Limit sets and commensurability of Kleinian groups

Details Limit sets and commensurability of Kleinian groups Limit sets and commensurability of Kleinian groups

2010-04-10

arxiv.org/abs/1004.1751v1

Bulletin of the Australian Mathematical Society (2010), 82:1-9

Mathematics

Geometric Topology

9 pages

Scientific paper

10.1017/S0004972710000146

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.

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