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Conformal Geometry and Dynamics

ISSN 1088-4173



Conformal fractals for normal subgroups of free groups

Author: Johannes Jaerisch
Journal: Conform. Geom. Dyn. 18 (2014), 31-55
MSC (2010): Primary 37C45, 30F40; Secondary 37C85, 43A07
Published electronically: March 7, 2014
MathSciNet review: 3175016
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Abstract: We investigate subsets of a multifractal decomposition of the limit set of a conformal graph directed Markov system which is constructed from the Cayley graph of the free group $F_{d}$ with at least two generators. The subsets we consider are parametrised by a normal subgroup $N$ of $F_{d}$ and mimic the radial limit set of a Kleinian group. Our main results show that, regarding the Hausdorff dimension of these sets, various results for Kleinian groups can be generalised. Namely, under certain natural symmetry assumptions on the multifractal decomposition, we prove that, for a subset parametrised by $N$, the Hausdorff dimension is maximal if and only if $F_{d}/N$ is amenable and that the dimension is greater than half of the maximal value. We also give a criterion for amenability via the divergence of the Poincaré series of $N$. Our results are applied to the Lyapunov spectrum for normal subgroups of Kleinian groups of Schottky type.

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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: Kleinian groups, exponent of convergence, normal subgroups, amenability, conformal graph directed Markov systems
Received by editor(s): June 20, 2013
Published electronically: March 7, 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