Limit sets and commensurability of Kleinian groups

TL;DR

This paper investigates the relationship between limit sets and commensurability of Kleinian groups, proving that equal limit sets imply certain subgroup properties and finite index conditions.

Contribution

It establishes new criteria linking limit sets to subgroup commensurability and finite index in Kleinian groups, especially for finitely generated subgroups.

Findings

Equal limit sets imply subgroup commensurability.

Finitely generated subgroups with the same limit set are of finite index unless virtually fibered.

Provides conditions distinguishing virtually fibered subgroups.

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.