Measured Group Theory studies groups through their measurable actions. It investigates classification, rigidity and invariants of (actions of) discrete countable or locally compact groups and the ways these are encoded in the orbit structure of the actions.

The roots of the subject can be found in ergodic theory and operator algebras. Some milestones are Ornstein-Weiss' theorem on amenability and Zimmer's work on super-rigidity for lattices in higher rank Lie groups.

The subject gained a new trend when Gromov suggested measured group theory as a relative to geometric group theory and as a common framework to better understand the family of (cocompact and non-cocompact) lattices in a fixed Lie group.

In the last 20 years, measured group theory has known dramatic developments with the discovery of several new invariants such as cost and L

It has also established new and sometimes quite unexpected connections between various areas, including arithmetic groups, geometric group theory, von Neumann algebras, asymptotic group theory, Borel dynamics and combinatorics, percolation and graph convergence.

I will survey this theory with some digressions depending on the knowledge of the audience.

There is a rich interplay between the degree of regularity of a group action on the circle and the allowable algebraic structure of the group. In this series of lectures, I will outline some highlights of this theory, culminating in a construction due to Kim and myself of groups of every possible critical regularity α ∈ [1,∞).

The study of random walks and isoperimetry on discrete groups goes back to the works of Varopoulos in the early 80s. We will explain some basic isoperimetric inequalities, connections to behavior of random walks and tools such as comparison techniques for proving such inequalities. We will describe several constructions of groups with prescribed isoperimetric profiles. In the last lecture we will explain how isoperimetric considerations on induced Schreier graphs play a role in finding efficient probability measures with non-trivial Poisson boundary on Grigorchuk groups.

(joint with V. Guirardel and Koji Fujiwara) In studying the possible subgroups of the groups IET of all Interval Exchange Transformations, one can focus on the solvable ones. It turns out that, although the torsion-free solvable subgroups of IET are not so interesting, constuctions of the type of lamplighters produces many (uncountably many) non-isomorphic solvable subgroups, containing a lot of torsion.

I will present the classification of locally discrete finitely generated groups of analytic diffeomorphisms of the circle having the expansion property. If time permits I will survey the current knowledge on this problem when expansion does not hold.

In my talk I will address the famous question of M.Kac (traced back to L.Bers and A.Weyl) "Can one hear the shape of a drum?" in the context of groups viewed as geometric objects. I will explain why the answer in NO in a strong sense: there is a continuum family of 4-generated pairwise not quasi-isometric groups with the same spectrum of discrete Laplacian. Moreover, each of these groups has an uncountable family of amenable covering groups with the same spectrum.

The arguments will be based on the construction by the speaker of groups of intermediate growth (between polynomial and exponential), and the result in the spectral theory of graphs which somehow is related to the famous Hulanicki criterion of amenability of groups in terms of weak containment of unitary representations. The talk is based on joint results with A.Dudko and with T.Nagnibeda and A.Peres.

I will show that the alternating group of a topologically free action of a countably infinite amenable group on the Cantor set has property Γ (and in particular is inner amenable) and that there are large classes of such groups which are simple, finitely generated, and nonamenable. This is joint work with Robin Tucker-Drob.

Given a class of groups, on can ask to describe all possible homomorphisms between groups in the class. I will explain the answer to this question for various classes of groups of dynamical origin: topological full groups of Cantor minimal systems, a class groups of interval exchange transformations, the Higman-Thompson groups and their higher-dimensional cousins. After doing so, I will explain how these results are deduced from a common theorem, stated in the language of pseudogroups and étale groupoids.

Topological full groups of tail groupoids of Bratteli diagrams (a.k.a. AF groupoids) form a well-known class of locally finite groups. We will discuss how small modifications of these groupoids produce groups with interesting properties: infinite finitely generated torsion groups, groups of intermediate growth, non-elementary amenable groups, etc.. The precise conditions on the modifications producing these properties are not well understood, and several open questions will be discussed.

Invariant random subgroups of a topological (e.g. discrete) group are conjugation-invariant Borel probability measures on its Chabauty space of closed subgroups. They are thus in some sense "dynamical systems of group-theoretical origin". Their study in various contexts proves to be quite inspiring, and they have some spectacular applications to group theory and geometry. After a general introduction I will shortly discuss their known properties in the setting of Lie groups and their discrete subgroups, and outline some applications (most of this is joint work with many people, and some work where I was not involved).

Abstract: A positive cone in a group is the set of group elements which are positive in a total, left-multiplication invariant order of the group. Naturally, the set of all positive cones in a group is a compact set, and a basic question is to try to understand isolated points in that space and groups supporting such isolated orders. In this talk, I will review some of the results regarding the above questions, and try to relate/compare them to regularity properties of the languages describing such positive cones. We will put special attention when the underlying group is word-hyperbolic.

Thompson's group

References:

[1] R. Geoghegan and F. Guzmán, Associativity and Thompson's group, Contemporary Mathematics 394 (2006), 113-135

[2] M.V. Lawson, A class of subgroups of Thompson's group

A group is said to be strongly amenable if each of its proximal topological actions has a fixed point. We show that a finitely generated group is strongly amenable if and only if it is virtually nilpotent. More generally, a countable discrete group is strongly amenable if and only if none of its nontrivial quotients have the infinite conjugacy class property.

Joint with Joshua Frisch and Pooya Vahidi Ferdowsi.