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Orbihedra of nonpositive curvature

  • Werner Ballmann
  • Michael Brin
Article

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.

Keywords

Closed Geodesic Geodesic Segment Geodesic Curvature Geodesic Flow Nonpositive Curvature 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Publications Mathématiques de L’I.É.E.S. 1995

Authors and Affiliations

  • Werner Ballmann
    • 1
  • Michael Brin
    • 2
  1. 1.Mathematisches Institut der Universität BonnBonnGermany
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA

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