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Fractal models for normal subgroups of Schottky groups

Author: Johannes Jaerisch
Journal: Trans. Amer. Math. Soc. 366 (2014), 5453-5485
MSC (2010): Primary 37C45, 30F40; Secondary 37C85, 43A07
Published electronically: May 20, 2014
MathSciNet review: 3240930
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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.

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Additional Information

Johannes Jaerisch
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka, 560-0043 Japan

Keywords: Normal subgroups of Kleinian groups, exponent of convergence, graph directed Markov system, amenability, Perron-Frobenius operator, random walks on groups
Received by editor(s): December 15, 2012
Published electronically: May 20, 2014
Additional Notes: The author was supported by the research fellowship JA 2145/1-1 of the German Research Foundation (DFG)
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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