Fractal Models for Normal Subgroups of Schottky Groups

Details Fractal Models for Normal Subgroups of Schottky Groups Fractal Models for Normal Subgroups of Schottky Groups

2011-05-31

arxiv.org/abs/1106.0026v1

Mathematics

Dynamical Systems

33 pages

Scientific paper

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

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