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arXiv · 2508.09807

Brownian motion and orbit counting of Kleinian groups

Yanlong Hao·Beibei Liu
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Abstract

In this paper, we investigate the relationship between the divergence of Kleinian groups $Γ$ and the recurrence of simple random walks on the Schreier graph associated with $Γ$. In particular, we show that if $Γ$ is a subgroup of a lattice and is of divergence type, then the Schreier graph is recurrent. Our approach builds connections among the growth rate of the $Γ$-orbit, the volume growth rate of the quotient manifolds, and the growth rate of the Schreier graph. Using the connections, we construct abundant Kleinian groups of divergence type.

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Yanlong Hao, Beibei Liu. 2026-01-13. Brownian motion and orbit counting of Kleinian groups. https://arxiv.org/abs/2508.09807

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