ACTION NOW WANDERING SEMINAR

Let G be a group and mu a probability measure on G. A mu stationary random subgroup (SRS) of G is a probability measure on the space of closed subgroups of G which is stationary with respect to the mu random walk.

In joint work with Arie Levit, we show that for mu either a finitely supported measure on a hyperbolic group or the discretization of Brownian motion on a rank 1 simple Lie group, any mu-SRS which is almost surely infinite and discrete is supported on groups whose exponential growth rate is at least the entropy divided by twice the drift of the mu random walk. This generalizes our earlier work for invariant random subgroups and is one in a long line of works showing that invariant and stationary random subgroups are quite large.

In mathematics in general, it is fruitful to allow randomness. Indeed, it is often easier to deal with random rather than deterministic objects. It seems miraculous, however, when we are able to say more about deterministic objects by treating them as random ones. This idea applies in particular to the theory of locally symmetric manifolds and discrete subgroups of Lie groups.

The theory of invariant random subgroups (IRS), which has been developed quite rapidly during the last decade, has been very fruitful to the study of lattices and their asymptotic invariants. However, restricting to invariant measures limits the scope of problems that one can approach (in particular since the groups involved are highly non-amenable). It was recently realised that the more general notion of stationary random subgroups (SRS) is still very effective and opens paths to deal with questions which were thought to be unreachable.

In the talk I will describe various old and new results concerning arithmetic groups and general locally symmetric manifolds of finite as well as infinite volume that can be proved using `randomness', e.g.: 1. Kazhdan-Margulis minimal covolume theorem.

2. Most hyperbolic manifolds are non-arithmetic (a joint work with A. Levit).

3. Higher rank manifolds of large volume have a large injectivity radius (joint with Abert, Bergeron, Biringer, Nikolov, Raimbault and Samet).

4. Margulis' infinite injectivity radius conjecture: For manifolds of rank at least 2, finite volume is equivalent to bounded injectivity radius (joint with M. Fraczyk).

I will present a curious example showing that a set of periodic orbits of diffeomorphisms of the two dimensional torus is exactly equal to the set of different fibrations of the Whithead link exterior. This seems to suggest there is a connection we do not understand between the concept of "forcing" in dynamical systems, which is an order relation on periodic orbits, and the concept of different fibrations of a link exterior corresponding to a face of the Thurston norm ball. This is joint work with Eiko Kin and is based on previous work with Bronek Wajnryb.

The Morse boundary of a metric space X is a topological space that encodes the "hyperbolic directions" of X. When X is not hyperbolic, its Morse boundary is not even metrisable, which makes it sound like it should be impossible to understand. As it turns out, however, there are various results that describe the Morse boundaries of various interesting groups, some even giving complete descriptions of the homeomorphism type. The talk will be an overview of these results.

We describe the characters of metablian and polycyclic groups and show they are all monomial. Relying on the criterion of Hadwin and Shulman, we find that stability questions for such groups boil down to purely dynamical questions. For example, Hilbert-Schmidt stability of polycyclic groups is equivalent to a question of toral dynamics, and thus related to classical conjectures on the manner. We deduce that all virtually nilpotent groups, free metabelian groups, wreath products (e.g lamplighter) and certain upper trianglular groups are Hilbert-Schmidt stable. This is a joint work with Arie Levit.