Boundaries of groups

Transcription

1 Boundaries of groups The American Institute of Mathematics The following compilation of participant contributions is only intended as a lead-in to the AIM workshop Boundaries of groups. This material is not for public distribution. Corrections and new material are welcomed and can be sent to workshops@aimath.org Version: Tue Oct 11 13:34:

2 2 Table of Contents A. Participant Contributions Bell, Robert 2. Bourdon, Marc 3. Charney, Ruth 4. Cordes, Matt 5. Dani, Pallavi 6. Davis, Mike 7. Groves, Daniel 8. Guilbault, Craig 9. Hagen, Mark 10. Haissinsky, Peter 11. Manning, Jason 12. Melnick, Karin 13. Moran, Molly 14. Osajda, Damian 15. Paoluzzi, Luisa 16. Puls, Michael 17. Sathaye, Bakul 18. Sisto, Alessandro 19. Stark, Emily 20. Tshishiku, Bena 21. Xie, Xiangdong

3 3 A.1 Bell, Robert Chapter A: Participant Contributions I hope to learn more about the latest techniques in the study of boundaries of CAT(0) groups. My primary objective for this workshop is to identify specific projects to which I can contribute. A problem which I find particularly compelling is finding conditions which imply that certain topological properties hold for every CAT(0) boundary of a group. The well-know examples of Croke and Kleiner and a generalization due to Mooney show that once a group has many intersecting flats (e.g. the right-angled Artin group defined by the path of four vertices), then the group can act cocompactly, isometrically, and properly on two different CAT(0) spaces, whose boundaries are non-homeomorphic. On the other hand, as shown by Hruska and Kleiner, if a CAT(0) group has the isolated flats property then any two boundaries are homeomorphic. My background includes some expertise in the study of Artin groups, Coxeter groups, and mapping class groups. A.2 Bourdon, Marc I am interested in boundaries of Gromov hyperbolic groups. More precisely, I am interested in their quasi-conformal geometry, and to related quasi-isometric invariants like the conformal dimension and the L p -cohomology. A.3 Charney, Ruth There have been numerous recent developments aimed at generalizing hyperbolic techniques to broader classes of groups, such as acylindrically hyperbolic groups. In my work with Harold Sultan, later generalized by Matt Cordes, we introduce a Morse boundary that identifies aspects of hyperbolic behavior in any geodesic metric space by restricting to rays satisfying a Morse property. I am interested in discussing properties and applications of the Morse boundaries for various classes of groups (see, for example, the work of Devin Murray on CAT(0) groups) as well as its relation with other types of boundaries. For example, how are topological properties of the Morse boundary, such as local connectivity, reflected in the group? In addition, I would like to explore possible measures on these spaces and dynamical properties of the group action. A.4 Cordes, Matt I study the Morse boundary of groups, a quasi-isometry invariant boundary that is nonempty for spaces with some aspect of negative curvature, i.e., if there exists a Morse geodesic. Sisto showed that every acylindrically hyperbolic group has a Morse geodesic. (The class of acylindrically hyperbolic groups includes MCG(S),Out(F n ), relatively hyperbolic groups, non-directly decomposable right-angled Artin groups, and more.) In the case of a hyperbolic group, this boundary is the normal Gromov boundary. There are many open questions about this boundary. It would be interesting to understand the relationship between the Morse boundary and the many other boundaries for the groups listed above or related spaces. And if so, can we extract some interesting information from the understanding the relationship. During the workshop, I would be happy to learn about some of these other boundaries.

