# Hyperbolic isometries in cocompact Hadamard (i.e. cat(0) proper simply connected) spaces

Swenson proved in "A cut point theorem for ${\rm CAT}(0)$ groups" that a locally compact Hadamard space with a geometric action by a group $G$ admits a hyperbolic isometry (that lie in $G$). Is it still true if we only assume the action to be cocompact? I tried to modify his proof but I didn't succeed...

Thanks

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Are you sure of your lat remark? They do not define proper action in their paper, but Bridson and Haefliger define an action to be proper in for every compact $K$ the number of $g$ in the group that satisfy $g(K)\cap K\neq\void$ is finite. With this definition, the isometry group of a homogeneous manifold is not proper! –  Aurelien Mar 29 '11 at 13:44