We will discuss a higher-degree version of property T and a recent proof that arithmetic lattices in a semisimple Lie group G satisfy it below the rank of G. The proof relies on functional analysis and the polynomiality of higher Dehn functions of arithmetic lattices below the rank. Along the way we will discuss rigidity results regarding unitary representations of semisimple groups and their lattices and present some relevant conjectures. If time permits, we will discuss some applications to dynamics, to character rigidity and to stability. Based on a joint work with Roman Sauer.

We say that a finitely generated group is self-simulable if every computable action of the group on an effectively closed zero-dimensional space (this means that both the space and the action can be described by a Turing machine in a specific way) is the topological factor of a subshift of finite type on said group. We will show that these groups do exist and that their class is stable under commensurability and quasi-isometries of finitely presented groups. We shall present several examples of well-known groups which are self-simulable, such as Thompson's V and higher-dimensional general lineal groups. We shall also show that Thompson's group F is self-simulable if and only if it is non-amenable, therefore giving a new characterization of this well-known open problem. Joint work with Mathieu Sablik and Ville Salo.

Actions of a group by measure preserving transformations on a probability space may have properties which are much stronger than ergodicity; two such properties are the existence of a spectral gap and strong ergodicity (in the sense of K. Schmidt). In general, amenability of the acting group is an obstruction for these properties. We will discuss these strong versions of ergodicity for actions on solenoids and nilmanifolds by groups of automorphisms. The results we will present, which are joint work with C. Francini and Y. Guivarc'h, will show that, in some sense, amenability is the only obstruction for the properties above.

Soficity is an intriguing approximation property for groups that was introduced in a seminal work of M. Gromov and singled out by B. Weiss. Like other approximation properties proceeding it, such as residual finiteness and amenability, soficity asks for the group to be approximated by finite groups, in a suitable sense. A weaker notion, called hyperlinearity, can be defined using approximation by unitary matrix groups. A big open question is to find examples (if they exist!) of non-sofic, or non-hyperlinear groups. On the other side of our story, several new rigidity properties of groups, colloquially referred to as stability, have gathered considerable attention. Among them is flexible Hilbert Schmidt stability: A group G is flexibly HS-stable if any approximate finite dimensional unitary representation of G is close to a compression of a genuine representation of possibly larger dimension. In this talk, we present our recent work giving conditional statements of the form "If G is flexibly HS-stable, then there exists a non hyperlinear group". This statement is shown to hold for various groups with property (T), such as some lattices in Lie groups, and Gromov random groups. The proof technique, based on the work of Ioana, Spaas and Wiersma, is operator algebraic in nature.

In any dimension, there is a hyperbolic space (simply connected Riemannian manifold of constant sectional curvature -1), even in infinite (countable) dimension. We will be interested in the isometry group of this infinite dimensional hyperbolic space. This group has a natural Polish topology and we will be interested in its actions on compact spaces. A comparison with the isometry group of a Hilbert space will be a common thread.

[abstract]

We say that two groups are orbit equivalent (OE) if they both act on a same probability space with the same orbits. A famous result of Ornstein and Weiss states that all infinite amenable groups are orbit equivalent to Z. In other words: orbit equivalence does not take into account the geometry of groups. Delabie, Koivisto, Le Maître and Tessera offer in a recent article to refine this relation with a quantitative version of OE. They obtain obstructions to the existence of such equivalence using the isoperimetric profile. After defining the quantitative version of OE we will focus in this talk on the "inverse problem", namely: can we find a group that is OE to a prescribed group with prescribed quantification? Using the diagonal products introduced by Brieussel and Zheng we will answer this question in the case of a prescribed OE with Z.

In this talk I will discuss a new criterion for amenability of a group based on the idea of partial actions. As an application I will show a conjecture of Cyr and Kra, namely that every 0 entropy minimal subshift has an amenable automorphism group.

We use random walks on a hyperbolic group to construct an expansive flow on a space with a Hoelder structure. We then relate the simplicial volume of the space to the conformal dimension of the Gromov boundary of the group.

Can one relate random walks on a group with random walks on a dense subgroup of it? We develop a technique to do it in some cases. This allows us to exhibit some new interesting phenomena in Furstenberg-Poisson boundary theory. Joint work with Michael Björklund and Hanna Oppelmayer.

In this talk, I will present a noncommutative analogue of Bader-Shalom factor theorem for lattices with dense projections in product groups. Combining with previous works, we obtain a noncommutative analogue of Margulis factor theorem for all irreducible lattices in higher rank semisimple algebraic groups. Namely, we give a complete description of all intermediate von Neumann subalgebras sitting between the group von Neumann algebra associated with the lattice and the group measure space von Neumann algebra associated with the action of the lattice on the Furstenberg-Poisson boundary. This is joint work with Rémi Boutonnet.

