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Home , Burnside problem, Diffeomorphism, Essentially unique, Finite group, Group action, Group theory

The structure of and diffeomorphism groups

Kathryn Mann ∗

My favorite opening to a math talk is from a 2014 fascinating world of the , especially the ho- lecture of Etienne Ghys. Part of a minicourse for motopy type, of such groups. I recommend Hatcher’s young researches in geometric , he be- A 50-year view of diffeomorphism groups [Hat12] as gins the lecture with “My second favorite group [dra- an introduction to what is known, and not known in matic pause...] is the group of all diffeomorphisms this direction. of a compact .” Beyond the obvious ques- tion what is your first favorite, then? (presumably this is answered by the rest of the lecture ) Why study diffeomorphism groups? this has another hook that I like even better. We understand mathematical objects by understand- Ghys’ statement is a bit like answering the question ing their – this is the essence of Felix “what is your favorite food” with “my favorite food Klein’s and a guiding principle in is dessert!” That’s a great response, I couldn’t agree many . Following this principle, more, but didn’t you just cheat there by naming an anyone who studies topological or smooth infinite of things in place of a single thing? For it should seek to understand the groups Homeo(M) has been known since the 1980s that varying the man- and Diff(M) of self- or, respectively, ifold M and even varying what you mean by diffeo- self-diffeomorphisms of a manifold M. Topologists (smooth, C1, C2,...) produces an infinite are also interested in these groups for many other family of pairwise non-isomorphic groups. In other reasons, including the role they play in surgery the- words, manifolds are completely classified by the al- ory, K-theory, and the theory of fiber bundles. To gebraic structure of their diffeomorphism groups. give a concrete and relatively elementary example, However, to Ghys’ credit, many of the tools we the of Homeo(M) and Diff(M) have to approach the study of these groups are broad (that is to say the cohomology of their classifying principles that can be applied to large classes of ex- ) gives the classes of topologi- amples. In fact, several structural results about dif- cal and smooth M-bundles, respectively, while their feomorphism groups also hold for groups of homeo- cohomology as discrete groups gives characteristic , although the absence of differentiability classes for flat bundles, or bundles with a typically necessitates a different toolkit for the proof. transverse to the fibers. Thus, both the My goal here is to give you a quick tour of what type and the of these groups play we know of the structure theory of homeomorphism a crucial role in classifying fiber bundles over an ar- and diffeomorphism groups and highlight a few re- bitrary base . cent developments both in this theory and in its re- Another source of motivation for the study of dif- lationship with the modern study of dynamical sys- feomorphism groups comes from dynamical systems. tems. To keep this piece within a reasonable scope, In its most general sense, dynamics is simply the I have chosen to focus on the algebraic and dynam- study of transformations under iteration. Classically, ical structural properties, leaving absent the equally dynamicists studied the behavior of a single transfor- ∗Kathryn Mann is an assistant professor of mathematics at mation and its iterates, that is to say a cyclic sub- Cornell University. Her email address is [email protected]. group of Homeo(M) or of Diff(M). In the past fifty

1 years, the field has widened to encompass the study of larger systems of transformations obeying some al- gebraic laws, i.e. very general (typically infinite but finitely generated) of homeomorphism or diffeomorphism groups. A major motivating ques- tion is to understand to what extent the algebraic structure of a G ⊂ Homeo(M) influences or constrains the possible dynamics of actions of G Figure 1: A local chart for Diff(M) to the space of on M. In this way we can start to build a dictio- vector fields on M nary between algebraic properties such as nilpotency, , subgroup distortion, amenability and so on, tion has only recently been answered if you replace and dynamical properties such as entropy, existence Homeo(M) with Diff(M); this is the solution to Zim- of fixed points, or stability under perturbation. mer’s of Brown, Hurtado and Fisher as ex- To give one further perspective, one can also (some- plained in Fisher’s recent article in this series [Fis20]. what tautologically) view the groups Homeo(M) and Diff(M) as examples of groups of a lo- A helpful organizing principle is to think of cally – namely, the manifold M homeomorphism and diffeomorphism groups as loose itself, with its topological or smooth structure. But analogs of simple Lie groups. The group Diff(M) these groups also act transitively on many natural ob- is in fact, in a precise sense, an infinite-dimensional jects associated with M, especially in the case where with a Frechet manifold structure, locally M is connected. These include the space of labeled modeled on the infinite-dimensional of or unlabeled n-tuples of points in the case where the smooth vector fields on M. A local chart from a manifold has at least two, the space of or- neighborhood of the identity in Diff(M) to the space dered tuples in the 1-dimensional case, spaces of em- of vector fields is given by mapping a diffeomorphism −1 bedded discs, of simple closed (non-separating g to the vector field X(p) := expp g(p) where expp is simple closed curves when M is a ), and so the Riemannian exponential on a neighborhood on. As such, they are also groups of of the identity in the Tp(M). Charts of the configuration spaces of points, curves, discs, based at other points than the identity may etc. The standard compact-open or C∞ topology be obtained by composing with left multiplication. makes Homeo(M) and Diff(M) examples of Polish This gives Diff(M) a smooth structure for which left (completely metrizable) spaces, fostering some recent multiplication and inverses are smooth maps, and its interactions with descriptive theory, which has a “Lie ” is the of vector fields on M. rich toolkit to study Polish groups and automorphism However, the local chart above is not the usual groups of a wide class of structures. Lie algebra exponential map! In fact, the Lie alge- bra exponential – assigning to a vector field the time A guiding principle: analogies with one map of the flow generated by this vector field – simple Lie groups is not surjective onto any neighborhood of the iden- tity, even in the simple case where M = S1. (It is The main barrier to understanding the groups a pleasant exercise to identify some such diffeomor- Homeo(M) and Diff(M) is the fact that these groups phisms that are not the time one map of any flow. In are just intractably huge, even when M itself is low- the case these are diffeomorphisms with some, dimensional or otherwise of low complexity. For in- but not all, points being periodic with finite period stance, fix some manifold M of dimension at least greater than one. See also Milnor’s wonderful survey 2. It is an open question whether there exists a [Mil84].) For this reason, the Lie group structure on finitely generated, torsion- that is not iso- Diff(M) is – perhaps counterintuitively – not usually morphic to some subgroup of Homeo(M). This ques- the source of analogy with finite dimensional sim-

