Critical exponent and Hausdorff dimension in pseudo-Riemannian hyperbolic geometry

TL;DR

This paper explores the geometry of limit sets in pseudo-Riemannian hyperbolic spaces, defining analogues of critical exponent and Hausdorff dimension, and establishing their equality and bounds, including a rigidity result in Lorentzian 3-space.

Contribution

It introduces a pseudo-Riemannian analogue of critical exponent and Hausdorff dimension, proving their equality and establishing bounds, along with a Lorentzian rigidity theorem.

Findings

Critical exponent and Hausdorff dimension are equal in this setting.

These quantities are bounded above by the classical Hausdorff dimension.

A Lorentzian rigidity theorem analogous to Bowen's theorem is proved.

Abstract

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 $\mathrm{PO}(p,q+1)$ introduced by Danciger, Gu\'eritaud 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 $\mathbb{H}^{2,1}=\mathrm{ADS}^3$ which can be understood as a Lorentzian version of a famous Theorem of R. Bowen in $3$-dimensional hyperbolic geometry.