Miklós Abért: Invariant random subgroups and their applications

Speaker: Miklós Abért

Title: Invariant random subgroups and their applications

Abstract: An invariant random subgroup (IRS) of a discrete group Gamma is a random subgroup of Gamma whose distribution is invariant under conjugation. IRS-s are natural generalizations of normal subgroups and every finite index subgroup (or, more generally, every lattice) gives rise to an IRS. IRS-s are also inherently related to probability measure preserving actions of the group.

It turns out that weak* convergence of IRS-s can be expressed as Benjamini-Schramm convegence of certain quotient spaces. We show two recent applications, one in graph theory, the other in the theory of Lie groups.
The first says that finite Ramanujan graphs have large essential girth, a joint work with Glasner and Virag. The second says that for a higher rank simple Lie group G, every sequence of locally symmetric G-spaces with volume tending to infinity must Benjamini-Schramm converge to the symmetric space of G. This is joint work with Bergeron, Biringer, Gelander, Nikolov, Raimbault and Samet.

All notions will be carefully explained.