2 ple Lie groups. Additionally, the group Homeo(M) in Diffc(M) and let B ⊂ M be an open . Then has no natural smooth structure or Lie algebra, but any in Diffc(B) lies in the normal clo- many of the known parallels between diffeomorphism sure of g. The same holds replacing Diff with Homeo. groups and simple Lie groups also hold here. Rather, I believe that the important feature shared by all of While I have deliberately stated these in parallel these groups is the fact that they are the automor- for Diff and Homeo, the proofs of 1 and 2 are entirely phism groups of highly homogeneous structures. different in the smooth and C0 case. Fragmentation The remainder of this article is organized to il- is easy for diffeomorphisms: an element of Diffc(M) lustrate specific instances of this analogy between is always the time one map of a compactly supported simple (and occasionally semisimple) Lie groups and time dependent vector field, and cutting that vector diffeomorphism groups, beginning with simplicity it- field off by smooth bump functions is the only tool re- self. To keep things within a reasonable scope, I quired for fragmentation. No such strategy works for have chosen to focus primarily on the identity com- homeomorphisms, rather, the proof is a deep result of ponents Homeo0(M) and Diff0(M) of the homeo- Edwards and Kirby [EK71] using the same machinery morphism and diffeomorphism groups of a mani- that goes into proving the . fold M, as discussion of the mapping class groups Similarly, perfectness of Homeoc(B) is an old trick Homeo(M)/ Homeo0(M) (and analogously for Diff) (perhaps first appearing in this form in a paper of would take us much further afield. R.D Anderson [And58]) but for Diffc(B) it is a hard theorem of Thurston [Thu74a] tied to the cohomology Simplicity of classifying spaces, and it uses results of Herman for diffeomorphisms of the circle and the that rely Recall that a Lie group is simple if its Lie alge- heavily on KAM theory. bra is nonabelian and has no nontrivial proper ide- The common point is that localized perfectness im- als. The best analog of this for topological groups plies simplicity. This is a short argument which uses a is to ask whether the identity of the clever trick of “displacing supports”. Given a home- group has no nontrivial proper normal subgroups. omorphism g (other than the identity), one wishes When M is a non-compact manifold, the subgroup of to show that any other homeomorphism f can be compactly supported homeomorphisms or diffeomor- written as a product of conjugates of g. I’ll give phisms – those which pointwise fix the complement of the essence of the argument here, in the special case some compact set – is an obvious , so where f has support on some small ball B such that the relevant object of study is its , g(B) ∩ B = ∅. If g is not the identity, such a ball denoted Homeoc(M) or Diffc(M). Of course, when is guaranteed to exist, and stepping from here to the M is compact, these are just the groups Homeo0(M) general case is not too difficult. and Diff0(M) themselves. So suppose f is such a homeomorphism, and that f In both the smooth and topological cases, simplic- can be written as a commutator f = aba−1b−1 (the ity (in the algebraic sense described above) was es- argument is essentially the same if f is a product tablished in the 1970s by Edwards, Kirby [EK71] , of several – this is where we are using Thurston [Thu74a] and others as a consequence of local perfectness.) Going forward, I’ll use the stan- the following three general properties: dard notation [a, b] for the commutator aba−1b−1. 1. Fragmentation. Let O be an open cover of Recall that g(B) ∩ B = ∅. Take any homeomor- M. Then Diffc(M) and Homeoc(M) are generated phism h that pointwise fixes B and moves g(B) dis- by homeomorphisms supported on elements of O. joint from B ∪ g(B). One now checks by hand that 2. Localized perfectness. Let B be an open ball in f = [a, g], [b, hgh−1] which, if you write it out and M. Any element of Diffc(B) or Homeoc(B) can be unpack all the commutators, is easily seen to be a written as a product of commutators. product of conjugates of g. The trick to showing that 3. Simplicity of the commutator group. Let g 6= id f agrees with this nested commutator is the elemen-

3 absolutely simple real Lie groups, is necessarily con- tinuous.

