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Orbihedra of nonpositive curvature

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Abstract

A 2-dimensional orbihedron of nonpositive curvature is a pair (X, Γ), where X is a 2-dimensional simplicial complex with a piecewise smooth metric such that X has nonpositive curvature in the sense of Alexandrov and Busemann and Γ is a group of isometries of X which acts properly discontinuously and cocompactly. By analogy with Riemannian manifolds of nonpositive curvature we introduce a natural notion of rank 1 for (X, Γ) which turns out to depend only on Γ and prove that, if X is boundaryless, then either (X, Γ) has rank 1, or X is the product of two trees, or X is a thick Euclidean building. In the first case the geodesic flow on X is topologically transitive and closed geodesics are dense.

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Authors and Affiliations

  1. Mathematisches Institut der Universität Bonn, Wegelerstrasse 10, 53115, Bonn, Germany

    Werner Ballmann

  2. Department of Mathematics, University of Maryland, 20742, College Park, MD, USA

    Michael Brin

Authors
  1. Werner Ballmann
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  2. Michael Brin
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Partially supported by MSRI, SFB256 and University of Maryland.

Partially supported by MSRI, SFB256 and NSF DMS-9104134.

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Ballmann, W., Brin, M. Orbihedra of nonpositive curvature. Publications Mathématiques de l’Institut des Hautes Scientifiques 82, 169–209 (1995). https://doi.org/10.1007/BF02698640

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  • DOI: https://doi.org/10.1007/BF02698640

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