Bounded orbit equivalence (BOE) for pmp actions of countable groups is a strong quantitative version of orbit equivalence. In this talk we will discuss some results related to the following broad question: given a countable group, what can be said about its pmp actions up to BOE? I will introduce the notion of isometric orbit equivalence, which is defined for pmp actions of finitely generated groups and implies BOE. I will present a general construction of isometric orbit equivalences for any finitely generated group G, and show that it leads to interesting nontrivial BOE when G is a free group.

The talk is concerned with the general problems of determining ideal and trace structures of C*-algebras associated to minimal actions of discrete groups G on locally compact spaces X. We state a uniqueness theorem for a class of morphisms in the category of C*-dynamical systems, and use that to give a complete characterization of simplicity and existence/uniqueness of traces on C*-algebras generated by covariant representations arising from stabilizer subgroups. We will give applications in concrete examples. This is joint work with Eduardo Scarparo.

Irwin and Solecki introduced a projective Fraïssé limit, a dual concept to the (injective) Fraïssé limit from model theory. They used it to construct the pseudo-arc, a one-dimensional homogeneous continuum (compact and connected space), which is generic among the continua in an Euclidean space of dimension at least 2. They considered the class of finite linear (combinatorial) graphs together with epimorphisms preserving the edge relation, and showed that the topological realization of its projective Fraïssé limit is the pseudo-arc. We show that finite connected graphs with confluent epimorphism form a projective Fraïssé class and we investigate the continuum obtained as the topological realization of its projective Fraïssé limit. We show that this continuum is indecomposable, but not hereditarily indecomposable, as arc-components are dense. It is one-dimensional, pointwise self-homeomorphic, but not homogeneous, and each point is the top of the Cantor fan. Moreover, it is hereditarily unicoherent, in particular, it does not embed a circle; however, it embeds the universal solenoid and the pseudo-arc. This is joint work with W. J. Charatonik and R. P. Roe.

The finitely generated groups Γ for which we have a comprehensive understanding of all commensurated subgroups of Γ are rare. We prove a theorem that relates the commensurated subgroups of a group with the topological dynamics of its micro-supported actions on compact spaces. As an application we obtain a criterion to exclude the existence of non-trivial commensurated subgroups in certain classes of groups. Examples include topological full groups of minimal Z-actions, or more generally topological full groups of amenable groups. This is joint work with Pierre-Emmanuel Caprace.

In this talk, I will present the following result (joint with Marcin Sabok), at the intersection of geometric group theory and descriptive set theory: given a hyperbolic group G, the orbital equivalence relation induced by the G-action on its Gromov boundary is hyperfinite. I will start by explaining and motivating the concepts in this statement, and will then sketch an idea of its proof.

Many groups are known to act on the real line by homeomorphisms; indeed it is well-known that the existence of a faithful such action is equivalent to left-orderability. When this is the case, it is natural to try to understand the possible actions of the group on the real line. From a Borel perspective, one way to formalise this question consists in asking for which group G the semi-conjugacy equivalence relation on the space of actions on the real line of G is smooth. I will address this and some finer related questions for some classes of left-orderable groups, including finitely generated solvable groups and Thompson’s group. The talk is based on joint works with J. Brum, C. Rivas and M. Triestino.

It is a significant problem in the theory of C*-algebras to determine which topological dynamical systems give rise to classifiable crossed products. For the positive direction, the best available method is to show that the action is almost finite (the notion introduced by Kerr), which in the case of zero-dimensional systems reduces to showing that the space can be partitioned into clopen Rokhlin towers. We give a short proof of the fact that the action of G on X is almost finite if the action restricted to some infinite normal subgroup of G is almost finite.

We introduce and analyse a cycle-cutting algorithm on a weighted locally finite graph G, which yields a weight-maximal subforest F of G. The main feature of this subforest is that if a G-component C has ≥3 nonvanishing ends, then so does at least one F-component in C. For a quasi-pmp Borel graph G with ≥3 nonvanishing ends per component, this subforest witnesses the nonamenability of G (according to a recent result of Robin Tucker-Drob and myself). We also introduce a random version of this subforest, which generalizes the Free Minimal Spanning Forest, to capture nonunimodularity in the context of percolation theory. This is joint work with Ruiyuan Chen and Grigory Terlov.

We give a comprehensive structural analysis of amenable subrelations of a treed quasi-measure-preserving equivalence relation. The main philosophy is to understand the behavior of the Radon-Nikodym cocycle in terms of the geometry of the amenable subrelation within the treeing. This allows us to extend structural results that were previously only known in the measure-preserving setting, e.g., we show that every nowhere smooth amenable subrelation is contained in a unique maximal amenable subrelation. Two of the main ingredients are an extension of Carrière and Ghys's criterion for nonamenability, along with a new Ping-Pong-style argument we call the "Paddle-ball lemma" that we use to apply this criterion in our setting. This is joint work with Anush Tserunyan.

In this talk we will discuss some aspect of the question: given a group, what is the range of the Furstenberg entropy of ergodic stationary actions of it? For the special linear group SL(d,R) and its lattices, constraints on this spectrum come from Nevo-Zimmer structure theorems; and entropy values can be realized based on random walks on random stationary graphs.