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

TL;DR

This paper extends the concept of joint spectral radius to groups acting on non-positively curved spaces, providing geometric versions of key results and applications to uniform growth, including a new proof of a known theorem.

Contribution

It introduces a geometric framework for joint spectral radius in non-positively curved spaces and generalizes classical results to hyperbolic and symmetric spaces.

Findings

Established geometric versions of Berger-Wang and Bochi results for hyperbolic spaces

Produced hyperbolic elements in various geometric settings

Provided a new proof and generalization of a theorem on 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.