Séminaire de Géométrie, Groupes et Dynamique

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

Europe/Paris
435 (UMPA)

435

UMPA

Description
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.