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Orbihedra of nonpositive curvature. (English) Zbl 0866.53029
In this paper the authors generalize some results on smooth Riemannian manifolds of nonpositive sectional curvature to the much more general class of metric spaces with nonpositive curvature in the sense of Alexandrov and Busemann. For a good treatment of such spaces see the book of W. Ballmann [Lectures on spaces of nonpositive curvature (DMV Seminar 25, Birkhäuser, Basel) (1995; Zbl 0834.53003)].
Let \(X\) denote a simply connected topological space, and let \(\Gamma\) denote a group of homeomorphisms of \(X\) that acts properly discontinuously. The pair \((X,\Gamma)\) is called an orbispace and is compact if the quotient space \(X/\Gamma\) is compact. An orbispace \((X,\Gamma)\) is an orbihedron if \(X\) admits a \(\Gamma\)-invariant triangulation. An orbispace \((X,\Gamma)\) is said to have rank 1 if there exists a geodesic \(\gamma\) of \(X\) and an isometry \(\varphi\) of \(\Gamma\) such that \(\varphi\) translates \(\gamma\) and \(\gamma\) does not bound a flat half plane in \(X\).
If \(X/\Gamma\) is a smooth Riemannian manifold of nonpositive sectional curvature, then the rank rigidity theorem proved by the first author says that if \((X,\Gamma)\) has rank at least two, then either \(X\) splits as a Riemannian product or \(X\) is a symmetric space of noncompact type. The main result of this paper is an analog for compact 2-dimensional orbihedra of the rank rigidity theorem. Theorem. Let \((X,\Gamma)\) be a compact 2-dimensional orbihedron without boundary and with a piecewise smooth metric of nonpositive curvature. Then either a) \((X,\Gamma)\) has rank 1, or b) \(X\) is the product of two trees endowed with the product metric of two interior metrics, or c) \(X\) is a thick Euclidean building of type \(A_2\), \(B_2\) or \(G_2\), endowed with its natural metric.
In addition the authors extend to rank 1 orbispaces or 2-dimensional orbihedra a number of results that were previously proved by the first author in the case that \((X,\Gamma)\) is a smooth Riemannian manifold. They also obtain two generalizations of a result of J. Tits about the existence of free subgroups of \(\Gamma\) with two or more generators. Theorem. Let \((X,\Gamma)\) be a compact orbispace that is either a Euclidean building or a 2-dimensional orbihedron without boundary and a piecewise smooth metric of nonpositive curvature. Then either \(\Gamma\) contains a nonabelian free group or \(X\) is isometric to a Euclidean space in the first case and the Euclidean plane in the second case. In the first case \(\Gamma\) is assumed to be a group of automorphisms of \(X\).

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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