On the joint spectral radius for isometries of non-positively curved spaces and uniform growth

2018/04/02 by Emmanuel Breuillard, Koji Fujiwara, Breuillard, Emmanuel +1 · 6 citations
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Group Theory (math.GR) #math.GR

paper · pdf · doi:10.48550/arxiv.1804.00748

55 pages

arxiv created 2018/04/02 · openalex publication_date 2018/04/02 · arxiv updated 2018/04/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We recast the notion of joint spectral radius in the setting of groups acting by isometries on non-positively curved spaces and give geometric versions of results of Berger-Wang and Bochi valid for δ-hyperbolic spaces and for symmetric spaces of non-compact type. This method produces nice hyperbolic elements in many classical geometric settings. Applications to uniform growth are given, in particular a new proof and a generalization of a theorem of Besson-Courtois-Gallot.

Citations

• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products
• Discrete isometry groups of symmetric spaces

Cited by

• Uniform spectral gaps, non-abelian Littlewood-Offord and anti-concentration for random walks
• The Tits alternative for visibility spaces
• Extensions of hyperbolic groups have locally uniform exponential growth
• Continuity and approximability of competitive spectral radii
• Minimal volume entropy of simplicial complexes
• Periodic approximation of topological Lyapunov exponents and the joint spectral radius for cocycles of mapping classes of surfaces

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• Growth of quotients of groups acting by isometries on Gromov hyperbolic spaces
• Spectral gaps, critical exponents and representations of negatively curved groups
• Quasi-Isometries, Boundaries and JSJ-Decompositions of Relatively Hyperbolic Groups
• Hyperbolic spaces not quasi-isometric to curve complexes