Invariant random subgroups in hyperbolic reflection groups

TL;DR

This paper constructs ergodic invariant random subgroups with uncountably many isomorphism types in hyperbolic reflection groups, using Coxeter polytopes, and addresses questions posed by Thomas, Glasner, and Hase.

Contribution

It demonstrates the existence of complex invariant random subgroups in hyperbolic reflection groups, expanding understanding of subgroup diversity in these groups.

Findings

Existence of uncountably many isomorphism types of subgroups

Construction of invariant random subgroups using Coxeter polytopes

Applications to questions by Glasner and Hase

Abstract

We prove that the Fuchsian (4,4,4) triangle group and also right-angled reflection groups of hyperbolic spaces in higher dimensions admit ergodic invariant random subgroups having uncountably many isomorphism types of subgroups in their support (in most cases we actually prove a stronger statement), providing an answer to a question of S. Thomas. We also give similar constructions in higher-dimensional spaces. Our constructions are based on Coxeter polytopes in hyperbolic spaces. We also provide examples of invariant random subgroups related to questions of Y. Glasner and A. Hase through a similar construction.