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An Invitation to Coarse Groups

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An Invitation to Coarse Groups 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. Version 0.0 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)i2I 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 2 I. When this is the case, any two sequences (xi)i2I and (yi)i2I with xi; yi 2 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 2 R so that dX( f1(z); f2(z)) ≤ C for all z 2 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; 1) ! [0; 1) 0 0 0 such that dY ( f (x); f (x )) ≤ ρ dX(x; x ) for every x; x 2 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
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