Limit sets and commensurability of Kleinian groups

Abstract

In this paper, we obtain several results on the commensurability of two Kleinian groups and their limit sets. We prove that two finitely generated subgroups $G_1$ and $G_2$ of an infinite co-volume Kleinian group $G \subset \Isom(\mathbf{H}^3)$ having $\Lambda(G_1) = \Lambda(G_2)$ are commensurable. In particular, it is proved that any finitely generated subgroup $H$ of a Kleinian group $G \subset \Isom(\mathbf{H}^3)$ with $\Lambda(H) = \Lambda(G)$ is of finite index if and only if $H$ is not a virtually fiber subgroup.