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PreprintCC BY-SA 4.0arXiv

On 2-Sphere Bowditch Boundaries Attaining Conformal Dimension

  • Abhijit Pal
  • Rana Sardar
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Abstract

Bonk and Kleiner proved that if G is a Gromov hyperbolic group whose boundary ∂∞G is homeomorphic to an Ahlfors Q-regular metric $2$-sphere Z, and the Ahlfors regular conformal dimension of Z is attained and equal to Q, then G acts discretely, cocompactly, and isometrically on ℍ3. In this article, we extend the Bonk-Kleiner theorem to the setting of relatively hyperbolic groups. More precisely, we prove that if (G,ℋ) is a relatively hyperbolic group whose Bowditch boundary is homeomorphic to an Ahlfors Q-regular metric $2$-sphere Z, with the Ahlfors regular conformal dimension of Z attained and equal to Q, then G acts discretely and isometrically on ℍ3, and every subgroup in ℋ is virtually ℤ2.

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Cite

BibTeXRIS
BibTeX
@misc{pal20262sphere,
  title = {{On 2-Sphere Bowditch Boundaries Attaining Conformal Dimension}},
  author = {Abhijit Pal and Rana Sardar},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28346},
  eprint = {2608.28346},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28346},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - On 2-Sphere Bowditch Boundaries Attaining Conformal Dimension
AU  - Abhijit Pal
AU  - Rana Sardar
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.28346
UR  - https://arxiv.org/abs/2608.28346
AB  - Bonk and Kleiner proved that if G is a Gromov hyperbolic group whose boundary ∂_∞G is homeomorphic to an Ahlfors Q-regular metric $2$-sphere Z, and the Ahlfors regular conformal dimension of Z is attained and equal to Q, then G acts discretely, cocompactly, and isometrically on ℍ^3. In this article, we extend the Bonk-Kleiner theorem to the setting of relatively hyperbolic groups. More precisely, we prove that if (G,ℋ) is a relatively hyperbolic group whose Bowditch boundary is homeomorphic to an Ahlfors Q-regular metric $2$-sphere Z, with the Ahlfors regular conformal dimension of Z attained and equal to Q, then G acts discretely and isometrically on ℍ^3, and every subgroup in ℋ is virtually ℤ^2.
ER  -