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Affine maps between CAT(0) spaces

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Abstract

We study affine maps between CAT(0) spaces with cocompact group actions, and show that they essentially split as products of dilations and linear maps (on the Euclidean factor). This extends known results from the Riemannian case. Furthermore, we prove a splitting lemma for the Tits boundary of a CAT(0) space with cocompact group action, a variant of a splitting lemma for geodesically complete CAT(1) spaces by Lytchak.

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Acknowledgments

We thank Dan Guralnik and Eric Swenson for their inspiring paper [12], and Russell Ricks and Ben Schmidt for discussions of this paper. Ricks in particular suggested the simple proof of Lemma 3.7 and the reference for Theorem 2.3. We also thank the referee for a thorough report. The first two authors held postdoctoral fellowships at the University of Michigan, and are thankful for the excellent working environment provided.

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Authors and Affiliations

  1. University of Michigan, Ann Arbor, MI, USA

    Hanna Bennett & Ralf Spatzier

  2. Epic, Verona, WI, 53593, USA

    Christopher Mooney

Authors
  1. Hanna Bennett
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  2. Christopher Mooney
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  3. Ralf Spatzier
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Corresponding author

Correspondence to Ralf Spatzier.

Additional information

H. Bennett is Supported in part by an EAF Grant for the IBL Center at the University of Michigan

C. Mooney is Supported in part by the NSF EMSW21 Grant RTG-0602191 and a Caterpillar Fellowship

R. Spatzier is Supported in part by the NSF Grants DMS-0906085, DMS-1307164 and DMS-0932078 000, the latter while this author was in residence at MSRI in Berkeley during the Spring 2015 semester.

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Bennett, H., Mooney, C. & Spatzier, R. Affine maps between CAT(0) spaces. Geom Dedicata 180, 1–16 (2016). https://doi.org/10.1007/s10711-015-0087-3

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