Detecting product splittings of CAT(0) spaces

TL;DR

This paper establishes conditions under which a proper CAT(0) space with a cocompact isometry group splits as a product, based on invariant subsets of its boundary at infinity.

Contribution

It proves that certain invariant subsets of the boundary at infinity imply a product splitting of the space, extending understanding of CAT(0) space decompositions.

Findings

Presence of an invariant subset of circumradius π/2 implies a product splitting.

Proper discontinuity of the group action yields additional equivalent conditions for splitting.

Conditions involving invariant sets intersecting round spheres characterize product decompositions.

Abstract

Let $X$ be a proper CAT($0$) space and $G$ a cocompact group of isometries of $X$ without fixed point at infinity. We prove that if $\partial X$ contains an invariant subset of circumradius $\pi/2$, then $X$ contains a quasi-dense, closed convex subspace that splits as a product. Adding the assumption that the $G$-action on $X$ is properly discontinuous, we give more conditions that are equivalent to a product splitting. In particular, this occurs if $\partial X$ contains a proper nonempty, closed, invariant, $\pi$-convex set in $\partial X$; or if some nonempty closed, invariant set in $\partial X$ intersects each round sphere $K \subset \partial X$ inside a proper subsphere of $K$.