Strong subgroup recurrence and the Nevo–Stuck–Zimmer theorem

Let (Formula presented.) be a countable group and (Formula presented.) its Chabauty space, namely, the compact (Formula presented.) -space consisting of all subgroups of (Formula presented.). We call a subgroup (Formula presented.) a boomerang subgroup if for every (Formula presented.), (Formula presented.) for some subsequence (Formula presented.). Poincaré recurrence implies that (Formula presented.) -almost every subgroup of (Formula presented.) is a boomerang, with respect to every invariant random subgroup (Formula presented.) of (Formula presented.). We establish for boomerang subgroups many density-related properties, most of which are known to hold almost surely for invariant random subgroups. Let (Formula presented.) be a number field, (Formula presented.) its ring of integers, (Formula presented.) a finite set of valuations including all the Archimedean valuations, and (Formula presented.) an absolutely almost simple, connected, algebraic group defined over (Formula presented.). Our main result is that if (Formula presented.), then any (Formula presented.) that is commensurable to the (Formula presented.) -arithmetic group (Formula presented.) has very few boomerang subgroups. Namely, every boomerang in (Formula presented.) 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 (Formula presented.) is commensurable to (Formula presented.).