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# Forum Mathematicum

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Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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# A rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries

Russell Ricks
Published Online: 2019-06-14 | DOI: https://doi.org/10.1515/forum-2018-0133

## Abstract

We prove the following rank rigidity result for proper $\mathrm{CAT}\left(0\right)$ spaces with one-dimensional Tits boundaries: Let Γ be a group acting properly discontinuously, cocompactly, and by isometries on such a space X. If the Tits diameter of $\partial X$ equals π and Γ does not act minimally on $\partial X$, then $\partial X$ is a spherical building or a spherical join. If X is also geodesically complete, then X is a Euclidean building, higher rank symmetric space, or a nontrivial product. Much of the proof, which involves finding a Tits-closed convex building-like subset of $\partial X$, does not require the Tits diameter to be π, and we give an alternate condition that guarantees rigidity when this hypothesis is removed, which is that a certain invariant of the group action be even.

Keywords: CAT(0); dimension; rank rigidity

MSC 2010: 53C20; 53C24; 20F65

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Revised: 2019-04-06

Published Online: 2019-06-14

Funding Source: National Science Foundation

Award identifier / Grant number: NSF 1045119

This material is based upon work supported by the National Science Foundation under grant number NSF 1045119.

Citation Information: Forum Mathematicum, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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