Fixed point property for a CAT(0) space which admits a proper cocompact group action

Details Fixed point property for a CAT(0) space which admits a proper cocompact group action Fixed point property for a CAT(0) space which admits a proper cocompact group action

2011-02-03

arxiv.org/abs/1102.0729v2

Mathematics

Metric Geometry

Scientific paper

We prove that a geodesically complete CAT(0) space which admits a proper cocompact isometric action of a group and a complete locally doubling CAT(0) space satisfy a certain geometric condition obtained in the author's previous paper. Suppose that $\mathcal{Y}$ is a finite family of geodesically complete CAT(0) spaces each of which admits a proper cocompact isometric action of a group. Then, combining our result with a theorem due to Izeki, Kondo, and Nayatani, it follows that a random group of the graph model has a common fixed point when it acts isometrically on any (finite or infinite) Cartesian product of CAT(0) spaces each of which is isometric to some $Y\in\mathcal{Y}$. It also follows from our result that a sequence of expanders does not embed coarsely into such a product. The same results for a Cartesian product of complete CAT(0) spaces each of which is locally doubling with a common doubling constant also follow.

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