Uniform growth of groups acting on Cartan-Hadamard spaces
  Let $X$ be an $n$-dimensional simply connected manifold of pinched sectional curvature $-a^2 \leq K \leq -1$. There exist a positive constant $C(n,a)$ such that for any finitely generated discrete group $\Gamma$ acting on $X$, then either $\Gamma$ is virtually nilpotent or the algebraic entropy $Ent (\Gamma) \geq C(n,a)$. 
G\'erard Besson (IF), Gilles Courtois (CMLS-EcolePolytechnique),   Sylvain Gallot (IF)