Determining subgroups via stationary measures
Dongryul M. Kim, Andrew Zimmer
TL;DR
The paper develops a probabilistic framework to detect subgroup commensurability via stationary measures of random walks on isometry groups of general metric spaces. It introduces an abstract, well-behaved random-walk setup with bordifications and a boundary-to-quasi-geodesic map, proving a universal rigidity theorem: if two random walks have non-singular forward hitting measures on the boundary and positive drifts, then their generated subgroups are commensurable. The results are instantiated for separable Gromov hyperbolic spaces and Teichmüller spaces (via isometry groups and moduli actions), with concrete corollaries distinguishing subgroups such as fiber subgroups in fibered hyperbolic 3-manifolds using Cannon–Thurston maps and boundary measures. The work further connections to normal subgroups show how stationary measures can be realized on subgroups, with explicit fibered-3-manifold examples yielding absolute continuity and Lebesgue-stationarity on boundary spheres. Overall, the paper provides a robust probabilistic criterion for subgroup distinction and commensurability, unifying geometric, dynamical, and boundary-measure perspectives.
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 Teichmüller 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.