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arXiv · 1804.06374

Detecting product splittings of CAT(0) spaces

Russell Ricks
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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 $K$.

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Russell Ricks. 2018-04-17. Detecting product splittings of CAT(0) spaces. https://arxiv.org/abs/1804.06374

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