Further notable work in this direction was done by Borel and Tits in the 1970s. Much more recently, L. Figure 2: Schematic for the simplicity argument. The Kramer removed the hypothesis that the target group commutator [a, b] can be written as [a, g], [b, hgh−1] be a Lie group, showing:

Theorem 0.2 (Kramer, [Kra11]). An abstract iso- tary fact that homeomorphisms with disjoint support morphism between an absolutely simple real Lie commute. The conjugate ga−1g−1 has support on group and a locally compact, σ- is nec- g(B), so [a, g] has support on B ∪ g(B) and agrees essarily continuous. with a on B. Similarly, [b, hgh−1] agrees with b on B and has support on B ∪ hg(B). Thus, on g(B) As a consequence of this, one can show that the and on hg(B) these two commutators commute, so standard Lie group topology is the unique locally [a, g], [b, hgh−1] is supported on B and agrees with compact, σ-compact group topology on such a group [a, b] there. Variations, some quite sophisticated, of (a result proved earlier for compact Lie groups by this trick of displacing supports to produce commut- Kallman). The hypothesis “absolutely simple” pre- ing elements appear throughout the literature cited vents such counterexamples as a discontinuous al- below. gebraic automorphism of R obtained from a linear Indicative of the subtleties involved in the ques- transformation of R as a vector space over Q, or a tion of simplicity, there is one case which we still discontinuous algebraic automorphism of SLn(C) in- do not know how to treat. In between Homeoc(M) duced by a wild automorphism of C. But these are, r r in a sense made precise by Kramer, the only kinds of and Diffc(M) lie the groups Diffc(M) of C diffeo- morphisms for r = 1, 2, 3, ... etc. Mather, also in pathologies that can occur. the 1970s, showed with yet a different strategy that 0.1 and 0.2 are instances of automatic r the groups Diffc(M) are also perfect – except in the continuity results, promoting an algebraic morphism case r = dim(M) + 1, where it remains open! See to a topological one. While their proofs rely on Lie [Mat74, Mat75] or the survey [CKK]. To this day, theoretic techniques, Hurtado recently showed that 2 1 whether Diff0(S ) is a remains an open a form of automatic continuity holds for diffeomor- question; the sticking point here is exactly the prob- phism groups as well. lem of local perfectness. Further references and dis- cussion can be found in [Ban97]. Theorem 0.3 (Hurtado [Hur15]). Let M be a com- pact manifold, and N any manifold. Any homomor- phism Φ : Diff0(M) → Diff0(N) is necessarily con- Automatic continuity tinuous. A broad question, applicable to any , is to what extent does the algebraic structure Homeomorphism groups satisfy an even stronger of the group determine its possible group ? version of automatic continuity, analogous to Along this line, in the 1930’s and 40’s, Cartan, Kramer’s result. van der Waerden, Freudenthal and others studied Theorem 0.4 (Rosendal, Solecki [RS07], Mann abstract between simple (and also [Man16]). Let M be a manifold, compact or homeo- semisimple) Lie groups. A sample consequence of morphic to the interior of a compact manifold with their work is the following: boundary, and G any separable topological group. Theorem 0.1. An abstract algebraic Then any Φ : Homeo(M) → G is between compact semisimple Lie groups, or between necessarily continuous.

4 This says the algebraic structure of the group “re- this, a result of Kallman from the 1980’s, states that members” the topology, in a very strong way. Sep- the Cr topology is the unique Polish topology on arability of the target G is needed as a hypothe- Diffr(M) [Kal74]. This implies in particular that sis to eliminate obvious counterexamples such as the there can be no discontinuous automorphisms of such (discontinuous) identity map from Homeo(M) (with a group. But in fact, a much stronger statement its usual compact-open topology) to Homeo(M) about automorphisms was known even earlier: any equipped with the discrete topology. automorphism of Diffr(M) is inner. This kind of Other recently found to sat- rigidity of the group structure is best framed as a isfy strong forms of automatic continuity include “reconstruction” result, which brings us to our next the group of an infinite-dimensional, separable, com- topic. plex with the strong topol- ogy, the group of the Urysohn universal Reconstruction theorems space, and automorphism groups of Lebesgue probability spaces. (These are results of Tsankov While automatic continuity says that the algebraic [Tsa13], Sabok [Sab19], and Ben Yaacov – Beren- structure of a “remembers” stein – Melleray [BYBM13], respectively.) Many of its group topology, it was known decades earlier that these results rely on the technique of ample generics the algebraic structure remembers the underlying in Polish groups developed by Kechris and Rosendal. manifold. This is a 1963 result of Whittaker [Whi63], Rosendal’s survey [Ros09], though by now slightly which states that the existence of an abstract isomor- out of date, is a good invitation to the subject. phism Φ : Homeo0(M) → Homeo0(N) implies that While the statements of Theorems 0.3 and 0.4 mir- M is homeomorphic to N and Φ is induced by con- ror each other – taking G = Homeo(N) for some jugation by a homeomorphism. other manifold N in the statement of 0.4 gives the Whittaker’s work also applies to more general C0 counterpart of Hurtado’s smooth version – as was topological spaces, and does not require that man- the case for simplicity, the proofs of these two re- ifolds be compact. However, some care is needed in sults are essentially different. The C0 case, with the hypotheses because (for example) Homeo0(R) is the exception of the line and the circle, relies on a abstractly isomorphic to the homeomorphism group kind of diagonal argument common in these proofs of a closed – one simply forgets the endpoints for other automorphism groups, together with com- of the closed interval and restricts the homeomor- mutator tricks like the one in the proof of simplicity phisms to the interior – but of course the line and explained above. The smooth case uses the idea of av- the interval are not homeomorphic. eraging a metric under a of diffeomorphisms Nearly twenty years later, Filipkiewicz extended to show that any sequence tending to the identity in this to diffeomorphism groups, showing that in this Diff0(M) converges along a subsequence to an isome- case the algebraic structure of the group also remem- try in Diff0(N). This is then improved to convergence bers the regularity of the diffeomorphisms. This gives to the identity, from which one concludes continuity. us the “infinite family of desserts” advertised above. Establishing such convergence requires maintaining Theorem 0.5 (Filipkiewicz [Fil82]). Let M and N control on of diffeomorphisms, motivating be compact manifolds and suppose there is an iso- ideas that would later lead Hurtado to work on the morphism Φ : Diffr(M) → Diffs(N). Then r = s, for diffeomorphisms of the 0 0 the manifolds M and N are Cr diffeomorphic, and Φ and the Zimmer program explained in the last section is induced by a Cr-diffeomorphism. of this article. Interestingly, automatic continuity (either in Hur- In both this and Whittaker’s theorem, the basic tado’s sense with a restricted target, or the very gen- strategy is to promote an isomorphism between trans- r eral sense) for the groups Diff0(M) when 0 < r < ∞ formation groups to a bijection between manifolds by r remains open. As some positive evidence towards associating subgroups of Diff (M) or Homeo0(M) to

