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Hausdorff dimension and complex hyperbolic Schottky groups: a simplification

Abstract

In the present work, we study the Hausdorff dimension of the limit set of Schottky groups on the boundary of the complex hyperbolic group via the Eigenvalue algorithm as in [17]. The visual geometry on \(\overline{{\mathbb {H}}^2_{\mathbb {C}}}\) allows a direct application of the Eigenvalue algorithm, but at the same time, it hardens the computations. Using the Heisenberg structure on \(\partial {\mathbb {H}}^2_{\mathbb {C}}\setminus \{\infty \}\) and the Cygan metric, we propose a Markov partition associated with the group that simplifies the application of the Eigenvalue algorithm.

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Acknowledgements

The first author would like to thank Prof. Andres Sambarino for their fruitful conversation and comments. Both authors would like to thank the Referee’s comments and fruitful suggestions.

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Correspondence to Alejandro Ucan-Puc.

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First author thanks to CONACYT Project 740649 for founding this project. Second author thanks Bolsa Jovem Cientista do Nosso Estado No. E-26/201.432/2022.

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Ucan-Puc, A., Romaña, S. Hausdorff dimension and complex hyperbolic Schottky groups: a simplification. Geom Dedicata 216, 58 (2022). https://doi.org/10.1007/s10711-022-00718-2

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  • DOI: https://doi.org/10.1007/s10711-022-00718-2

Keywords

  • Complex Hyperbolic groups
  • Limit Set
  • Hausdorff dimension
  • Kleinian Groups

Mathematics Subject Classification

  • 37M99
  • 37F99
  • 32M99
  • 30F45
  • 20H10
  • 57M60