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

Abstract

Let $Y=(Y,d)$ be a $\textrm{CAT}(0)$ space which is either proper or of finite telescopic dimension, and $\Gamma$ a countable group equipped with a symmetric and nondegenerate probability measure $\mu$. Suppose that $\Gamma$ acts on Y via a homomorphism $\rho :\Gamma \rightarrow \textrm{Isom}({Y})$, where $\textrm{Isom}({Y})$ denotes the isometry group of Y, and that the action given by $\rho$ has finite second moment with respect to $\mu$. We show that if $\rho (\Gamma )$ does not fix a point in the boundary at infinity $\partial Y$ of Y and the rate of escape $l_{\rho }(\Gamma )=l_{\rho }(\Gamma ,\mu )$ associated to an action given by $\rho$ vanishes, then there exists a flat subspace in Y that is left invariant under the action of $\rho (\Gamma )$. Note that if the rate of escape does not vanish, then we know that there exists an equivariant map from the Poisson boundary of $(\Gamma ,\mu )$ into the boundary at infinity of Y by a result of Karlsson and Margulis. The key ingredient of the proof is $\mu$-harmonic functions on $\Gamma$ and $\mu$-harmonic maps from $\Gamma$ into Y. We prove a result similar to the above for an isometric action of $\Gamma$ on a locally finite-dimensional $\textrm{CAT}(0)$ space.