Fractal models for normal subgroups of Schottky groups

Abstract

For a normal subgroupNNof the free groupFd\mathbb {F}_{d}with at least two generators, we introduce the radial limit setΛr(N,Φ)\Lambda _{r}(N,\Phi )ofNNwith respect to a graph directed Markov systemΦ\Phiassociated toFd\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 ifΦ\Phiis symmetric and linear, then we have thatdimH⁡(Λr(N,Φ))=dimH⁡(Λr(Fd,Φ))\dim _{H}(\Lambda _{r}(N,\Phi ))=\dim _{H}(\Lambda _{r}(\mathbb {F}_d,\Phi ))if and only if the quotient groupFd/N\mathbb {F}_{d}/Nis amenable, wheredimH\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 ifFd/N\mathbb {F}_{d}/Nis non-amenable, thendimH⁡(Λr(N,Φ))>dimH⁡(Λr(Fd,Φ))/2\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.