  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.
