Isometric group actions with vanishing rate of escape on CAT(0) spaces

TL;DR

This paper investigates isometric group actions on CAT(0) spaces with vanishing escape rate, showing such actions either fix a boundary point or preserve a flat subspace, using harmonic maps in the analysis.

Contribution

It establishes a link between vanishing escape rate actions and invariant flat subspaces in CAT(0) spaces, extending understanding of group actions in geometric group theory.

Findings

Actions with vanishing escape rate either fix a boundary point or preserve a flat subspace.

Existence of invariant flat subspaces under certain isometric actions.

Harmonic maps are key tools in analyzing group actions on 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.