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Forum Mathematicum

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

Russell RicksORCID iD: https://orcid.org/0000-0002-2007-2285
Published Online: 2019-06-14 | DOI: https://doi.org/10.1515/forum-2018-0133


We prove the following rank rigidity result for proper CAT(0) 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 X equals π and Γ does not act minimally on X, then 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 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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About the article

Received: 2018-05-31

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, DOI: https://doi.org/10.1515/forum-2018-0133.

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