Groups and dynamics workshop 2025 Aussois

Genericity in topological dynamics

Given a countable group \(G\), the space of actions of \(G\) on the Cantor space has a natural Polish structure (and one may also study some of its subspaces, e.g., transitive actions or minimal actions). We will briefly discuss why, to understand the properties of this space of actions, one is led to study spaces of subshifts; and how an abstract, general criterion leads to notions of isolated/projectively isolated subshifts. Density of the latter set is connected to the existence of a comeager conjugacy class in the space of actions of \(G\) (this was first observed by Doucha; our general approach enables us to generalize this criterion to the space of minimal actions). Then we will discuss some more concrete questions, motivated by this general approach - in particular, we will investigate in some depth the case of non-finitely generated amenable groups, as well as the case of finitely generated free groups. In this latter case, it turns out that there exists a generic minimal action of any free group on the Cantor space, as well a a generic minimal measure-preserving action. If time allows, we will also discuss spaces of transitive actions, where the situation turns out to be quite different. Hopefully, we will have some time to discuss some related open problems, one of which is the following: does there exist a non-finitely generated countable group which admits a nontrivial minimal subshift of finite type?

No prerequisite in topological dynamics are expected. These two talks are based on joint work of Doucha, Melleray, and Tsankov.

Profinite rigidity of Kähler groups

A classical problem in complex algebraic geometry is understanding the topology of closed complex submanifolds of complex projective space, so-called smooth complex projective varieties, and, more generally, of compact Kähler manifolds. Two natural invariants to consider are the fundamental group and its profinite completion; the latter is also known as the algebraic fundamental group. In this mini course we will address the following questions: When is the fundamental group of a compact Kähler manifold determined by its profinite completion? And, when does the profinite completion even determine the homeomorphism type of the underlying manifold? In particular, we will explain positive answers to both questions in the case of a direct product of closed non-positively curved Riemann surfaces. As we will explain, this and other related rigidity results are a consequence of the more general profinite invariance of a certain universal morphism from a Kähler group to a direct product of orbisurface groups; the latter will be defined in the course. Our course is based on joint work with Hughes, Stover and Vidussi.

Dissociated permutation groups

Dissociation is a natural property that permutation groups can satisfy. It has equivalent formulations in both unitary representation theory and ergodic theory. First observed for automorphism groups of \(\omega\)-categorical structures as a consequence of the classification of their unitary representations, dissociation alone is enough to recover classification results. In this talk, I will give examples of dissociated permutation groups, describe methods to obtain dissociation and discuss the various consequences of this property regarding both unitary representation theory and ergodic theory. Joint work with Colin Jahel and Matthieu Joseph.

Additive invariants for near actions

A near action of a group on a set, is, roughly speaking, the same as an action, except that group elements acts modulo indeterminacy on finite subsets – in other words, it is a "germ at infinity" of an action on a set. It is natural, but hopeless for most groups, to aim at a classification of near actions of a given group. An additive invariant for near actions of a group \(G\), in an abelian group \(A\) is a map from the "set" of near \(G\)-sets to \(A\), that is additive under taking disjoint unions. There is always a "universal" additive invariant \(\Phi(G)\) for a group \(G\). In some vague sense, it classifies near \(G\)-sets modulo \(G\)-sets. For instance, when \(G\) is a finite group or more generally free product of finite groups, \(\Phi(G) = 0\). We will discuss \(\Phi(G)\) and variants (e.g., considering some special class of near \(G\)-sets), and also exhibit groups with \(\Phi(G)\) finite cyclic but not trivial.

On various profinite completions of groups acting on rooted trees

Groups that act faithfully on rooted trees can be completed in different ways to obtain profinite groups. The profinite completion of the group maps onto each of them. Determining the kernels of these maps is known as the congruence subgroup problem (by analogy with a similar problem in arithmetic groups). This problem has been studied by various authors over the years, most notably for self-similar groups and (weakly) branch groups. In the case of self-similar regular branch groups, much insight can be gained into this problem using a symbolic-dynamical point of view. After reviewing some of the known results on the congruence subgroup problem for various combinations of self-similar and (weakly) branch groups, I plan to talk on joint work with Zoran Sunic where we use the symbolic-dynamical approach describe explicitly one of the kernels of the congruence subgroup problem, for certain self-similar branch groups. Examples will be given. No previous knowledge of self-similar or branch groups is required.

