Geometric renormalisation and Hausdorff dimension for loop-approximable geodesics escaping to infinity

TL;DR

This paper proves that for certain Kleinian groups, the Hausdorff dimension of the set of geodesics escaping to infinity matches that of the entire group, extending previous results to a broader class of groups.

Contribution

It establishes a new equality of Hausdorff dimensions for limit sets of Kleinian groups under specific subgroup conditions, generalizing earlier work on Riemann surfaces.

Findings

Hausdorff dimension of the transient limit set equals that of the full limit set under given conditions

Extension of Fernandez and Melián's results to Kleinian groups with loxodromic elements

Provides geometric renormalization techniques for analyzing geodesics escaping to infinity

Abstract

The main result of this paper is to show that if $\H$ is a normal subgroup of a Kleinian group $G$ such that $G/\H$ contains a coset which is represented by some loxodromic element, then the Hausdorff dimension of the transient limit set of $\H$ coincides with the Hausdorff dimension of the limit set of $G$. This observation extends previous results by Fern\'andez and Meli\'an for Riemann surfaces.