Schedule for: 17w5094 - Structure and Geometry of Polish groups

I will establish the ergodic dimension dimension of some families of countable groups such as the generalized Baumslag-Solitar groups, braid groups B n, surface braid groups B n(S g), outer automorphism groups of free groups Out(F n),.... Partial results and open questions will be presented for instance for the mapping class groups MCG(S g) and SL(n,Z).

We show that if m is a conjugation invariant mean on a group G acting minimally and nonelementarily on a tree T then the m-measure of every vertex stabilizer is 1. This is joint work with Yair Hartman and Phillip Wesolek.

I will talk about $L_1$ full groups of ergodic measure-preserving transformations, which are measurable analogues of topological full groups of minimal homeomorphisms of the Cantor space. After describing some of the basic properties of these groups, I will present a short proof that the index map takes values into $\mathbb Z$ which was found with Todor Tsankov. Finally, I will mention some results on the topological rank of $L_1$ full groups.

I will survey a few papers concerning the following question: What can one learn about $G$ from its representation theory into $S$. Where $S=S_\infty$ is the Polish group of all permutations of a countable set. $\text{Hom}(G,S)$ is a Polish space in its own right. We focus on two aspects. How a (Baire) generic representation of $G$ looks like. And the existence of representations with special transitivity properties such as faithful primitive actions, or faithful highly transitive actions of the group.

We provide a direct proof of a recent theorem of Ben-Yaacov, Melleray, and Tsankov. If $G$ is a Polish group and $X$ is a minimal, metrizable $G$-flow with all orbits meager, we use $X$ to produce a new $G$- ow $SG(X$) which is minimal and non-metrizable.

I will report on the progress of a program to understand the finitely generated subgroups of Richard Thompson's group $F$, equipped with the embeddability relation.  We show that this relation contains a well ordered chain of length $\epsilon_0 + 1$.  We also prove that for each $\alpha < \epsilon_0$, there is a finitely generates EA subgroup of $F$ of class $\alpha+2$.  This is joint work with Collin Bleak and Matt Brin.

(Joint work with A. Le Boudec) We begin by giving a detailed overview of the tree almost automorphism groups and describing their relationship to Higman-Thompson groups and topological full groups. We then show each almost automorphism has one of two possible types, corresponding to the dynamics of the action on the boundary. We next consider the subgroups such that every element is contained in a compact subgroup; such groups are the topological analogue of torsion subgroups. We characterize these subgroups in terms of the dynamics of their action on the boundary and deduce that they are indeed locally elliptic - i.e. every finite set is contained in a compact subgroup. We finally consider the commensurated subgroups of almost automorphism groups; these subgroups generalize normal subgroups. We show every commensurated closed subgroup of an almost automorphism group is either finite, compact and open, or equal to the entire group.

Let $G$ be a locally compact group, let $K$ be a closed subgroup of $G$, and let $H$ be a group of automorphisms of $G$ such that $h(K) = K$ for all $h in H$. When is the action of $H$ on $G/K$ a small invariant neighbourhoods (SIN) action, i.e. when is there a basis of neighbourhoods of the trivial coset consisting of $H$-invariant sets? In general, the SIN property is a strong restriction, but when $G$ is totally disconnected and $H$ is compactly generated, it turns out to be equivalent to the seemingly weaker condition that the action of $H$ on $G/K$ is distal on some neighbourhood of the trivial coset. (The analogous statement is false in the connected case: compact nilmanifolds give rise to counterexamples.) This has some general consequences for the structure of t.d.l.c. groups: for example, given any compact subset $X$ of a t.d.l.c. group $G$, there is an open subgroup containing $X$ that is the unique smallest such up to finite index.

The scale of the endomorphism $\alpha$ of the totally disconnected, locally compact (t.d.l.c.) group $G$ is a positive integer defined to be the minimum of the indices $[\alpha(U) : \alpha(U)\cap U]$, where $U$ ranges over the compact open subgroups of $G$. Existing methods for computing the scale draw on analogies with computing eigenvalues in linear algebra. These methods generally do not match the effectiveness of linear algebra however, the principal obstacle being the lack of a general description of t.d.l.c.~groups.

For locally compact groups amenability and Kazhdan's property (T) are mutually exclusive in the sense that a group having both properties is compact. This is no longer true for more general Polish groups. However, a weaker result still holds for SIN groups (topological groups admitting a basis of conjugation-invariant neighbourhoods of identity): if such a group admits sufficiently many unitary representations, then it is precompact as soon as it is amenable and has the strong property (T) (i.e. admits a finite Kazhdan set). If an amenable topological group with property (T) admits a faithful uniformly continuous representation, then it is maximally almost periodic. In particular, an extremely amenable SIN group never has strong property (T), and an extremely amenable subgroup of unitary operators in the uniform topology is never a Kazhdan group. This leads to first examples distinguishing between property (T) and property (FH) in the class of Polish groups. Disproving a 2003 conjecture by Bekka, we construct a complete, separable, minimally almost periodic topological group with property (T), having no finite Kazhdan set. Finally, as a curiosity, we observe that the class of topological groups with property (T) is closed under arbitrary infinite products with the usual product topology. A large number of questions about various particular topological groups remain open. The talk is based on the preprint arXiv:1512.01572v3 [math.GR], to appear in Trans. Am. Math. Soc., never before presented at a conference.

