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

Breuillard, Emmanuel; Fujiwara, Koji

Annales de l'Institut Fourier, Tome 71 (2021) no. 1, pp. 317-391.

Nous généralisons la notion de rayon spectral joint dans le cadre des actions de groupes par isométries sur les espaces à courbure négative ou nulle et nous donnons des versions géométriques des résultats de Berger–Wang et de Bochi valables dans tout espace $\delta$-hyperbolique ainsi que dans les espaces symétriques de type non-compact. Cette méthode permet de produire des éléments hyperboliques dans de nombreuses situations géométriques classiques. Nous donnons par ailleurs des applications à la croissance uniforme ainsi qu’une nouvelle preuve et une généralisation d’un théorème de Besson–Courtois–Gallot.

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.

Classification : 20F65, 20M20, 15A30
Keywords: joint spectral radius, $\mathrm{CAT}(0)$ spaces, Gromov hyperbolic spaces, uniform exponential growth, affine isometric actions, ping-pong
Mot clés : rayon spectral joint, espaces $\mathrm{CAT}(0)$, espaces hyperboliques au sens de Gromov, croissance exponentielle uniforme, actions isométriques affines, ping-pong

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