Amenability, Critical Exponents of Subgroups and Growth of Closed\n Geodesics

2014/11/25 by Rhiannon Dougall, Dougall, Rhiannon, Richard Sharp +2 · 1 citation
Mathematics · #37B10 #37C30 #37D20 (Secondary) #37D35 #37D40 (Primary) #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1411.6817

openalex publication_date 2014/11/25 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

Let \Γ be a (non-elementary) convex co-compact group of isometries of a\npinched Hadamard manifold X. We show that a normal subgroup \Γ₀ has\ncritical exponent equal to the critical exponent of \Γ if and only if\n\Γ / \Γ₀ is amenable. We prove a similar result for the exponential\ngrowth rate of closed geodesics on X / \Γ. These statements are analogues\nof classical results of Kesten for random walks on groups and of Brooks for the\nspectrum of the Laplacian on covers of Riemannian manifolds.\n

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