Brownian motion and orbit counting of Kleinian groups

TL;DR

This paper explores the connection between divergence properties of Kleinian groups and the recurrence of random walks on associated Schreier graphs, linking geometric growth rates to probabilistic behavior.

Contribution

It establishes that divergence type Kleinian groups of sublattice type have recurrent Schreier graphs and constructs many such groups using these connections.

Findings

Divergence type groups lead to recurrent Schreier graphs

Growth rates of orbits and volume are interconnected

Constructs numerous divergence type Kleinian groups

Abstract

In this paper, we investigate the relationship between the divergence of Kleinian groups $\Gamma$ and the recurrence of simple random walks on the Schreier graph associated with $\Gamma$. In particular, we show that if $\Gamma$ 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 $\Gamma$-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.