Invariant random subgroups over non-Archimedean local fields

TL;DR

This paper demonstrates that sequences of irreducible lattices in higher rank semisimple groups over non-Archimedean local fields converge to the Bruhat-Tits building, extending known results from real Lie groups to linear groups over local fields.

Contribution

It extends convergence results of lattices and invariant random subgroups from real Lie groups to higher rank linear groups over non-Archimedean local fields.

Findings

Benjamini-Schramm convergence of complexes to Bruhat-Tits building

Convergence of relative Plancherel measures and Betti numbers

Extension of Borel density theorem for invariant random subgroups

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 of Abert, Bergeron, Biringer, Gelander, Nokolov, Raimbault and Samet 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.