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PreprintCC BY 4.0arXiv

Higher dimensional Kleinian groups of small Hausdorff dimension are classical Schottky groups II

  • Yong Hou
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Abstract

This is the higher dimensional continuation of our works on Hausdorff dimension of Kleinian groups \cite{Hou2010}, \cite{Hou2020}. In this paper we prove that for all r≥ 2 there exists λr>0, which is independent of the dimension N of the hyperbolic space, such that any finitely generated non-elementary torsion-free Kleinian group of Isom(ℍN) generated by r elements with limit set of Hausdorff dimension <λr is a classical Schottky group. For two generators the bound can be taken to be 1/3. The proof relies on our previous works in \cite{Hou2010}, \cite{Hou2020} and on a metric carrier graph reduction. Our result can also be considered as the complete converse to the well-known theorem of Phillips-Sarnak \cite{PS} for spaces of any dimension N.

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Cite

BibTeXRIS
BibTeX
@misc{hou2026higher,
  title = {{Higher dimensional Kleinian groups of small Hausdorff dimension are classical Schottky groups II}},
  author = {Yong Hou},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03335},
  eprint = {2611.03335},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03335},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Higher dimensional Kleinian groups of small Hausdorff dimension are classical Schottky groups II
AU  - Yong Hou
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03335
UR  - https://arxiv.org/abs/2611.03335
AB  - This is the higher dimensional continuation of our works on Hausdorff dimension of Kleinian groups \cite{Hou2010}, \cite{Hou2020}. In this paper we prove that for all r≥ 2 there exists λ_r>0, which is independent of the dimension N of the hyperbolic space, such that any finitely generated non-elementary torsion-free Kleinian group of Isom(ℍ^N) generated by r elements with limit set of Hausdorff dimension <λ_r is a classical Schottky group. For two generators the bound can be taken to be 1/3. The proof relies on our previous works in \cite{Hou2010}, \cite{Hou2020} and on a metric carrier graph reduction. Our result can also be considered as the complete converse to the well-known theorem of Phillips-Sarnak \cite{PS} for spaces of any dimension N.
ER  -