Uniform growth of groups acting on Cartan-Hadamard spaces

Abstract

Let $X$ be an $n$-dimensional simply connected manifold of pinched sectional curvature $-a^2\le K\le -1$. 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 $Ent(\Gamma )\ge C(n,a)$.