A Lower Bound for the Exponent of Convergence of Normal Subgroups of Kleinian Groups

TL;DR

This paper proves that for non-elementary Kleinian groups, the convergence exponent of any non-trivial normal subgroup is at least half that of the group, with strict inequality in divergence type cases, providing a new proof of this bound.

Contribution

It offers a concise new proof establishing a lower bound on the convergence exponent of normal subgroups in Kleinian groups, enhancing understanding of their geometric properties.

Findings

The exponent of convergence of normal subgroups is at least half that of the parent group.

Strict inequality holds if the group is of divergence type.

Provides a simplified proof of a known bound.

Abstract

We give a short new proof that for each non-elementary Kleinian group $\Gamma$, the exponent of convergence of an arbitrary non-trivial normal subgroup is bounded below by half of the exponent of convergence of $\Gamma$, and that strict inequality holds if $\Gamma$ is of divergence type.