The Poisson boundary of Thompson's T is not the circle
(joint with Cosmas Kravaris and Eduardo Silva)

The Poisson boundary of a countable group G endowed with a probability measure μ is a probability G-space that encodes all significant asymptotic information encoded in the random walk driven by μ. Given a "natural" action of G on a measurable space equipped with a μ-stationary probability measure, one can ask whether this action is a model for the Poisson boundary. For groups G acting proximally on the circle S¹, the circle with its unique μ-stationary measure is an example of such an action, and for discrete subgroups of PSL₂(R) this space coincides with the Poisson boundary. We prove that for a large class of groups acting proximally on S¹ (that share features with Thompson's group T) this is not the case, thus answering a question of Deroin and Navas. The purpose of this talk is to define the previous terms and to introduce some tools in the proof of the result, such as the conditional entropy criterion of Kaimanovich.

Automorphisms of hyperbolic groups and growth

Let \(G\) be a torsion-free hyperbolic group, let \(f\) be an automorphism of \(G\), and let \(g\) be an element of \(G\). We will study the possible growth types for the word length of \(f^n(g)\), as \(n\) goes to infinity. This is a joint work with Rémi Coulon, Arnaud Hilion and Gilbert Levitt.

Invariant random sub-von Neumann algebras

The notion of IRS (invariant random subgroup) is well-studied in dynamics on groups. We extend this notion to von Neumann algebras, more precisely to the group von Neumann algebra \(LG\) of a discrete countable group \(G\). We call this concept IRA (invariant random sub-algebra). In particular, we study the case of amenable IRAs, i.e. almost every sub-von Neumann algebra of \(LG\) is amenable. This generalises a result of Bader-Duchesne-Lécureux about amenable IRSs. This is joint work with Tattwamasi Amrutam and Yair Hartman.

Geometry of groups and profinite completion, after M. Bridson, A. Reid et al.

The general context of this presentation is the following question: what are the groups which are characterized by the list of their finite quotients? The associated equivalence relation is the one that puts in the same class groups with the same profinite completion. A group is said to be (absolutely) profinitely rigidi if it is the only element in its equivalence class; in other words, if any infinite finitely generated group with the same profinite completion is isomorphic to it. The problem of exhibiting many classes of groups enjoying this rigidity property is still in its beginnings. We will present constructions of groups possessing the property of absolute profinite rigidity, as well as results on a weakened version of it (relating to a restricted class of groups put into comparison). The techniques used are relevant to representation theory, to the topology and geometry of low-dimensional manifolds, and to arguments from the study of arithmetic groups.

Groups of cellular automata on groups

We study the (topological) automorphism group \(\operatorname{Aut}(A^G)\) of a full shift \(A^G\), where \(A\) is a finite alphabet and \(G\) is a finitely-generated infinite group acting on \(A^G\) by the shift map. These are also known as groups of cellular automata. Even in the case \(G = \mathbf{Z}\), these groups exhibit complicated behavior. We exhibit nontrivial obstructions to embeddability between \(\operatorname{Aut}(A^G)\) and \(\operatorname{Aut}(A^H)\) for distinct \(G\), \(H\). These come from growth rate and quantitative residual finiteness properties of \(G\) and \(H\). In particular, \(\operatorname{Aut}(A^{\mathbf{Z}^{d+1}})\) does not embed into \(\operatorname{Aut}(B^{\mathbf{Z}^d})\). We also show that groups \(\operatorname{Aut}(A^G)\) for different choices of G provide natural examples of groups with word problems far above polynomial time.

Continuity of asymptotic entropy on groups

The asymptotic entropy of a random walk on a countable group is a non-negative number that determines the existence of non-constant bounded harmonic functions on the group. A natural question to ask is whether the asymptotic entropy, seen as a function of the step distribution of the random walk, is continuous. In this talk, I will explain two recent results on the continuity of asymptotic entropy: one for groups whose Poisson boundaries can be identified with a compact metric space carrying a unique stationary measure, and another for wreath products \(A \wr \mathbf{Z}^d\), where \(A\) is a countable group and \(d \geq 3\).

Uniformly Amenablity and Hyperfiniteness of Treeable Equivalence Relations

It is well-known that hyperfinite countable Borel equivalence relations, shortly CBERs, are treeable (that is, the equivalence classes can be realized as connected components of an acylcic Borel graph) and amenable. It is an important open problem whether amenable CBERs are hyperfinite, and not much is known even if one additionally assumes treeability. In this talk I will address this problem and introduce the notion of uniform amenability for CBERs, a property that implies amenability and which is automatic for orbit equivalence relations of continuous amenable actions on compact Polish spaces, and for Borel actions of amenable groups. I will then show that if a CBER is uniformly amenable and treeable by a tree with finite degree, then it is hyperfinite. The result that I will present is actually stronger and will rely on the notion of Borel asymptotic dimension. As corollaries, I will show that the orbit equivalence relation of a free, amenable, continuous action by a finitely generated free group on a sigma-compact Polish space is hyperfinite, and that the orbit equivalence relation of a Borel action by an amenable group is hyperfinite, if it is treeable. The material presented is part of a joint work with Petr Naryshkin.