Determining subgroups via stationary measures

Dongryul M. Kim; Andrew Zimmer

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arXiv · 2512.12966

Determining subgroups via stationary measures

Dongryul M. Kim·Andrew Zimmer

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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 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.

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BibTeX RIS

Dongryul M. Kim, Andrew Zimmer. 2025-12-15. Determining subgroups via stationary measures. https://arxiv.org/abs/2512.12966

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