4 4 Another open area of study would be to understand more about the dynamical properties of the Morse boundary. In the CAT(0) setting, Devin Murray has explored some of these results and it would be nice to see which generalize, which do not, and why. Furthermore, it would be interesting to investigate if there is a connection between the topology of the Morse boundary of a group and algebraic information about that group. (The topology can be pretty wild; Murray showed that the Morse boundary of the group Z Z 2 does not have a locally countable basis!) A.5 Dani, Pallavi Recently I have been investigating the quasi-isometry classification of right-angled Coxeter groups (RACGs)in collaboration with Anne Thomas. We used the (visual) boundaries of RACGs to give a description of the Bowditch JSJ tree in the class of 2-dimensional hyperbolic RACGs for which this tree is defined, leading to a complete quasi-isometry classification of a large class of hyperbolic RACGs. I am interested in understanding Morse boundaries for RACGs, with a view to applying them to the q.i.-classification question. This should have broader appeal, as it will lead to large collection of tractable examples of Morse boundaries. I am also interested in comparing different boundaries for RACGs. A.6 Davis, Mike The action dimension of Artin groups The action dimension of a discrete group G is the smallest dimension of a contractible manifold on which it can act properly. Bestvina, Kapovich and Kleiner [bkk] have showed how to use van Kampen s embedding obstruction to compute the action dimension of G. If G has a Z-set boundary, G, and if K is a finite complex embedded in G such that the van Kampen obstruction for embedding K in S m is nonzero, then the action dimension of G is m+2. When G is not known to have a Z-set boundary, one replaces the condition that K G, by the condition that there is a uniform proper embedding of the 0-skeleton of Cone(K) into G. In [ados, dj] my collaborators and I used this idea to compute the action dimension in the case when G is an Artin group. Bibliography [ados] G. Avramidi, M.W. Davis, B. Okun and K. Schreve) The action dimension of rightangled Artin groups, Bull. London Math. Soc. 48 (1) (2016), [bkk] M. Bestvina, M. Kapovich and B. Kleiner, Van Kampen s embedding obstruction for discrete groups, Invent. Math. (2) 150 (2002), [dj] M.W.Davis and J. Huang, Determining the action dimension of an Artin group by using its complex of abelian subgroups, arxiv: A.7 Groves, Daniel My recent interest in boundaries has arisen in joint work with Jason Manning and Manning and Sisto, where we begin to understand what happens to the(bowditch) boundary of a relatively hyperbolic group under the operation of group-theoretic Dehn filling. This has lead to a study of the topological and analytic properties of these boundaries. I have a number of questions about boundaries that I think are interesting, and that I would like to know the answer to. Here are a few:

5 Question 1. Can the new theories of hyperbolicity, such as acylindrical hyperbolicity be characterized in terms of convergence group actions of some kind? Note that the Gromov boundary of many of the spaces arising in this theory are not compact. Question 2. Is there some way of using the boundary of a hyperbolic group to develop a group-theoretic notion of drilling which complements the theory of relatively hyperbolic Dehn filling? Question 3. Suppose that G is a hyperbolic group such that G has no local cut points. Is there some homeomorphism η[0, ) [0, ) so that for any ǫ > 0 we have: A. there exist quasicircles in G of quality η; and B. for some family F of such quasicircles (of quality η), each of which induce a quasi-isometry H 2 G whose image comes uniformly close to 1 so that for any pair Q 1,Q 2 F there is a chain of quasicircles between Q 1 and Q 2 which are pairwise distance at most ǫ apart? Note: The distance between two quasicircles should come from integrating (around S 1 ) the distance between points Q 1 (t),q 2 (t) for t S 1 with respect to a visual metric on G based at 0. I would be interested in this question in case G is a 2 sphere. In this case, would such a condition already imply that G has enough regularity so that the Cannon conjecture is true? [In which case we could take F to be the set of round circles whose convex hull includes a fixed basepoint of H 2, interpreting the geometry of G as H 3.] We remark that Bonk and Kleiner proved that all boundaries of one-ended hyperbolic groups admit some quasicircle and, under the assumption of no local cutpoints, Mackay proved that a quasicircle can be made to go through any finite collection of points. Neither of these seem like enough to answer the above question, which asks for many more quasicircles. A.8 Guilbault, Craig Group boundaries provide a bridge between the classical and ongoing study of ends of spaces and the subject of geometric group theory. From that perspective, I find Bestvina s axiomatic approach to group boundaries particularly appealing. That approach is based on an established concept from topology that of a Z-set. A closed subset A of a space B is a Z-set if U A U is a homotopy equivalence for every open U B. For example, the boundary of a manifold M n is a Z-set in that manifold. A compactification X = X Z of a space X is a Z-compactification provided Z is a Z-set in X. Examples include the addition of a boundary to an open manifold and the addition of the visual boundary to a proper CAT(0) space. According to Bestvina, a Z-structure on a group G occurs when one Z-compactifies a (nice) space X on which G acts properly and cocompactly in such a way that G-translates of compactabecomesmallastheyarepushedtowardz. Iftheactionextendsto X, theresultis called an EZ-structure. CAT(0) and word hyperbolic groups are the canonical examples, but many other groups admit Z- and EZ-structures. It is an open question whether every group with a finite K(G, 1) admits a Z-structure. Theorems from geometric topology provides a number of intermediate goals to aim for. The overlap with geometric topology is particularly interesting when the K(G, 1) is a closed manifold. 5

