Séminaire Géométrie et groupes discrets

Confined Subgroups of Semisimple Lie Groups

by Mikołaj Frączyk (Jagiellonian University, Krakow)

Amphithéâtre Léon Motchane (IHES)

Amphithéâtre Léon Motchane


Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette

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.

Organized by

Fanny Kassel