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PreprintCC BY 4.0arXiv

Markov maps and quasi-conformal deformations associated to geometrically finite Kleinian groups

  • Shengjin Huo
  • Wanting Shao
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Abstract

It is known that the dynamical properties of Kleinian groups with parabolic elements are much more complex than those without parabolic elements. In this paper, we mainly study the quasi-conformal deformation properties of geometrically finite Kleinian groups with parabolic elements. We show that, for a Kleinian group G with parabolic elements, there exists a subset K of the limit set Λ(G) such that the first return map fK is (eventually) expending. Furthermore, the expending property is invariant under quasiconformal deformations. As an application, we show that the geometrically finite Kleinian groups have Ruelle's property. Finally, we use eigenvalue algorithm to calculate the precise Hausdorff dimensions of Hecke groups, which serves as an example to support our results.

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Cite

BibTeXRIS
BibTeX
@misc{huo2026markov,
  title = {{Markov maps and quasi-conformal deformations associated to geometrically finite Kleinian groups}},
  author = {Shengjin Huo and Wanting Shao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19533},
  eprint = {2610.19533},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.19533},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Markov maps and quasi-conformal deformations associated to geometrically finite Kleinian groups
AU  - Shengjin Huo
AU  - Wanting Shao
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.19533
UR  - https://arxiv.org/abs/2610.19533
AB  - It is known that the dynamical properties of Kleinian groups with parabolic elements are much more complex than those without parabolic elements. In this paper, we mainly study the quasi-conformal deformation properties of geometrically finite Kleinian groups with parabolic elements. We show that, for a Kleinian group G with parabolic elements, there exists a subset K of the limit set Λ(G) such that the first return map f_K is (eventually) expending. Furthermore, the expending property is invariant under quasiconformal deformations. As an application, we show that the geometrically finite Kleinian groups have Ruelle's property. Finally, we use eigenvalue algorithm to calculate the precise Hausdorff dimensions of Hecke groups, which serves as an example to support our results.
ER  -