Séminaire de Géométrie, Groupes et Dynamique

Yair Glasner: "Boomerang subgroups and the Nevo-Stuck-Zimmer theorem"

435 (UMPA)



Let \Gamma be a countable group and \mathrm{Sub}(\Gamma) its Chabauty space, namely the compact \Gamma-space of all subgroups of \Gamma. We call a subgroup \Delta \in \mathrm{Sub}(\Gamma) a boomerang subgroup if for every \gamma \in \Gamma, \gamma^{n_i} \Delta \gamma^{-n_i} \rightarrow \Delta for some subsequence \{n_i \} \subset \mathbb{N}. Poincar\'{e} recurrence implies that \mu-almost every subgroup of \Gamma is a boomerang, with respect to every invariant random subgroup \mu. I will discuss boomerang subgroups and in particular, I will prove that every boomerang subgroup in \Gamma = \mathrm{SL}_n(\mathbb{Z}), \ n \ge 3 is either finite and central or of finite index. This gives a simple new proof for the Nevo-Stuck-Zimmer theorem in this case. More generally this method applies for lattices of the form G(\mathbb{Z}) where G is a Chevaley group over \mathbb{Q}.  
This is a joint work with Waltraud Lederle.