Orbihedra of nonpositive curvature

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