Piecewise Euclidean structures and Eberlein’s Rigidity Theorem in the singular case

Davis, Michael W¹ ; Okun, Boris² ; Zheng, Fangyang¹

¹ Department of Mathematics, The Ohio State University, Columbus, Ohio 43201, USA
² Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37400, USA

Geometry & topology, Tome 3 (1999) no. 1, pp. 303-330.

Résumé

In this article, we generalize Eberlein’s Rigidity Theorem to the singular case, namely, one of the spaces is only assumed to be a CAT(0) topological manifold. As a corollary, we get that any compact irreducible but locally reducible locally symmetric space of noncompact type does not admit a nonpositively curved (in the Aleksandrov sense) piecewise Euclidean structure. Any hyperbolic manifold, on the other hand, does admit such a structure.

Détail

DOI : 10.2140/gt.1999.3.303

Keywords: piecewise Euclidean structure, CAT(0) space, Hadamard space, rigidity theorem

Bibliographie

[1] W Ballmann, Lectures on spaces of nonpositive curvature, DMV Seminar 25, Birkhäuser Verlag (1995)

[2] W Ballmann, M Gromov, V Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics 61, Birkhäuser (1985)

[3] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften 319, Springer (1999)

[4] S S Chen, P Eberlein, Isometry groups of simply connected manifolds of nonpositive curvature, Illinois J. Math. 24 (1980) 73

[5] R Charney, M Davis, Singular metrics of nonpositive curvature on branched covers of Riemannian manifolds, Amer. J. Math. 115 (1993) 929

[6] R Charney, M Davis, The polar dual of a convex polyhedral set in hyperbolic space, Michigan Math. J. 42 (1995) 479

[7] R Charney, M Davis, G Moussong, Nonpositively curved, piecewise Euclidean structures on hyperbolic manifolds, Michigan Math. J. 44 (1997) 201

[8] M W Davis, T Januszkiewicz, Hyperbolization of polyhedra, J. Differential Geom. 34 (1991) 347

[9] P Eberlein, Rigidity of lattices of nonpositive curvature, Ergodic Theory Dynam. Systems 3 (1983) 47

[10] P Eberlein, Isometry groups of simply connected manifolds of nonpositive curvature II, Acta Math. 149 (1982) 41

[11] P Eberlein, Euclidean de Rham factor of a lattice of nonpositive curvature, J. Differential Geom. 18 (1983) 209

[12] P Eberlein, Lattices in spaces of nonpositive curvature, Ann. of Math. $(2)$ 111 (1980) 435

[13] A Eskin, B Farb, Quasi-flats and rigidity in higher rank symmetric spaces, J. Amer. Math. Soc. 10 (1997) 653

[14] M Gromov, Hyperbolic groups, from: "Essays in group theory", Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75

[15] M Gromov, W Thurston, Pinching constants for hyperbolic manifolds, Invent. Math. 89 (1987) 1

[16] B Kleiner, B Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. Hautes Études Sci. Publ. Math. (1997)

[17] B Leeb, A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry, Bonner Mathematische Schriften 326, Universität Bonn Mathematisches Institut (2000)

[18] O Loos, Symmetric spaces II: Compact spaces and classification, W. A. Benjamin, New York-Amsterdam (1969)

[19] G D Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies 78, Princeton University Press (1973)

[20] D Montgomery, L Zippin, Topological transformation groups, Interscience Publishers, New York-London (1955)

[21] P Planche, Structures de Finsler invariantes sur les espaces symétriques, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995) 1455

[22] M S Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 68, Springer (1972)

[23] A Selberg, Recent developments in the theory of discontinuous groups of motions of symmetric spaces, from: "Proceedings of the Fifteenth Scandinavian Congress (Oslo, 1968)", Lecture Notes in Mathematics 118, Springer (1970) 99