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

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 $\delta$-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.

arXiv comment55 pages

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Cited by in corpus (11)

• Product set growth in groups and hyperbolic geometry
• The joint spectrum
• Groups acting on CAT(0) cube complexes with uniform exponential growth
• Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces
• Minimal volume entropy of simplicial complexes
• Ping-pong in Hadamard manifolds
• Geometric finiteness in negatively pinched Hadamard manifolds
• Product set growth in Burnside groups
• Locally elliptic actions, torsion groups, and nonpositively curved spaces
• Extensions of hyperbolic groups have locally uniform exponential growth
• Large deviation principles for non-elementary random walks on hyperbolic spaces