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

Abstract

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

References

  • [1]

    W. Ballmann, Lectures on Spaces of Nonpositive Curvature, DMV Seminar 25, Birkhäuser, Basel, 1995. Google Scholar

  • [2]

    W. Ballmann and M. Brin, Orbihedra of nonpositive curvature, Publ. Math. Inst. Hautes Études Sci. (1995), no. 82, 169–209. Google Scholar

  • [3]

    W. Ballmann and M. Brin, Rank rigidity of Euclidean polyhedra, Amer. J. Math. 122 (2000), no. 5, 873–885. CrossrefGoogle Scholar

  • [4]

    W. Ballmann and S. Buyalo, Periodic rank one geodesics in Hadamard spaces, Geometric and Probabilistic Structures in Dynamics, Contemp. Math. 469, American Mathematical Society, Providence (2008), 19–27. Google Scholar

  • [5]

    A. Balser and A. Lytchak, Centers of convex subsets of buildings, Ann. Global Anal. Geom. 28 (2005), no. 2, 201–209. CrossrefGoogle Scholar

  • [6]

    H. Bennett, C. Mooney and R. Spatzier, Affine maps between CAT(0) spaces, Geom. Dedicata 180 (2016), 1–16. Google Scholar

  • [7]

    M. R. Bridson and A. Haefliger, Metric Spaces of Non-positive Curvature, Grundlehren Math. Wiss. 319, Springer, Berlin, 1999. Google Scholar

  • [8]

    K. Burns and R. Spatzier, Manifolds of nonpositive curvature and their buildings, Publ. Math. Inst. Hautes Études Sci. 65 (1987), 35–59. CrossrefGoogle Scholar

  • [9]

    P.-E. Caprace and M. Sageev, Rank rigidity for CAT(0) cube complexes, Geom. Funct. Anal. 21 (2011), no. 4, 851–891. Google Scholar

  • [10]

    R. Charney and A. Lytchak, Metric characterizations of spherical and Euclidean buildings, Geom. Topol. 5 (2001), 521–550. CrossrefGoogle Scholar

  • [11]

    D. P. Guralnik and E. L. Swenson, A “transversal” for minimal invariant sets in the boundary of a CAT(0) group, Trans. Amer. Math. Soc. 365 (2013), no. 6, 3069–3095. Google Scholar

  • [12]

    B. Kleiner, The local structure of length spaces with curvature bounded above, Math. Z. 231 (1999), no. 3, 409–456. CrossrefGoogle Scholar

  • [13]

    B. Leeb, A Characterization of Irreducible Symmetric Spaces and Euclidean Buildings of Higher Rank by Their Asymptotic Geometry, Bonner Math. Schriften 326, Universität Bonn, Bonn, 2000. Google Scholar

  • [14]

    A. Lytchak, Rigidity of spherical buildings and joins, Geom. Funct. Anal. 15 (2005), no. 3, 720–752. CrossrefGoogle Scholar

  • [15]

    P. Papasoglu and E. Swenson, Boundaries and JSJ decompositions of CAT(0)-groups, Geom. Funct. Anal. 19 (2009), no. 2, 559–590. Google Scholar

  • [16]

    R. Ricks, Flat strips, Bowen–Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces, Ergodic Theory Dynam. Systems 37 (2017), no. 3, 939–970. Google Scholar

  • [17]

    R. Ricks, Boundary conditions detecting product splittings of CAT(0) spaces, Groups Geom. Dyn., to appear; https://arxiv.org/abs/1804.06374.

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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