Conformal Fractals for Normal Subgroups of Free Groups

TL;DR

This paper studies the Hausdorff dimension of certain fractal subsets associated with normal subgroups of free groups, generalizing results from Kleinian groups and linking dimension properties to group amenability.

Contribution

It extends multifractal analysis of limit sets to free groups' normal subgroups, establishing criteria for maximal Hausdorff dimension and amenability.

Findings

Hausdorff dimension is maximal iff the quotient group is amenable

Dimension exceeds half of the maximum under certain conditions

Amenability characterized by divergence of Poincaré series

Abstract

We investigate subsets of a multifractal decomposition of the limit set of a conformal graph directed Markov system, which is constructed from the Cayley graph of a free group with at least two generators. The subsets we consider are parametrised by a normal subgroup $N$ of the free group and mimic the radial limit set of a Kleinian group. Our main results show that, regarding the Hausdorff dimension of these sets, various results for Kleinian groups can be generalised. Namely, under certain natural symmetry assumptions on the multifractal decomposition, we prove that, for a subset parametrised by $N$, the Hausdorff dimension is maximal if and only if $\F_{d}/N$ is amenable and that the dimension is greater than half of the maximal value. We also give a criterion for amenability via the divergence of the Poincar\'{e} series of $N$. Our results are applied to the Lyapunov spectrum for normal subgroups of Kleinian groups of Schottky type.