5 of M, eventually showing that the ping class group Homeo(S)/ Homeo0(S) of homeo- (point stabilizer) subgroup Gx of a point x ∈ M is morphisms isotopy. This was recently extended mapped under an isomophism Φ to the isotropy sub- by Bavard–Dowdall–Rafi to infinite-type noncompact group of some point of N. A key tool is commuta- surfaces as well [BDR20]. The second example that tion relations. To give a toy example, note that Gx comes to mind is a recent result of Cantat that Cre- is characterized by the following property: A non- mona groups of projective spaces determine the di- trivial diffeomorphism g lies in Gx if and only if, for mension of the space [Can14]. Surely the reader can any B containing x, the subgroup of diffeo- draw a few examples from their own field of study! morphisms supported on B does not commute with its conjugate by g. Though this is far from a proof Optimal regularity that point stabilizers are mapped to point stabiliz- r ers, it is a hint that one might be able to pick out Filipkiewicz’s theorem says that Diff0(M) and s such subgroups by algebraic relations (in this case, Diff0(M) are non-isomorphic when r 6= s. But it commutation). leaves no practical way to distinguish them alge- The relationship between point stabilizers in M braically. As a concrete step in this direction, one and those in N gives a point-to-point map τ : M → could ask: r N. Furthermore, this map is compatible with the For a given manifold M, can the groups Diff0(M) and r s in the sense that, for any f ∈ Diff0(M), Diff0(M) be distinguished by their finitely generated −1 we have Gf(x) = f(Gx)f from which it follows that subgroups? τ(f(x)) = Φ(f)τ(x)Φ(f)−1. This compatibility al- Navas posed this question explicitly for one- lows one to show fairly easily that τ is a homeomor- dimensional manifolds in a 2017 problems list phism; Filipkiewicz then appeals to deep results of [Nav18]. As he notes, questions of distinguishing reg- Montgomery and Zippin [MZ55] to conclude that it ularity have a long history among dynamicists that is a Cr diffeomorphism. can be traced back to work of Denjoy on the rela- Analogous theorems for groups of symplecto- tionship between regularity and topological dynam- morphisms, contact diffeomorphisms, and volume ics. Denjoy showed that, while there are many C1 preserving diffeomorphisms were later obtained by counterexamples, if f is any C2 diffeomorphism of the Banyaga. An expository account of this and Filip- circle without periodic or fixed points, then for every kiewicz’s theorem can be found in [Ban97]. M. Rubin point x, the {f n(x): n ∈ Z} is a dense [Rub89] generalized Whittaker’s results in a differ- of the circle. A typical C1 counterexample is illus- ent direction, extending it to more general classes of trated in Figure 3: there is an topological spaces and subgroups of their automor- (red in the figure) on which the diffeomorphism be- phism groups using a more model-theoretic frame- haves like a . To visualize the dynamics, the work. Both show that the algebraic structure of the figure shows the of such a homeomor- is sufficient to recover the un- phism, with points on the leftmost circle connected derlying manifold or space. to their images on a copy of the circle to the right. Such reconstruction results can be interpreted as Dense orbits and existence or non-existence of peri- instances of the broad principle that any sufficiently odic points are both properties invariant under con- rich object can be recovered, within its class, from jugation by homeomorphisms, so Denjoy’s theorem its automorphism group. To give a sense of the implies that there exist C1 diffeomorphisms which breadth of this principle, let me mention two other cannot be topologically conjugate to any C2 diffeo- recent examples (with a geometric topological fla- morphism, an observation later generalized by Harri- vor, apologies for my bias). The first is the folk- son to higher dimensional manifolds and higher reg- lore theorem that the homeomorphism class of a ularity. compact surface S can, with a few low-complexity Denjoy’s theorem takes as input a weak dynamical exceptions, be completely recovered from the map- assumption (no periodic points) and a regularity hy-

