# Convergence of quasi-Fuchsian groups using critical exponent

## TL;DR

This paper proves that a sequence of quasi-Fuchsian groups with critical exponents approaching the boundary dimension converges to a totally geodesic representation, revealing a link between critical exponent limits and geometric structure.

## Contribution

It establishes a convergence result for quasi-Fuchsian groups based on the behavior of their critical exponents, connecting geometric limits to boundary dimensions.

## Key findings

- Convergence of quasi-Fuchsian groups when critical exponent approaches boundary dimension.
- Limit groups are totally geodesic representations.
- Provides a criterion for geometric convergence based on critical exponents.

## Abstract

We prove that a sequence of quasi-Fuchsian representations for which the critical exponent converges to the topological dimension of the boundary of the group (larger than 2), converges up to subsequence and conjugacy to a totally geodesic representation.