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

We recast the notion of joint spectral radius in setting groups acting by isometries on non-positively curved spaces and give geometric versions results Berger–Wang Bochi valid for δ-hyperbolic symmetric non-compact type. This method produces nice hyperbolic elements many classical settings. Applications to uniform growth are given, particular a new proof generalization theorem Besson–Courtois–Gallot.