6 6 As for applications: a Z-boundary contains information about cohomology of G with ZG-coefficients; groups that admit Z-structures satisfy the Novikov Conjecture; and a Z- boundary has implications for semistability of G. In work with my recent PhD student Molly Moran, I have become interested in the relationship between topological and metric properties of a group boundary and the large-scale geometry of G. One needn t leave the realm of CAT(0) groups to find hard and interesting questions. A.9 Hagen, Mark My interest in boundaries of groups has so far been in boundaries of spaces (and groups acting on them) that exhibit various forms of nonpositive or generalised negative curvature, broadly construed. Some examples of the types of questions I m interested in are below. All the different boundaries of a given space. FormanyspacesX, therearemany naturally associated boundaries, so a basic question is: how are all the different boundaries of X related (are there reasonable maps between them etc.)? The case where X is a CAT(0) cube complex is very interesting, since here many boundaries have been defined: the visual boundary and Tits boundary for the CAT(0) metric, the contracting boundary recording the rank-one part of the visual boundary (but with a more tractable topology), the Roller boundary, simplicial boundary, and hierarchically hyperbolic boundary for the combinatorial metric/hyperplane structure, etc. The relationships between these different boundaries are partly, but not entirely, understood. (E.g. is the simplicial boundary some kind of Poincare dual of the Roller boundary?) For a surface S, one can ask a similar question about the relationship between the (intrinsically defined) hierarchically hyperbolic space boundary of MCG(S) or that of T (S) and the Thurston compactification of T (S). Boundary invariance. The Gromov boundary of a hyperbolic space has the important feature that quasi-isometries induce boundary homeomorphisms; in particular, one can therefore define the Gromov boundary of a hyperbolic group. This famously breaks down for visual boundaries of CAT(0) spaces, which Croke and Kleiner exhibited using a rightangled Artin group. However, there s some hope that in the context of generalised negative curvature, some kind of boundary invariance is available. For example, Charney-Sultan showed that the contracting boundary of a CAT(0) space depends only on its quasi-isometry type, and more generally this is true of the Morse boundary. These boundaries capture the hyperbolic-like directions in a space. What if we try to throw in some non-hyperbolic directions? One context for asking this question is the setting of hierarchically hyperbolic spaces, which can be compactified using the HHS boundary mentioned above. The question is: given two hierarchically hyperbolic structures on a fixed (qi class of) space(s), are the associated boundaries homeomorphic? Subgroups with good limit sets. Given a group G acting geometrically on a space X with a nice boundary X, which subgroups H G have nice limit sets in X? For instance, when we know that H also acts geometrically on a space Y which has a nice boundary Y (nice in the same sense as X), when is there an H equivariant embedding Y X? A.10 Haissinsky, Peter I am mostly interested in the properties of (word hyperbolic) groups that can be read from their boundaries. Prototypical examples include Mostow s rigidity theorem and Paulin s theorem which characterizes quasi-isometry classes of word hyperbolic groups.

