# Boundary conditions detecting product splittings of CAT(0) spaces

### Russell Ricks

Binghamton University, USA

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## 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 $K$.

## Cite this article

Russell Ricks, Boundary conditions detecting product splittings of CAT(0) spaces. Groups Geom. Dyn. 14 (2020), no. 1, pp. 283–295

DOI 10.4171/GGD/544