Groups and Dynamics Seminar

Date: October 24, 2017

Time: 3:00PM - 4:00PM

Location: BLOC 220

Speaker: Rostyslav Kravchenko, Northwestern University

Title: Characteristic random subgroups and their applications

Abstract: The invariant random subgroups (IRS's) were implicitly used by Stuck and Zimmer in 1994 in their study of lattices in simple Lie groups of higher rank. Around year 2010 there appeared several important papers that dealt with IRS's, among them papers by Vershik, Grigorchuk and Bowen. The name itself was coined by Abert, Glasner and Virag, who generalized a classical theorem of Kesten to the case of IRS's. Since then IRS's were actively studied, in particular Bowen has investigated the set of IRS's of free groups of finite rank, and Glasner studied it for linear groups. We define the notion of characteristic random subgroups (CRS's) which are a natural analog of IRS’s for the case of the group of all automorphisms. We determine CRS's for free abelian groups of infinite rank and for free groups of finite rank. Using our results on CRS's of free groups we show that for groups of geometrical nature (like hyperbolic groups, mapping class groups and outer automorphisms groups) there are infinitely many continuous ergodic IRS's. This is a joint work with L. Bowen and R. Grigorchuk.