6 tional dynamical input from earlier work of Bonatti, Monteverde, Navas and Rivas. A different approach to this question, inspired by work of Ghys, was pur- sued by the author and Wolff, with the aim of even- tually addressing the question for higher dimensional manifolds. While we found a short proof somewhat Figure 3: Dynamics of a C1 “Denjoy counterexam- complimentary in technique to that of Kim-Koberda, ple” diffeomorphism of the circle the higher dimensional case is still out of reach!

pothesis (twice continuously differentiable), and gives Structure theorems for group actions as output the strong dynamical conclusion that all or- The reconstruction results of the previous sections bits are dense. Thus, it reveals something about the came out of a classification of between interplay between and regular- large transformation groups. Classifying homomor- ity. When one has a group action rather than a single phisms between them is a significantly harder prob- diffeomorphism, the algebraic structure of the group lem. A homomorphism Homeo0(M) → Homeo(N) is can also enter the picture. A well known and early in- simply a group action of the group Homeo0(M) on stance of a result along these lines is Thurston’s gen- the space N, so going forward I will use the language eralization of the [Thu74b]. of group actions to describe such maps. (Again, we 1 Thurston’s theorem says that a group of C diffeo- are restricting to the identity component Homeo0(M) morphisms of a manifold with a common fixed point to avoid actions that factor through an action of the at which the diffeomorphisms are all first equal .) to the identity map (that’s a regularity + dynamical There are many natural examples of such actions. assumption) necessarily has the (algebraic) local in- The groups Homeo0(M) and Diff0(M) obviously act dicability property that every finitely generated sub- on the product manifold M × W for any manifold group surjects to Z. The search for similar theorems W . They also have a natural action on Confk(M) = of the form ((M ×...×M)−4)/Symn, the “configuration space” [dynamical assumption] + [regularity] of k distinct, unlabeled points in M. In some in- ⇒ [algebraic conclusion] stances, such as when M is a Riemannian mani- fold of negative , or more generally when (or any thereof) remains an active and π1(M) has trivial , the action of Homeo0(M) exciting area. This is the context for Navas’ question or Diff0(M) on M can also be shown to lift to an ac- above. tion on covers of M. Finally, to give some examples Several results along these lines are known to hold special to the differentiable case, Diff0(M) also acts when M is one-dimensional, and a positive answer to naturally on the tangent of M, the projec- Navas’ question was given by Kim and Koberda not tivized of M, and various bundles long after the appearance of his problems list [KK20]. over M. With the exception of the product action Their proof uses a highly refined version of the dif- on M × W and the tangent bundle action, all of the feomorphisms with disjoint supports commute line of examples above have a single orbit, indicating that tricks to encode the regularity of a diffeomorphism by even a classification of transitive actions is a nontriv- the rate at which it moves, and therefore contracts, ial and potentially difficult problem. small sub-intervals (forced to be supports of group This classification problem has been posed, in var- elements) towards a fixed point. Obtaining enough ious forms, at least since the time of Whittaker’s control on supports to pull off this argument from work. Part of the challenge in the problem is sim- purely algebraic hypotheses is quite a technical feat, ply stating a clear conjectural picture. Rubin, in and even so, in low regularity they use some addi- the 1989 article [Rub89] briefly mentioned above,

7 Figure 4: The configuration space of two labeled points on S1 is an open annulus, that of two unlabeld 1 points is topologically a mobius band. Both inherit an action of Homeo0(S ).

asks “Are there any reasonable assumptions on the groups and Lie groups, one could frame the problem type of the so that the embeddability of classifying morphisms between such groups as the of Homeo(X) in Homeo(Y ) will imply that X is analog of for these groups – a some kind of continuous of Y ?” (emphasis perspective which I myself have advertised. But a mine). Embeddability here is meant in the algebraic more useful framing is to draw parallels with the clas- sense of the existence of an injective homomorphism sical theory of compact (or locally compact) transfor- Homeo(X) → Homeo(Y ), and Rubin is speaking not mation groups pursued by Bredon, Conner, Floyd, just of manifolds, but of quite general topological Montgomery, Zippin and others in the 1950s-70s. spaces. Two years after that, Ghys asked whether Montgomery and Zippin [MZ55] frame one aspect of embeddability of Diff0(M) into Diff0(N) implies that this program as a search for the similarities between the dimension of N must be at least as large as that of actions of a compact group on a manifold and stan- M [Ghy91] . This was answered positively by Hur- dard linear or other model geometric actions. Years tado in 2014 using his automatic continuity result later, Bredon [Bre72], in broad terms, says that he [Hur15], improving on a very low dimensional case I views theory of compact transformation groups as had proved earlier (but that case did not have auto- “a generalization of the theory of fiber bundles” 1. matic continuity!). Hurtado also gives a more precise Indeed, the general starting point for this theory is picture of what a classification might look like. Al- typically to first show that only finitely many orbit though he does not give an explicit conjecture, he types may occur, then to understand and classify or- notes that Diff0(M) acts naturally on the tangent bits, and then to say that the neighborhood of any bundle of M (as well as various Grassmanians, etc.) orbit looks something like a , giving a and remarks that all known examples of actions of local bundle-like structure. However, complete solu- Diff0(M) on other manifolds are built from pieces tions to the classification problem for compact group that have the form of natural bundles over M or con- actions are generally only attainable with some re- figuration spaces of points on M. If one believes this striction on the types of orbits that appear, whether is an exhaustive list, then one could refine Rubin’s by explicit assumption, or by restricting the dimen- question to ask for a stratification of the target man- sion of the space and the group. A comprehensive ifold into understandable pieces that map continu- 1 ously either to M or to tuples of points on M. While [MZ55,Bre72] are old references, I do not know of a more modern comprehensive introduction, likely just because the study of transformation groups has fractured into many Pursuing the analogy between large transformation different directions.

