# zbMATH — the first resource for mathematics

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 $$X$$.

##### MSC:
 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
Full Text:
##### References:
 [1] S. B. Alexander andR. L. Bishop, The Hadamard-Cartan theorem in locally convex metric spaces,L’Enseignement math.,36 (1990), 309–320. · Zbl 0718.53055 [2] A. D. Alexandrov, A theorem on triangles in a metric space and some applications,Trudy Math. Inst. Steklov,38 (1951), 5–23 (Russian). [3] W. Ballmann, Axial isometries of manifolds of nonpositive curvature,Math. Ann.,259 (1982), 131–144. · Zbl 0487.53039 · doi:10.1007/BF01456836 [4] W. Ballmann,Lectures on Spaces of Nonpositive Curvature, Birkhäuser Verlag, 1995. · Zbl 0834.53003 [5] W. Ballmann, Nonpositively curved manifolds of higher rank,Annals of Math.,122 (1985), 597–609. · Zbl 0585.53031 · doi:10.2307/1971331 [6] W. Ballmann andM. Brin, Polygonal complexes and combinatorial group theory,Geometriae Dedicata,50 (1994), 165–191. · Zbl 0832.57002 · doi:10.1007/BF01265309 [7] W. Ballmann, M. Brin andP. Eberlein, Structure of manifolds of nonpositive curvature, I,Annals of Math.,122 (1985), 171–203. · Zbl 0589.53047 · doi:10.2307/1971373 [8] W. Ballmann, M. Brin andR. Spatzier, Structure of manifolds of nonpositive curvature, II,Annals of Math.,122 (1985), 205–235. · Zbl 0598.53046 · doi:10.2307/1971303 [9] W. Ballmann andS. Buyalo,Nonpositively curved metrics on 2-polyhedra, Math. Zeitschrift (to appear). · Zbl 0843.53036 [10] K. S. Brown,Buildings, Springer-Verlag, 1988. [11] F. Bruhat andJ. Tits, Groupes réductifs sur un corps local, I, Données radicielles valuées,Publications Math. IHES,40 (1972), 5–251. [12] K. Burns andR. Spatzier, Manifolds of nonpositive curvature and their buildings,Publications Math. IHES,65 (1987), 35–59. · Zbl 0643.53037 [13] H. Busemann, Spaces with nonpositive curvature,Acta Mathematica,80 (1948), 259–310. · Zbl 0038.10005 · doi:10.1007/BF02393651 [14] S.-S. Chen andP. Eberlein, Isometry groups of simply connected manifolds of nonpositive curvature,Ill. J. Math.,24 (1980), 73–103. · Zbl 0413.53029 [15] I. P. Cornfeld, S. V. Fomin and Ya. G.Sinai,Ergodic Theory, Springer-Verlag, 1982. [16] P. Eberlein, Lattices in spaces of nonpositive curvature,Annals of Math.,111 (1980), 435–476. · Zbl 0432.53023 · doi:10.2307/1971104 [17] P. Eberlein, Geodesic rigidity in compact nonpositively curved manifolds,Trans. Amer. Math. Soc.,268 (1981), 411–443. · Zbl 0526.53045 · doi:10.1090/S0002-9947-1981-0632536-5 [18] P. Eberlein andJ. Heber, A differential geometric characterization of symmetric spaces of higher rank,Publications Math. IHES,71 (1990), 33–44. · Zbl 0722.53041 [19] M. Gromov,Hyperbolic groups, Essays in Group Theory (S. M. Gersten, ed.), MSRI Publ. 8, Springer-Verlag, 1987, 75–264. · Zbl 0634.20015 [20] M. Gromov,Structures métriques pour les variétés riemanniennes, rédigé par J. Lafontaine et P. Pansu, Cedic/Fernand Nathan, 1981. [21] B. Kleiner,A metric characterization of Tits buildings, in preparation (1995). [22] G. Knieper, Das Wachstum der Äquivalenzklassen geschlossener Geodätischen in kompakten Mannigfaltigkeiten,Archiv. Math.,40 (1983), 559–568. · Zbl 0504.53041 · doi:10.1007/BF01192824 [23] R. Lyndon andP. Schupp,Combinatorial Group Theory, Springer-Verlag, 1977. · Zbl 0368.20023 [24] M. Morse, A fundamental class of geodesics on any surface of genus greater than one,Trans. Amer. Math. Soc.,26 (1924), 25–60. · JFM 50.0466.04 · doi:10.1090/S0002-9947-1924-1501263-9 [25] J. Tits, Free subgroups in linear groups,J. Algebra,20 (1972), 250–270. · Zbl 0236.20032 · doi:10.1016/0021-8693(72)90058-0 [26] J. Tits, Buildings of spherical type and finite BN-pairs,Lecture Notes in Math.,386, Springer-Verlag, 1974. · Zbl 0295.20047 [27] A. Wald, Begründung einer koordinatenlosen Differentialgeometrie der Flächen,Ergebnisse math. Kolloquium Wien,7 (1935), 24–46.
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.