7 Boundaries of word hyperbolic groups are endowed not only with a canonical topology but also with a canonical geometric structure (the conformal gauge). An open problem in which I am interested in is to understand when the topology of the boundary determines its geometric structure. More generally, I would like to describe the analytic properties of the boundaries of some subclasses of groups. With Cyril Lecuire, we were able to use boundary methods to prove the quasi-isometric rigidity of (finitely generated) Kleinian groups. This enabled us to complete the quasiisometric rigidity of the class of fundamental groups of compact 3-manifolds. A.11 Manning, Jason I am most familiar with boundaries of hyperbolic and relatively hyperbolic groups. Recently with Groves and with Groves and Sisto I have been investigating how the operation of Dehn filling affects these boundaries. With Groves, we showed that certain connectivity properties are usually preserved by Dehn filling. With Groves and Sisto we used some of the metric properties of the boundary to pin down the topology of some boundaries arising in a relative version of the Cannon conjecture. I m very interested in learning more about other boundaries, including the Morse boundary and the boundary which can be associated to a hierarchically hyperbolic space. A.12 Melnick, Karin I am interested in boundaries of hyperbolic groups and the action of these groups on their boundaries. A.13 Moran, Molly My goals and interests for this workshop are two-fold. First, I am interested in metrics on visual boundaries of CAT(0) spaces. I have studied two possible natural metrics and some of their properties, including extension of the metric to the whole space and how a CAT(0) group acts on the boundary. I would like to fill in some of the gaps that exist in my current list of properties and explore other unanswered questions about these metrics. While the question of metrics on CAT(0) boundaries is interesting and not widely studied in its own right, I also have some hope that understanding the linearly controlled dimension of the boundary (which requires a specific metric) may lead to answering the open question of whether or not CAT(0) groups have finite asymptotic dimension. My second interest is Z-structures. The rich study of boundaries of CAT(0) and hyperbolic groups led Bestvina to formalize the concept of a group boundary by defining a Z-structure on a group. A few classes of groups are know to admit Z-structures (CAT(0), δ-hyperbolic, Baumslag-Solitar, systolic, and certain relatively δ-hyperbolic groups) and it would be great to add to this list. One could then potentially focus on the study of group boundaries via these more generalized boundaries rather than on a case by case basis. A.14 Osajda, Damian My recent interest in boundaries of groups concern the following topics: - using Gromov boundaries for proving combination theorems for complexes of hyperbolic groups (joint with Alexandre Martin); - constructions of Gromov boundaries with interesting analytical properties (Combinatorial Loewner Property; with Antoine Clais); - constructions of Z-structures for new classes of groups; - using the topology of the boundary for constructions of high 7

8 8 dimensional hyperbolic groups not containing Poincare duality groups of dimension above two; - using boundaries for quasi-isometric rigidity results for some combinatorially nonpositively curved (CNPC) groups (with Jingyin Huang); - constructions of Poisson boundaries for some CNPC groups. A.15 Paoluzzi, Luisa I recently suggested the following problem to a PhD candidate as a possible subject for his thesis. Unfortunately, the candidate was not granted a fellowship. What are the compacta of dimension 2 (connected, without local cut points) that can arise as boundaries of convex cocompact Kleinian groups in dimension 4, besides the 2-sphere? Of course, trying to classify possible 2-dimensional boundaries of generic hyperbolic groups seems to be unfeasible. One might hope, though, that with the extra constrain that the groups are Kleinian, so that their boundaries embed in the 3-sphere, one can obtain at least some partial result. The problem of classification can be initially reduced to two subproblems: On one hand, one can try and provide examples of such boundaries (this can be interesting in its own right, for a limited number of examples of hyperbolic manifolds of higher dimension, i.e. > 3, are known) and on the other, one can manage to exclude certain compacta. A.16 Puls, Michael Let 1 < p < and let G be a finitely generated infinite amenable group. A conjecture posed by Gromov during the 1980 s states: The first reduced l p -cohomology space of G vanishes. It is now known that this conjecture is true for 1 < p 2; however the case p > 2 remains open. The first reduced l p -cohomology space of G vanishes if and only if the only p-harmonic functions on G with gradient is l p are the constant functions. This last statement is true precisely when the p-harmonic boundary of G is empty or consists of exactly one point. Hence the following problem is of great interest. Problem: Let 2 < p < and let G be a finitely generated infinite amenable group. Is the cardinality of the p-harmonic boundary of G equal to zero or one? A.17 Sathaye, Bakul Recently I have been interested in the construction of new examples of CAT(0) 4- manifolds that do not support a Riemannian smoothing. The construction uses a nontrivial knot in 3-sphere. The 4-manifold, M, constructed using the knot contains a knotted torus, giving rise to a flat that is knotted locally in the universal cover M. It turns out that this local knottedness goes out to the boundary to give a global knottedness. In particular, the boundary of a flat in the manifold is a copy of S 1 in the boundary of M, which is homeomorphic to the 3-sphere. This gives a knot in M = S 3. Further, if we start with a link in S 3, we get a collection of linked tori, which gives a collection of flats in the universal cover, that are linked locally. I am interested in understanding the relationship between the local and global behavior of flats in CAT(0) manifolds. Particularly, if we have an unknot at every vertex, does the boundary of the flat also give an unknot in the boundary? Or if we start with a link, will the flats also give a non-trivial link in the boundary? How will the link in the boundary be related to the local linking?

