Brownian motion and orbit counting of Kleinian groups

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