Critical exponents of normal subgroups, the spectrum of group extended transfer operators, and Kazhdan distance

2017/02/20 by Rhiannon Dougall, Dougall, Rhiannon · 1 citation
Mathematics · #37B10 #37C30 #37D20 (Secondary) #37D35 #37D40 (Primary) #Advanced Operator Algebra Research #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · doi:10.48550/arxiv.1702.06115

openalex publication_date 2017/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a pinched Hadamard manifold X and a discrete group of isometries Γ of X, the critical exponent δ_Γ is the exponential growth rate of the orbit of a point in X under the action of Γ. We show that the critical exponent for any family N of normal subgroups of Γ₀ has the same coarse behaviour as the Kazhdan distances for the right regular representations of the quotients Γ₀/Γ. The key tool is to analyse the spectrum of transfer operators associated to subshifts of finite type, for which we obtain a result of independent interest.

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