Determining subgroups via stationary measures

TL;DR

This paper investigates the properties of stationary measures for random walks on isometry groups of metric spaces, establishing conditions under which subgroups are commensurable and applying results to hyperbolic and Teichmüller spaces.

Contribution

It introduces criteria linking stationary measures to subgroup commensurability for random walks on isometry groups of general metric spaces.

Findings

Non-elementary random walks with non-singular stationary measures imply subgroup commensurability.

Results apply to Gromov hyperbolic and Teichmüller spaces.

Proves singularity of stationary measures for different fiber subgroups in hyperbolic 3-manifolds.

Abstract

In this paper, we consider random walks on the isometry groups of general metric spaces. Under some mild conditions, we show that if two non-elementary random walks on a discrete subgroup of the isometry group have non-singular stationary measures, then subgroups generated by the random walks are commensurable. This result in particular applies to separable Gromov hyperbolic spaces and Teichm\"uller spaces. As a specific application, we prove singularity between stationary measures associated to random walks on different fiber subgroups of the fundamental group of a hyperbolic 3-manifold fibering over the circle.