Fractal Models for Normal Subgroups of Schottky Groups

Abstract

For a normal subgroup $N$ of the free group $\F_d$ with at least two generators we introduce the radial limit set $\Lr(N,Φ)$ 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. Our main result states that if $\Phi$ is symmetric and linear, then we have that $\dim_{H}(\Lr(N,Φ))=\dim_{H} \Lr(\F_d,Φ))$ if and only if the quotient group $\F_d /N$ 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 $\F_d /N$ is non-amenable then $\dim_{H}(\Lr(N,Φ))>\dim_{H}(\Lr(\F_d,Φ))/2$, which extends results by Falk and Stratmann and by Roblin.

30 pages

References in corpus (2)

• An extension of Kesten's criterion for amenability to topological Markov chains
• A Lower Bound for the Exponent of Convergence of Normal Subgroups of Kleinian Groups

Cited by in corpus (4)

• An extension of Kesten's criterion for amenability to topological Markov chains
• A Lower Bound for the Exponent of Convergence of Normal Subgroups of Kleinian Groups
• Group Extended Markov Systems, Amenability, and the Perron-Frobenius Operator
• Conformal Fractals for Normal Subgroups of Free Groups