Divergence-type semigroups in Kleinian groups and Hausdorff dimension

TL;DR

This paper proves that the Hausdorff dimension of a specific limit set of a Kleinian group equals its critical exponent, offering a new proof and developing related theoretical tools.

Contribution

It provides a novel proof connecting Hausdorff dimension and critical exponent for certain limit sets, and constructs divergence-type subsemigroups within Kleinian groups.

Findings

Hausdorff dimension equals critical exponent for the sublinearly conical Myrberg limit set

Constructed divergence-type subsemigroups of Kleinian groups

Developed Patterson--Sullivan theory in this context

Abstract

Let $\Gamma$ be a non-elementary, non-convex-cocompact Kleinian group acting on $\mathbb{H}^{d}$. We show that the Hausdorff dimension of the sublinearly conical Myrberg limit set of $\Gamma$ is equal to the critical exponent of $\Gamma$. This gives a different proof of a theorem by M. Mj and W. Yang. Along the way, we construct subsemigroups of $\Gamma$ of divergence type and develop the Patterson--Sullivan theory.