9 In this workshop, my aim is to learn more about the current study on the boundaries of CAT(0) groups, and to identify specific open questions and problems to work on. A.18 Sisto, Alessandro I am interested in various kinds of boundaries of groups. The main ones are Bowditch boundaries of relatively hyperbolic groups and boundaries of hierarchically hyperbolic groups. Regarding boundaries of relatively hyperbolic groups, the problem that lately I m focusing on the most (together with Daniel Groves and Jason Manning) is describing what happens to the boundary when performing peripheral fillings. For example, we are able to show the following algebraic version of the classical hyperbolic Dehn filling theorem. Let G be hyperbolic relative to a copy H of Z 2, and suppose that the Bowditch boundary is homeomorphic to the 2 sphere S 2. Then for all but finitely many elements g of H, the group G/ g is hyperbolic and has Gromov boundary homeomorphic to S 2. We use this theorem to show that the relatively hyperbolic version of Cannon s Conjecture can be reduce to the usual non-relative one. Regarding hierarchically hyperbolic groups (HHG), this is a class of groups that includes mapping class groups and groups acting properly and cocompactly on CAT(0) cube complexes. The main point of a hierarchically hyperbolic structure is that one can reduce the study of a given group to the study of a specified family of hyperbolic spaces, as well as many fundamental groups of 3 manifolds. Together with Matt Durham and Mark Hagen, we defined a notion of boundary for HHGs and started to explore its properties. Perhaps one of the most promising directions is developing a theory of geometrically finite subgroups of HHGs, which in the case of mapping class groups should include interesting subgroups like convex-cocompact subgroups, Veech subgroups and Leininger-Reid combinations of Veech subgroups. All such subgroups come with a well-defined boundary map to the boundary of the mapping class group as an HHG (which is not always the case for the compactification given by projective measured laminations). Other applications of the HHG boundary include a new proof rank rigidity for cube complexes, which involves considering stationary measures on the boundary associated to random walks. A.19 Stark, Emily My research is in geometric group theory and low-dimensional topology, and I am interested in the visual boundary of certain hyperbolic groups. With Yael Algom-Kfir, I am studying the visual boundary of hyperbolic free-by-cyclic groups. It was shown by Kapovich Kleiner that the boundary of such a group is homeomorphic to the Menger curve, and we seek to understand the finer structure of this space. In particular, if G is such a group, then the free group embeds in G, and it was shown by Mitra that there is a continuous Cannon-Thurston map from the boundary of the free group onto the boundary of G. The structure of this map was further investigated by Kapovich Lustig, and we are working to refine their analysis. I am very interested in learning more about the structure and existence of Cannon-Thurston maps, the large-scale geometry of free-by-cyclic groups, and ways in which the visual boundary may be used to distinguish quasi-isometry classes. In addition, I hope to better understand the boundary of groups that are relatively hyperbolic. For example, in a project with Chris Hruska and Hung Tran, I am studying 9

10 10 the geometry of relatively hyperbolic surface group amalgams and related right-angled Coxeter groups. We have characterized the quasi-isometry classes among these groups, and I am interested in understanding structures on the boundary of these CAT(0) groups. For example, if one group of this type acts on two different CAT(0) spaces, do these spaces have homeomorphic visual boundaries? A.20 Tshishiku, Bena I m interested in mapping class groups, negatively curved manifolds, and related topics. Here is one question: Thurston compactified Teichmuller space Teich(S) with the space of projective measured foliations PMF(S). The action of the mapping class group Mod(S) on Teich(S) by isometries extends to an action on PMF(S) by homeomorphisms. Associated to this action are Pontryagin classes, since PMF(S) is a sphere. How do these Pontryagin classes relate to the stable cohomology of Mod(S)? Do the Pontryagin classes coincide with the even MMM classes? A.21 Xie, Xiangdong I am interested in the connection between the properties of the boundaries and the properties of the groups (spaces). More specifically, I am interested in the quasiconformal structure of the boundary of Gromov hyperbolic spaces, contracting and Tits boundary of CAT(0) spaces, and their applications to rigidity questions.