Strong subgroup recurrence and the Nevo-Stuck-Zimmer theorem

Abstract

Let $\Gamma$ be a countable group and $\mathrm{Sub}(\Gamma)$ its Chabauty space, namely the compact $\Gamma$-space consisting 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 $\mu$ of $\Gamma$. We establish for boomerang subgroups many density related properties, most of which are known to hold almost surely for invariant random subgroups. Let $\mathbb{K}$ be a number field, $O$ its ring of integers, $S$ a finite set of valuations including all the Archimedean valuations, and $\mathbb{G}$ an absolutely almost simple group defined over $\mathbb{K}$. Our main result is that if $\mathrm{rk}_{\mathbb{K}} \mathbb{G} \ge 2$ then any $\Gamma$ which is commensurable to the $S$-arithmetic group $\mathbb{G}(O_S)$ has very few boomerang subgroups. Namely, every boomerang in $\Gamma$ is either finite and central or of finite index. In particular we recover Margulis' normal subgroup theorem as well as the Nevo-Stuck-Zimmer theorem for such lattices. We include a short, accessible proof for the above theorem in the case that $\Gamma$ is commensurable to $\mathrm{SL}_n(\mathbb{Z}), \ n \ge 3$.