AI 中文总结
本文将Bonk-Kleiner定理推广到相对双曲群情形，证明若相对双曲群的Bowditch边界是满足条件的Ahlfors正则2-球面，则该群在双曲空间ℍ³上离散等距作用，且其相对子群为几乎ℤ²。
AI 中文摘要
Bonk与Kleiner证明，若G是Gromov双曲群，其边界∂∞G同胚于Ahlfors Q-正则度量2-球面Z，且Z的Ahlfors正则共形维数可达且等于Q，则G在ℍ³上作离散、紧且等距作用。本文将Bonk-Kleiner定理推广到相对双曲群情形，具体证明：若(G,ℋ)是相对双曲群，其Bowditch边界同胚于Ahlfors Q-正则度量2-球面Z，且Z的Ahlfors正则共形维数可达且等于Q，则G在ℍ³上作离散且等距作用，且ℋ中每个子群都是几乎ℤ²。
英文摘要
Bonk and Kleiner proved that if $G$ is a Gromov hyperbolic group whose boundary $\partial_{\infty}G$ 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 $\mathbb{H}^3$. In this article, we extend the Bonk-Kleiner theorem to the setting of relatively hyperbolic groups. More precisely, we prove that if $(G,\mathcal{H})$ 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 $\mathbb{H}^3$, and every subgroup in $\mathcal{H}$ is virtually $\mathbb Z^2$.
Comments19 pages, no figure
