Orbihedra of nonpositive curvature
type: Item
instance of: scholarly article
title: Orbihedra of nonpositive curvature
zbMATH Open document ID: 0866.53029
DOI: 10.1007/BF02698640
author: Michael Brin
author: Werner Ballmann
published in: Publications Mathématiques
publication date: 1997-07-20T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104108
review text: 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 \textit{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 …
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 910527
zbMATH Keywords: rank
zbMATH Keywords: rigidity for rank at least 2
zbMATH Keywords: orbihedron
zbMATH Keywords: nonpositive curvature
zbMATH Keywords: Euclidean building
Wikidata QID: Q55896011
reviewed by: Patrick Eberlein
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 910527