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

Barré, Sylvain

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.

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

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.