Fractal models for normal subgroups of Schottky groups

Abstract:

For a normal subgroup $N$ of the free group $\mathbb {F}_{d}$ with at least two generators, we introduce the radial limit set $\Lambda _{r}(N,\Phi )$ of $N$ with respect to a graph directed Markov system $\Phi$ associated to $\mathbb {F}_{d}$. These sets are shown to provide fractal models of radial limit sets of normal subgroups of Kleinian groups of Schottky type. Our main result states that if $\Phi$ is symmetric and linear, then we have that $\dim _{H}(\Lambda _{r}(N,\Phi ))=\dim _{H}(\Lambda _{r}(\mathbb {F}_d,\Phi ))$ if and only if the quotient group $\mathbb {F}_{d}/N$ is amenable, where $\dim _{H}$ denotes the Hausdorff dimension. This extends a result of Brooks for normal subgroups of Kleinian groups to a large class of fractal sets. Moreover, we show that if $\mathbb {F}_{d}/N$ is non-amenable, then $\dim _{H}(\Lambda _{r}(N,\Phi ))>\dim _{H}(\Lambda _{r}(\mathbb {F}_d,\Phi ))/2$, which extends results by Falk and Stratmann and by Roblin.