8 introduction to such problems can be found in Bre- don’s book [Bre72]. That one might similarly be able to classify ac- tions of homeomorphism or diffeomorphism groups is a comparatively recent idea, perhaps first advo- cated for by Ghys, although I personally learned this question from Benson Farb. However, it was auto- matic continuity theorems that provided the catalyst Figure 5: A quotient of Σe × S1 by a diagonal action needed to make major progress in this direction. Us- of π (Σ) is a circle bundle over Σ which inherits an ing automatic continuity for actions of Homeo(S1), 1 action of Homeo (Σ). The trivial action on S1 gives in 2012 E. Militon gave a complete classification of 0 a product bundle with closed orbits as shown, while actions of Homeo (S1) on the closed annulus and 2- 0 typical actions have dense orbits. torus [?Militon]. His classification gives a decompo- sition of the torus or annulus into orbits, each homeo- morphic to either the circle with the standard action, – for instance, any hyperbolic structure on the sur- or to an open annulus with the action coming from gives a discrete, faithful representations of π1(Σ) identifying the annulus with the configuration space into PSL(2, R) which acts naturally on RP 1 =∼ S1 by of two (labeled) points in S1. See Figure 6. These fractional linear transformations. Fix such an action, two types of orbits may be glued together in a mul- and take the quotient of Σe×S1 by the diagonal action titude of different ways, leading to an uncountable of π1(Σ) by deck transformations on the first factor, family of nonconjugate examples which Militon de- and homeomorphisms of S1 on the second. The re- scribes completely. sult is a S1 bundle over Σ with a continuous action Recent work of Lei Chen and the author gives a of Homeo0(Σ) by homeomorphisms. Provided the ac- 1 general classification of orbit types in all . tion of π1(Σ) on S doesn’t factor through that of a finite group, the orbits of the Homeo0(Σ) action on Theorem 0.6 (Chen–Mann [CM]). Suppose that this quotient space will accumulate on themselves. Homeo0(M) acts on a manifold N. Then each or- bit is the continuous, injective, image of a cover of a Interestingly, if one takes the representation of configuration space Confn(M). π1(Σ) into PSL(2, R) described above, then the re- sulting circle bundle over Σ is, topologically, the While locally embedded, such orbits are not nec- tangent bundle (or projectivized tangent bundle) of essarily globally embedded. The simplest example the surface. While the group of diffeomorphisms of where this fails is the following bundle construction. the surface has an obvious action on this space, I find it quite surprising that the group of homeomorphisms Example 0.7. Suppose Σ is a surface of genus at also does since a homeomorphism has no obvious in- least two. Then Homeo0(Σ) lifts to act continuously, duced action on the tangent space to a point! by homeomorphisms on the universal cover Σ,e which Because of automatic continuity, Theorem 0.6 re- is topologically an open disc, and this action com- duces to the problem of classifying certain closed mutes with the action of the deck group. Extend subgroups of Homeo0(M). (In the case where M this to an action on Σe × S1 by homeomorphisms that is noncompact, automatic continuity does not quite are constant on the second factor. Orbits of this ac- apply and a little more work is needed, so for sim- tion are embedded open discs, but we can change this plicity I’ll assume here that M is compact.) If by passing to a quotient of the product space by an ρ : Homeo0(M) → Homeo(N) is any action, then action of π1(Σ) that is nontrivial on the second fac- the stabilizer Gy := {g ∈ Homeo0(M): ρ(g)(y) = y} tor. To be precise, the π1(Σ) ad- of some point y ∈ N is a closed subgroup, and the mits many actions by homeomorphisms on the circle orbit map gives a continuous, injective map of the

