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Let be a proper CAT(0) space and a group of isometries of acting cocompactly without fixed point at infinity. We prove that if contains an invariant subset of circumradius , then contains a quasi-dense, closed convex subspace that splits as a product.
Adding the assumption that the -action on is properly discontinuous, we give more conditions that are equivalent to a product splitting. In particular, this occurs if contains a proper nonempty, closed, invariant, -convex set in ; or if some nonempty closed, invariant set in intersects every round sphere inside a proper subsphere of .
Cite this article
Russell Ricks, Boundary conditions detecting product splittings of CAT(0) spaces. Groups Geom. Dyn. 14 (2020), no. 1, pp. 283–295