Hausdorff dimension, diverging Schottky representations and the infinite dimensional hyperbolic space

TL;DR

This paper investigates the limit sets of diverging Schottky groups in hyperbolic spaces, generalizing Bowen's theorem and introducing a method to relate diverging sequences to almost converging sequences in infinite-dimensional hyperbolic space.

Contribution

It generalizes Bowen's theorem on Hausdorff dimensions and develops a new method to connect diverging Schottky groups with almost converging sequences in infinite-dimensional hyperbolic space.

Findings

Generalized Bowen's theorem for limit sets of Schottky groups

Developed a method to transform diverging sequences into almost converging sequences

Extended previous work by Mehmeti and Dang, with applications to McMullen's example

Abstract

One of our main goals in this paper is to understand the behavior of limit sets of a diverging sequence of Schottky groups in the group of isometries of the N-dimensional hyperbolic space. This leads us to a generalization of a classical theorem of Bowen on variations of Hausdorff dimension of limit sets; and to a method of transforming a diverging sequence of Schottky groups into an almost converging sequence in the group of isometries of the infinite dimensional hyperbolic space. Our results apply in particular to an example of McMullen and generalize a previous work by Mehmeti and Dang.