Using results on the structure of the topological full groups of minimal subshifts, we prove that the isomorphism relation on the space of finitely generated complete groups is not smooth. 

A dendrite is a continuum (i.e. a compact connected metrizable space) that is locally connected and such that any two points are joined by a unique arc. T. Wazewski introduced a universal dendrite in which any other dendrite can embedded. Its construction can be generalized to yield an uncountable family of non-homeomorphic dendrites. Their homeomorphism groups are closed subgroups of $S_\infty$ and thus are Polish groups. Moreover some of them are oligomorphic. In this talk, we will be interested in structural and topological properties of these groups like simplicity, Bergman property, property $(T)$, existence of a dense or comeager conjugacy class, automatic continuity or small index property.

In this talk I will present joint work with Andreas Thom on bounded normal generation (BNG) for projective unitary groups of von Neumann algebras. We say that a group has (BNG) if the conjugacy class of every nontrivial element and of its inverse generate the whole group in finitely many steps. After explaining how one can prove (BNG) for the projective unitary group of a finite factor, I will present applications to automatic continuity of homomorphisms with SIN target groups. The talk will be closed with recent results on the Bergman property for unitary groups of II_1 factors and countable cofinality for compact connected Lie groups.

Let ${\rm Aut}(M)$ be the automorphism group of a countable structure $M$. We show several new results about conjugacy classes, topological similarity classes, and cyclically dense conjugacy classes of groups ${\rm Aut}(M)$ and of groups $L_0({\rm Aut}(M))$ of Lebesgue measurable functions defined on $[0,1]$ with values in ${\rm Aut}(M)$. This is joint work with Maciej Malicki.

Generalizing classical work of Day and Følner for discrete groups, I will present characterizations of amenability of (not necessarily locally compact) topological groups in terms of the existence of almost invariant vectors and almost invariant finite subsets, and discuss some applications of these results, e.g., concerning the coarse geometry of Polish groups. This is joint work with Andreas Thom. Furthermore, linked with concentration of measure, the mentioned amenability criteria provide sufficient conditions for (a strong form of) extreme amenability, which can be used to prove the extreme amenability of topological groups of measurable maps. This is joint work with Vladimir Pestov.

We will discuss a recent approach to non amenability based on studying Liouville actions. We give a reformulation of Liouville property, which in its turn implies that this approach can not be used for strongly transitive actions. This is a joint work with Tianyi Zheng.

An old result of Dougherty, Jackson and Kechris implies that the boundary action of the free group F2 induces a hyperfinite equivalence relation. During the talk, I will discuss generalizations of this theorem to the class of hyperbolic groups. The examples discussed will include groups acting properly and cocompactly on CAT(0) cube complexes. This is joint work with Jingyin Huang and Forte Shinko

Using the metric version of the KPT correspondence, we prove that the automorphisms groups of several limits of finite dimensional operator spaces and systems are extremely amenable, including the Gurarij space and its non-commutative version $\mathbb{NG}$. Dually, we prove that the universal minimal flow of the Poulsen simplex $\mathbb P$ is $\mathbb P$ itself, and again similarly for its non-commutative version $\mathbb{NP}$. The approximate Ramsey properties (ARP) we find are consequence of the dual Ramsey Theorem (DRT) by Graham and Rothschild. In a similar way, we will see present an approximate Ramsey property for quasi-equipartitions and how to use it to deduce the ARP of the family $\{\ell_p^n\}_n$, $1\le p\neq 2<\infty$. We will also discuss the reformulation of the (ARP) of $\{\ell_p^n\}_{n\in \N}$ as a weak version of a multidimensional Borsuk-Ulam theorem. Finally, we will see that
- the DRT is a particular case of a factorization theorem for 0-1 valued matrices,
- the Graham-Leeb-Rothschild Theorem on grassmannians over a finite field $\mathbb F$ is a particular case of a factorization theorem for matrices with values in $\mathbb F$, and
- the ARP of $\{\ell_p^n\}_n$ is a particular case of a factorization theorem for matrices with values in $\mathbb R$ or $\mathbb C$.
This is a joint work with D. Bartosova, M. Lupini and B. Mbombo, and with V. Ferenczi, B. Mbombo and S. Todorcevic.

I will discuss novel applications of continuous logic to ergodic theory, particularly to the study of rigidity phenomena associated with strongly ergodic actions of countable groups. This is joint work in progress with François Le Maître and Todor Tsankov.

After Gromov's results on Geometric Group Theory, large scale geometry of finitely generated groups has been widely studied during last decades. The theory was successfully extended to countable groups thanks to the work of Dranishnikov and Smith. However, in order to study the large scale properties of uncountable (topological) groups, one has to use coarse structures. After a brief introduction of Coarse Geometry on groups, the first steps of a joint work with D. Dikranjan are presented. In particular, we introduce the category of coarse groups and we propose an enlargement of its quotient category by means of quasi-morphisms.

Let G be a locally compact group and let Sub(G) denote the compact space of closed subgroups of G. G acts naturally on Sub (G) by conjugation. A uniformly recurrent subgroup (URS) of G is a closed, minimal subset of Sub (G). To every minimal topological dynamical system of G one can naturally associate its stabilizer URS and Glasner and Weiss asked whether every URS can be obtained in this manner. We answer this question in the affirmative by constructing, for a fixed URS H, a G-flow with stabilizer URS equal to H and universal for all minimal flows whose stabilizer URS is subordinate to H. This is joint work with Nicolas Matte Bon.