Critical exponent and Hausdorff dimension in pseudo-Riemannian\n hyperbolic geometry

2016/06/17 by Olivier Glorieux, Glorieux, Olivier, Daniel Monclair +1 · 1 citation
Mathematics · #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.1606.05512

openalex publication_date 2016/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The aim of this article is to understand the geometry of limit sets in\npseudo-Riemannian hyperbolic geometry. We focus on a class of subgroups of\n\PO(p,q+1) introduced by Danciger, Gu 'eritaud and Kassel, called\n\ℍ^p,q-convex cocompact. We define a pseudo-Riemannian analogue of\ncritical exponent and Hausdorff dimension of the limit set. We show that they\nare equal and bounded from above by the usual Hausdorff dimension of the limit\nset. We also prove a rigidity result in \ℍ^2,1=\ADS³ which\ncan be understood as a Lorentzian version of a famous Theorem of R. Bowen in\n3-dimensional hyperbolic geometry.\n

Cited by

• Geometric components of representation spaces via robust families of submanifolds

Related

• Hausdorff dimension of directional limit sets for self-joinings of hyperbolic manifolds
• Hausdorff Dimension of non-conical and Myrberg limit sets
• Mesures de Hausdorff de l'ensemble limite de groupes kleiniens géométriquement finis
• Critical Exponents on Hyperbolic Surfaces with Long Boundaries and the Asymptotic Weil-Petersson Form
• Critical exponents on hyperbolic surfaces with long boundaries and the asymptotic Weil-Petersson form