frag-coarse embedding, 135 coarse image, 33 coequalizer, 132 frag-coarsely surjective, 135 coarse intersection, 35 colimit, 133 frag-bounded, 122 coarse inverse, 25 complete category, 133 frag-coarsified set-group, 130 left, 25 completely fragmented fragment of a coarse space, right, 25 fragmentary coarse 121 coarse kernel, 80 structure, 121 fragmentary close functions, coarse map, 26 conjugate equi-left-invariant 121

144 fragmentary coarse minimal left-equivariant quotient coarse action, 73 equivalence, 122 group coarse quotient object, 34 fragmentary coarse map, 122 structure, 45 quotient of coarse group by fragmentary coarse space, monic, 30 relations, 62 121 monomorphism, 30 fragmentary coarse space of move section, 20 coarse equivalences, addition, 99 composition, 20 130 cancellation, 99 set-groups, 37 fragmentary coarse structure, small category, 133 normalized, 67 121 subobject, 33 normally generates, 42 fragmented union, 135 subspace coarse structure, 25 HS-quasimorphism, 109 orbit map, 68 subword, 99 equivalent, 109 ending, 99 precoarse category, 137 proper, 99 intrinsically bounded, 90 prefrag-coarse category, 137 starting, 99 inversion function, 38 pro-Q coarse structure, 94 isomorphic coarse actions, 65 product coarse structure, 28 topologically proper action, length space, 72 product of coarse subspaces, 119 82 transposed function, 126 metrically-proper coarse proper, 25 trivial coarse action, 63 action, 71 function, 117 trivially coarse space, 27 metrizeable coarse structure, proper coarse action, 71 90 pull-back of coarse structure, ultrametric, 138 minimal group coarse 30 structure, 43 unit, 38 minimal group coarse quasi-isometric embedding, structure containing 25 word distance, 42 R, 44 quasi-isometry, 25 word length, 42

145 List of Symbols diag(a) block diagonal, page 21 S St(A, b) B {B ∈ b | B ∩ A , ∅} star of A with respect to b, page 20 G/A coarse coset space, page 75

Eas associative relation, page 43

Ue(E) bounded neighborhoods of the identity, page 48

Eid identity relation, page 43

Einv inverse relation, page 43

ﬁb Ecpt compact ﬁber coarse structure, page 22

Ecpt compact coarse structure, page 22

| - |× cancellation length, page 42

E ⊗ F product coarse structure, page 28

E-bounded bounded with respect to the coarse structure E, page 22

E, F relations on X, page 20

mod Eprop coarse structure whose controlled/bounded sets are described by the property “prop”, possibly with the modiﬁer “mod”., page 22

Ed induced coarse structure on a metric space (X, d), page 21

EQ pro-Q coarse structure, page 94

asymptotic, page 31

I equi-asymptotic, page 32

E T F B {F E | E ∈ E} exponential fragmentary coarse structure, page 124

HomCrs(G, H) set of coarse homomorphisms between G and H , page 52

−1 −1 [-]G , [-] inverse function in a group, page 38 Ker( f) coarse kernel, page 80

α: G y Y,[α]:(G, E) y (Y, F ) coarse action, page 63 f g multiplication in MorCrs(X, G), page 38

G/hhRii B (G, E(R)) quotient of coarse group by relations, page 62 G · y coarse orbit, page 68

X/R coarse quotient by relations, page 34

Y1 ∩ · · · ∩ Yn coarse intersection, page 35

146 CE(X, Y) fragmentary coarse space of coarse equivalences, page 130

A ∗ B product of coarse subspaces, page 82 f −1(Z) coarse preimage, page 36 f1 × f2 product of coarse maps, page 37

H E G coarsely normal subgroup, page 67 4 coarse containment, page 31

4I equi-coarsely contained, page 32 eG, e unit in a coarse group, page 38

∆X diagonal, page 21

MorFrCrs(X, Y) fragmentary coarse maps between X and Y, page 122 a, b families of subsets of X, page 20

C(E) family of controlled partial coverings, page 23

CI(E) I-indexed controlled partial coverings for a coarse structure E on X, page 31 idX identity morphism, page 26

Emax maximal coarse structure, page 21

Emin minimal coarse structure, page 21 Egrp min minimal group coarse structure, page 43 Eop B {(y, x) | (x, y) ∈ E} symmetric relation, page 20 red(w) reduced word, page 98

St(a, b) B {St(A, b) | A ∈ a} star of a with respect to b, page 20 grp ER minimal group coarse structure containing R, page 44

left ER minimal left-equivariant group coarse structure, page 45

Eτ equi-bi-invariant coarse group structure on Z with topology τ, page 93 E fin finite off-diagonal coarse structure, page 21 E ⊗ F product of relations, page 28

E(Z) relation on subset Z ⊂ Y, page 20 f ≈ g close functions, page 24 f ∗(F ) pull back of coarse structure, page 30

FE relation on controlled functions, page 123 g - E → hg is related to h by E, page 43

147 Coarse category of coarse spaces, page 26

CrsGroup category of coarse groups, page 54

FragCrs category of fragmentary coarse spaces, page 122

148