Homological dimension and critical exponent of Kleinian groups

TL;DR

This paper establishes an upper bound on the relative homological dimension of Kleinian groups based on their critical exponent and explores geometric conditions under which the limit set is a round sphere.

Contribution

It proves a new inequality linking homological dimension and critical exponent, and characterizes limit sets as round spheres under specific dimensional conditions.

Findings

Homological dimension of Kleinian groups is at most 1 plus their critical exponent.

If the limit set's topological and Hausdorff dimensions coincide, it must be a round sphere.

Provides a geometric criterion for the shape of limit sets in Kleinian groups.

Abstract

We prove that the relative homological dimension of a Kleinian group G does not exceed 1 + the critical exponent of G. As an application of this result we show that for a geometrically finite Kleinian group G, if the topological dimension of the limit set of G equals its Hausdorff dimension, then the limit set is a round sphere.