Invariant random subgroups over non-Archimedean local fields

Abstract

Let G be a higher rank semisimple linear algebraic group over a non-Archimedean local field. The simplicial complexes corresponding to any sequence of pairwise non-conjugate irreducible lattices in G are Benjamini–Schramm convergent to the Bruhat–Tits building. Convergence of the relative Plancherel measures and normalized Betti numbers follows. This extends the work (Abert et al. in Ann Math 185(3):711–790, 2017) from real Lie groups to linear groups over arbitrary local fields. Along the way, various results concerning Invariant Random Subgroups and in particular a variant of the classical Borel density theorem are also extended.