Isometric group actions with vanishing rate of escape on $$\textrm{CAT}(0)$$ spaces

Abstract

Let $\Gamma$ be a finitely generated group equipped with a symmetric and nondegenerate probability measure $\mu$ with finite second moment, and $Y$ a CAT(0) space which is either proper or of finite telescopic dimension. We show that if an isometric action of $\Gamma$ on $Y$ has vanishing rate of escape with respect to $\mu$ and does not fix a point in the boundary at infinity of $Y$, then there exists a flat subspace in $Y$ which is left invariant under the action of $\Gamma$. In the proof of this result, an equivariant $\mu$-harmonic map from $\Gamma$ into $Y$ plays an important role.