1995/01/07 by Werner Ballmann, Michael Brin · 3 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Geodesic #Mathematics #Curvature #Pure mathematics #Rank (graph theory) #Sectional curvature #Product (mathematics) #Metric (unit) #Piecewise #Mathematical analysis #Geometry #Combinatorics #Scalar curvature
paper · pdf · doi:10.1007/bf02698640
openalex publication_date 1995/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/23
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.