9 space Homeo0(M)/Gy into N. Thus, The- over M, i.e. there is a manifold F such that ∼ orem 0.6 becomes equivalent to determining which N = (Mf × F )/π1(M), where π1(M) acts diagonally closed subgroups of Homeo0(M) are “large enough” by deck transformations on Mf and on F by some so that their coset space can be locally embedded in representation to Homeo(F ). a finite-dimensional manifold. Classical invariance of says that Rn is not locally embeddable In fact, we can also describe the possible actions in a manifold of dimension less than n, so no Rn on N up to conjugacy: they are all obtained by a subgroup of Homeo0(M), that is to say no n-many construction as in Example 0.7. A related result for commuting flows, can survive the quotient map to diffeomorphism groups of compact manifolds can be Homeo0(M)/Gy if dim(N) < n. Applying this ob- obtained without the hypothesis that the action has servation to subgroups of Homeo0(M) with disjoint no global fixed points; one shows directly that in supports, and quoting local simplicity, with enough this setting no orbits can be singletons. This line work we eventually conclude that Gy contains the of work also allows one to complete the classification 1 identity component of the stabilizer of a finite set of of Homeo0(S ) actions on surfaces that was initiated points in M, hence Homeo0(M)/Gy is (up to taking a by Militon, filling in the case of the 2-sphere and open cover) can be identified with the configuration space annulus. However, it appears we have only yet seen of those points. the tip of the iceberg, and I hope to see much exciting The example of Diff0(M) acting on the projec- work in this direction in the near future. tivized tangent bundle of M shows that a similar classification for actions of diffeomorphism groups, Local and global rigidity of lattices even for actions on compact manifolds, must account for more kinds of orbit types. Remarkably, this kind Having advocated for the analogy between Lie groups of construction – taking quotients of jets of diffeo- and diffeomorphism or homeomorphism groups, this morphisms up to rth-order – is the only kind of new survey would be incomplete without a discussion of phenomenon that occurs: we show in [CM] that if the role of lattices in these groups. Diff0(M) acts on a manifold by smooth diffeomor- A Γ in a Lie group G is a discrete subgroup phisms, then every orbit is the continuous, injective such that G/Γ has finite volume with respect to the image of a fiberwise quotient of the r- over natural left-invariant on G giving Haar a cover of a configuration space of points on M. . A lattice is called co-compact or uniform if Theorem 0.6 confirms the picture suggested by G/Γ is additionally compact. In one sense, lattices Hurtado [Hur15] that N should be “built from pieces” are small subgroups, being discrete and, under appro- that look like bundles – the pieces are simply the priate hypotheses on G, they are also always finitely orbits of the action. Classifying all actions now generated groups. At the same time, that G/Γ is amounts to understanding how such pieces can be finite volume means that lattices are also large and glued together. We obtain some preliminary results this gives them some remarkable rigidity properties, under restrictions on dimension that simplify the constraining the ways they may be embedded into G types of orbits that may occur. (Such hypotheses are and into other Lie groups. The paradigm example commonplace in the classical literature on compact of such a result is Margulis’ theorem, transformation groups). A sample theorem, parallel- which essentially says that the representation theory ing classical “slice” or local product theorems for the of a lattice Γ in a semisimple Lie group G is the same compact group case, is as follows. as the representation theory of G itself: with a few technical caveats, any representation of Γ to a finite Theorem 0.8 (Chen – Mann [CM]). If M and N are dimensional extends to a continuous rep- connected manifolds with dim(N) < 2 dim(M) and resentation of G. Homeoc(M) acts on N without global fixed points, Infinite-dimensional linear representations of lat- then N has the structure of a generalized flat bundle tices have no such rigidity property. Remarkably,

10 nonlinear representations – that is, representations of lattices to groups of diffeomorphisms – do. That this might be true was an idea first suggested by R. Zim- mer in the late 1970s. Zimmer generalized other work of Margulis (specifically, on cocycle rigidity for group actions) to actions of lattices by measure-preserving diffeomorphisms of manifolds. He followed this with a series of broad about how lattices should act – or fail to act! – on manifolds, giving rise to what is now known as the Zimmer program. This program 2 1 2 2 is often summarized by the catchphrase “big groups Figure 6: The action of ( 1 1 ) ∈ SL(2, Z) on R /Z . don’t act on small manifolds” meaning that lattices The figure shows a and its im- in semisimple Lie groups that are large (specifically, age, translated back onto the fundamental domain. of high rank) should not act faithfully on manifolds of low-dimension. More broadly, the Zimmer program aims to clas- into invariant directions (here, eigendirections) that 2 sify all actions of higher rank semisimple Lie groups are contracted or expanded by iterates of the map. and their lattices on compact manifolds, the idea The most famous Anosov diffeomorphism is prob- 2 1 being that they should be constrained to a collec- ably ( 1 1 ) ∈ SL(2, Z), also known as Arnold’s “cat tion of natural or obvious examples (such as, but not map” after an illustration in Arnold’s Ergodic Prob- exclusively, those coming from actions of the ambi- lems in Classical Mechanics [AdA68] in which he ent Lie group as is the case in Margulis’s theorem). applies the map to a cartoon image of a cat. Af- While a complete classification is a lofty and prob- ter several iterations, the cat’s face is unrecognizably ably unattainable goal, there have been a stretched and distributed around the torus. This is of remarkable recent advances, for which I refer the a basic property of Anosov maps. Formally speak- reader to [Fis20] and Fisher’s other survey articles ing, volume preserving Anosov diffeomorphisms are referenced there. Here I will only very briefly high- ergodic, one consequence of this is what one might light one feature of this program, namely rigidity of call chaotic behavior: nearby points in the domain geometric examples. This is distinct from the “big have vastly different trajectories under iterates of the groups don’t act on small manifolds” tagline, looking map. instead for examples of groups that do act on spaces, Despite this chaos on the level of trajectories of in- but whose possible actions are highly constrained. Of dividual points, Anosov diffeomorphisms exhibit re- course, one expects techniques that answer this prob- markable global stability, in the sense that if you lem to apply to the “groups that don’t act” program nudge the whole system, the result is a system that as well, and vice versa. is qualitatively exactly the same – typically (depend- As one such example, consider the special linear ing on the regularity involved) conjugate to the origi- nal system. These rigidity type results for diffeomor- group SL(n, Z), a non-uniform lattice in SL(n, R). n phisms and 1-parameter families of diffeomorphisms Since the action of SL(n, Z) on R preserves the points with coordinates, it descends to an date back to work of Anosov himself in the 1960s. action on the torus Rn/Zn, which preserves both In the context of group actions, for groups other Lebesgue measure and the natural affine structure than R or Z, in many cases Anosov dynamics of a on the torus – instances of what I mean by a “ge- single group element can be promoted via an un- ometric example”. These linear actions also have a derstanding of the group structure to not only local strong dynamical property called Anosov dynamics: 2I am sweeping a few things under the rug here; the precise for any element with all eigenvalues off the unit cir- definition of Anosov is a formalization of this idea of invariant cle, the tangent space of the manifold splits globally splitting.

