Critical Exponent and Hausdorff Dimension in Pseudo-Riemannian Hyperbolic Geometry
Critical Exponent and Hausdorff Dimension in Pseudo-Riemannian Hyperbolic Geometry
复制标题
伪黎曼双曲几何中的临界指数和Hausdorff维数
DOI：
10.1093/imrn/rnz098
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发表时间：
2016
影响因子：
1
通讯作者：
Daniel Monclair
中科院分区：
文献类型：
--
作者：
Olivier Glorieux;Daniel Monclair
本文的目的是了解伪黎曼双曲几何中极限集的几何。我们主要研究了Danciger，Guéritaud和Kassel引入的一类子群：${\mathbb{H}}^{p，q}$-凸余紧.我们定义了临界指数的伪黎曼相似和极限集的Hausdorff维数。我们证明了它们是相等的，并且由极限集的通常的Hausdorff维有界。我们还证明了${\mathbb{H}}^{2，1}={\mathbb{A}}\textrm{d}{\mathbb{S}}^3$，中的一个刚性结果，它可以理解为$3$D双曲几何中R.Bowen的一个著名定理的洛伦兹版本。
The aim of this article is to understand the geometry of limit sets in pseudo-Riemannian hyperbolic geometry. We focus on a class of subgroups of $\textrm{PO}(p,q+1)$ introduced by Danciger, Guéritaud, and Kassel, called ${\mathbb{H}}^{p,q}$-convex cocompact. We define a pseudo-Riemannian analogue of critical exponent and Hausdorff dimension of the limit set. We show that they are equal and bounded from above by the usual Hausdorff dimension of the limit set. We also prove a rigidity result in ${\mathbb{H}}^{2,1}={\mathbb{A}}\textrm{d}{\mathbb{S}}^3$, which can be understood as a Lorentzian version of a famous Theorem of R. Bowen in $3$D hyperbolic geometry.