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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Conformal fractals for normal subgroups of free groups
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by Johannes Jaerisch PDF
Conform. Geom. Dyn. 18 (2014), 31-55 Request permission

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
  • Email: jaerisch@cr.math.sci.osaka-u.ac.jp
  • 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)
  • © Copyright 2014 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 18 (2014), 31-55
  • MSC (2010): Primary 37C45, 30F40; Secondary 37C85, 43A07
  • DOI: https://doi.org/10.1090/S1088-4173-2014-00263-X
  • MathSciNet review: 3175016