11 rigidity but global rigidity for the whole action. One Anosov is in terms of the induced action of a diffeo- important and influential instance of this is in the morphism on the tangent bundle, and hence requires work of Katok, Lewis, Zimmer, and others, starting an action to be C1, there are topological analogs of in the 1990s. To give a sample result, Katok, Lewis Anosov flows; the idea being that you can quantify and Zimmer showed that any measure-preserving ac- the rate of expansion and contraction of continuous tion of SL(n, Z) on Rn/Zn such that a single diffeo- trajectories even if you cannot meaningfully speak of morphism has Anosov dynamics is (up to passing to a splitting of the tangent space. These topological a finite index subgroup) smoothly conjugate to a lin- flows have many of the same dynamical properties, ear action [KLZ96]. Some mild hypothesis is required and C0 perturbations of Anosov diffeomorphisms also on the measure, although in other work this hypoth- exhibit similar stability properties. esis has been replaced with other hypotheses such as However, Zimmer’s conjecture that there should the existence of a finite orbit or a fixed point. See be “no actions of large groups on small manifolds”, Spatzier’s survey [Spa04] for a nice introduction to speaking here of discrete groups, is wide open in the the history of such results. C0 case, except when the manifold is very small: di- Rigidity of lattices remains an active and exciting mension one. These one-dimensional results rely on area. Recently, Brown, Rodriguez-Hertz and Wang the total ordering of points on the line, or cyclic or- proved a much broader theorem along the lines of dering of points on the circle, a property that can that of Katok–Lewis–Zimmer described above, but actually be promoted to a left-invariant - applicable to a much wider class of lattices and with ing on any group of homeomorphisms of the line or weaker hypotheses. As well obtaining global rigidity circle. As you might expect, such a total order on for actions of SL(n, Z) on tori Rm/Zm (note that m a group imposes strong algebraic constraints, and is not required to equal n here!) under suitable hy- this has been successfully leveraged to show certain potheses, they obtain rigidity of cocompact lattices groups, including lattices, do not act. Morris’ paper on , a class which includes tori: [Mor11] is a nice introduction; the state of the art can be found in a very recent paper of Deroin and Hur- Theorem 0.9 (Brown – Rodriguez-Hertz – Wang tado, who show non-orderability of all higher rank [BRHW17]). Suppose Γ is a irreducible cocompact lattices. lattice in a semisimple Lie group acting on a nilman- 0 ifold M by smooth diffeomorphisms. If some element There are also a few instance of C rigidity of has Anosov dynamics, then (up to possibly passing geometric examples in low dimension, even without to a finite index subgroup), this action is smoothly Anosov dynamics. It is less clear what “geometric” conjugate to a group of linear automorphisms of M. should mean for a group acting by homeomorphisms, but one possible definition, in the style of a model ge- Without passing to a finite index subgroup, one ob- ometry in the sense of , is that the action tains conjugacy to an affine action – still a geometric should be conjugate to that of a cocompact lattice in example. Their work also covers non-irreducible and a Lie group which acts transitively on the manifold. many nonuniform lattices, but in the interest of sim- With this definition, classifying geometric subgroups 1 plicity of the statement I have restricted to this case. of Homeo0(S ) is an easy exercise: any group acting However, differentiability is an essential component geometrically is (up to taking a finite, cyclic exten- of this result. While Brown, Rodriguez-Hertz and sion) the fundamental group π1(Σg) of a higher genus Wang can reduce the hypotheses from smooth to C1 surface, acting on the circle via a discrete, faithful or even Lipschitz actions at the cost of a weaker con- representation into PSL(2, R) or a finite, cyclic ex- clusion, the machinery going into the regularity of tension of PSL(2, R). Wolff and I showed these are the conjugacy is essentially a smooth phenomenon. precisely the rigid examples: That said, there are also a few known instances of “rigidity of geometric examples” for groups act- Theorem 0.10 (Mann, Mann – Wolff [MW] ing by homeomorphisms. Although the definition of [Man15]). All geometric actions of π1(Σg) are rigid,

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