Free vs. Locally Free Kleinian Groups

TL;DR

This paper proves that Kleinian groups with limit sets as Cantor sets of Hausdorff dimension less than 1 are free, and constructs examples of non-free groups with similar limit set dimensions, highlighting the relationship between group freeness and limit set complexity.

Contribution

It establishes a criterion for freeness of Kleinian groups based on the Hausdorff dimension of their limit sets and provides explicit examples of non-free groups with small limit set dimensions.

Findings

Groups with limit sets of Hausdorff dimension <1 are free.

Existence of non-free Kleinian groups with limit sets of dimension just above 1.

Construction of examples for any epsilon > 0.

Abstract

We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension $<1$ are free. On the other hand we construct for any $\epsilon>0$ examples of non-free purely hyperbolic Kleinian groups whose limit set is a Cantor set of Hausdorff dimension $<1+\epsilon$.