2022/08/31 by Glasner, Yair, Lederle, Waltraud · 2 citations
#20E42 #20F65 #37B20 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2208.14758
Let Γ be a countable group and Sub(Γ) its Chabauty space, namely the compact Γ-space consisting of all subgroups of Γ. We call a subgroup Δ∈ Sub(Γ) a boomerang subgroup if for every γ∈ Γ, γni Δγ-ni → Δ for some subsequence \ni \ ⊂ 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 \mathbbK be a number field, O its ring of integers, S a finite set of valuations including all the Archimedean valuations, and \mathbbG an absolutely almost simple group defined over \mathbbK. Our main result is that if rk_\mathbbK \mathbbG ≥ 2 then any Γ which is commensurable to the S-arithmetic group \mathbbG(OS) 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 SLn(ℤ), n ≥ 3.