arXiv2024
Let $Γ$ be a countable group and $\mathrm{Sub}(Γ)$ its Chabauty space, namely the compact $Γ$-space consisting of all subgroups of $Γ$. We call a subgroup $Δ\in \mathrm{Sub}(Γ)$ a boomerang subgroup if for every $γ\in Γ$, $γ^{n_i} Δγ^{-n_i} \rightarrow Δ$ for some subsequence $\{n_i \} \subset \mathbb{N}$. Poincaré recurrence implies that $μ$-almost every subgroup of $Γ$ is a boomerang, with respect to every invariant random subgroup $μ$ of $Γ$. 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 $Γ$ which is commensurable to the $S$-arithmetic group $\mathbb{G}(O_S)$ has very few boomerang subgroups. Namely, every boomerang in $Γ$ 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 $Γ$ is commensurable to $\mathrm{SL}_n(\mathbb{Z}), \ n \ge 3$.