Barré, Sylvain Two-dimensional finite non-positively curved polyhedra of rank two. (Polyèdres finis de dimension 2 à courbure \(\leq 0\) et de rang 2.) (French) Zbl 0831.53031 Ann. Inst. Fourier 45, No. 4, 1037-1059 (1995). Summary: We locally define the notion of polyhedra of rank two for two-dimensional finite non-positively curved polyhedra. We prove that the universal covering of such a space is either the product of two trees or an Euclidean Tits building of rank two. Cited in 1 ReviewCited in 7 Documents MSC: 53C35 Differential geometry of symmetric spaces 51E24 Buildings and the geometry of diagrams 51M20 Polyhedra and polytopes; regular figures, division of spaces 20E42 Groups with a \(BN\)-pair; buildings 05C12 Distance in graphs Keywords:rank two polyhedra; non-positively curved polyhedra; universal covering; Tits building PDF BibTeX XML Cite \textit{S. Barré}, Ann. Inst. Fourier 45, No. 4, 1037--1059 (1995; Zbl 0831.53031) Full Text: DOI Numdam EuDML OpenURL References: [1] W. BALLMANN, M. BRIN, Orbihedra of nonpositive curvature, preprint. · Zbl 0866.53029 [2] K.S. BROWN, Buildings, Springer-Verlag, New York, 1989. · Zbl 0715.20017 [3] K.S. BROWN, Five lectures on buildings, Congrès de Trieste 1990. · Zbl 0846.20032 [4] W. BALLMANN, M. BRIN, P. EBERLEIN, Structure of manifolds of nonpositive curvature. I, Annals of Mathematics, 122 (1985), 171-203. · Zbl 0589.53047 [5] W. BALLMANN, M. BRIN, R. SPATZIER, Structure of manifolds of nonpositive curvature. II, Annals of Mathematics, 122 (1985), 205-235. · Zbl 0598.53046 [6] K. BURNS, R. SPATZIER, Manifolds of nonpositive curvature and their buildings, Publications Mathématiques de l’IHES, n° 65, 1987. · Zbl 0643.53037 [7] W. BALLMANN, S. BUYALO, Nonpositive curved metrics on 2-polyedra, Bonn, février 1994. [8] M. BRIDSON, A. HAEFLIGER, Livre en préparation. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.