Publications Mathématiques de l'Institut des Hautes Études Scientifiques

, Volume 82, Issue 1, pp 169–209

Orbihedra of nonpositive curvature

  • Werner Ballmann
  • Michael Brin

DOI: 10.1007/BF02698640

Cite this article as:
Ballmann, W. & Brin, M. Publications Mathématiques de l’Institut des Hautes Scientifiques (1995) 82: 169. doi:10.1007/BF02698640


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.

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