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 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.Résumé. -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 δ-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.

The rates of growth in an acylindrically hyperbolic group

“…The proof is same verbatim after a suitable translation of notions, so we omit it (cf. [BrF,Theorem 1.4] for somewhat similar result). Note that if s ∈ S then |x − s(x)| ≤ L(S); and if s ∈ S 2 then L(s) ≤ |x − s(x)| ≤ 2L(S) by the definition of L(S) and the choice of x.…”

“…Put a metric on Y as before and denote it Ŷ . The result in [BrF,Theorem 1.13] applies since a lower bound of K implies the doubling property we assume in that theorem. Also the lemma holds too.…”

“…It suffices to show that if S ⊂ G is a finite set and H = S has exponential growth, then H contains a hyperbolic element on X. But as we said in that proof, by [BrF,Theorem 1.13], either S M , which is a subset of H, contains a hyperbolic element, or H = S is virtually nilpotent, which does not happen since H has exponential growth. We showed Θ(G) = Θ X (G), which implies that Θ(G) is well-ordered by Proposition 6.2.…”

“…It is known by [BrF,Theorem 1.13] that there exists a constant M that depends only on Y (in fact only on the dimension of Y in this case) such that for any finite set S ⊂ G, unless S is virtually nilpotent, S M contains a hyperbolic element on Y . We argue this element is hyperbolic on X.…”

Fixed points for group actions on 2-dimensional affine buildings

“…Parreau in [19,Corollaire 3] proved a similar result for subgroups Γ of connected reductive groups G over certain fields F , where Γ is generated by a bounded subset of G(F ) and the action is on the completion of the associated Bruhat-Tits building. Breuillard and Fujiwara established a quantitative version of Parreau's result for discrete Bruhat-Tits buildings [3,Theorem 7.16] and asked whether their result holds for the isometry group of an arbitrary affine building. Leder and Varghese in [11], using work of Sageev [22], obtained a similar result for groups acting on finite-dimensional CAT(0) cube complexes (these include discrete right-angled buildings, in particular a product of two simplicial trees).…”