Boundary conditions detecting product splittings of CAT(0) spaces

Abstract

Let $X$ be a proper CAT(0) space and $G$ a group of isometries of $X$ acting cocompactly 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 every round sphere $K\subset \partial X$ inside a proper subsphere of $K$.