Confined Subgroups of Semisimple Lie Groups

by Mikołaj Frączyk (Jagiellonian University, Krakow)

Monday Dec 4, 2023, 2:00 PM → 3:15 PM Europe/Paris

Amphithéâtre Léon Motchane (IHES)

Let G be a semisimple Lie group, e.g. G = SL(n,R). A subgroup Γ of G is called confined if there is a bounded neighborhood of the identity that contains a non-trivial element of every conjugate of Γ. For example, any normal subgroup of a co-compact lattice is confined. In joint work with Tsachik Gelander, we proved that when G has higher rank (e.g. G = SL(n,R) with n>2), a discrete subgroup of G is confined if and only if it is a lattice, which can be seen as an extension of Margulis' Normal Subgroup Theorem. The proof consists of two independent steps that I hope to explain in my talk: 1) the passage from discrete subgroups to stationary random subgroups and 2) the classification of discrete stationary random subgroups in higher rank. If time permits, I will also discuss some open questions related to this work.