Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Lipschitz precompactness for closed negatively curved manifolds

Author(s): Igor Belegradek
Journal: Proc. Amer. Math. Soc. 127 (1999), 1201-1208.
MSC (1991): Primary 53C20, 53C23; Secondary 20F32, 57R55
MathSciNet review: 1476116
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Abstract: We prove that, given a integer $n\ge 3$ and a group $\pi$ , the class of closed Riemannian -manifolds of uniformly bounded negative sectional curvatures and with fundamental groups isomorphic to $\pi$ is precompact in the Lipschitz topology. In particular, the class breaks into finitely many diffeomorphism types.

References:

[BGS]: W. Ballmann, M. Gromov, V. Schroeder, Manifolds of nonpositive curvature, Birkhauser, Progress in mathematics, vol. 61, 1985. MR 87h:53050
[Bel]: I. Belegradek, Intersections in hyperbolic manifolds, Geometry & Topology 2 (1998), 117-144, electronic: http://www.maths.warvick.ac.uk/gt/.
[Bes]: M. Bestvina, Degenerations of the hyperbolic space, Duke Math. J 56 (1988), 143-161. MR 89m:57011
[BF]: M. Bestvina and M. Feighn, Stable actions of groups on real trees, Invent. Math. 121 (2) (1995), 287-321. MR 96h:20056
[BS]: M. Burger and V. Schroeder, Amenable groups and stabilizers of measures on the boundary of a Hadamard manifold, Math. Ann. 276 (3) (1987), 505-514. MR 88b:53049
[Bow]: B. H. Bowditch, Discrete parabolic groups, J. Differential Geom. 38 (3) (1993), 559-583. MR 94h:53046
[FJ1]: F. T. Farrell and L. E. Jones, Negatively curved manifolds with exotic smooth structures, J. Amer. Math. Soc. 2 (4) (1989), 899-908. MR 90f:53075
[FJ2]: -, Complex hyperbolic manifolds and exotic smooth structures., Invent. Math. 117 (1) (1989), 57-74. MR 95e:57052
[FJ3]: -, Topological rigidity for compact non-positively curved manifolds;, Proc. Sympos. Pure Math., 54, Part 3, Amer. Math. Soc., 1993, pp. 229-274. MR 94m:57067
[F]: K. Fukaya, Theory of convergence for Riemannian orbifolds, Japan. J. Math. (N. S.) 12 (1) (1986), 121-160. MR 89e:53083
[G]: D. Gabai, On the geometric and topological rigidity of hyperbolic $3$ -manifolds, J. Amer. Math. Soc. 10 (1) (1997), 37-74. MR 97h:57028
[GMT]: D. Gabai, G. R. Meyerhoff, and N. Thurston, Homotopy Hyperbolic 3-Manifolds are Hyperbolic, MSRI Preprint No. 1996-058 (1996).
[GW]: R. E. Greene and H. Wu, Lipschitz convergence of Riemannian manifolds, Pacific J. Math. 131 (1) (1988), 119-141. MR 89g:53063
[KL]: M. Kapovich and B. Leeb, On asymptotic cones and quasi-isometry classes of fundamental groups of $3$ -manifolds, Geom. Funct. Anal. 5 (3) (1995), 582-603. MR 96e:57006
[KS]: R. S. Kirby and L. S. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Annals of Mathematics Studies, No. 88. Princeton University Press, Princeton, 1977. MR 58:31082
[KN]: S. Kobayashi and K. Nomidzu, Foundations of differential geometry, Vol. I, Interscience Publishers, a division of John Wiley & Sons, 1963. MR 97c:53001a; MR 27:2945
[P1]: F. Paulin, Topologie de Gromov équivariante, structures hyperboliques et arbres réels, Invent. Math. 94 (1) (1988), 53-80. MR 90d:57015
[P2]: -, Outer automorphisms of hyperbolic groups and small actions on ${\mathbb{R}}$ -trees, Arboreal group theory, Math. Sci. Res. Inst. Publ., 19, Springer, 1991, pp. 331-343. MR 92g:57003
[W]: F. Waldhausen, On irreducible $3$ -manifolds which are sufficiently large, Ann. Math. 87 (1968), 56-88.
[Y]: S.-T. Yau, On the fundamental group of compact manifolds of nonpositive curvature, Ann. Math. 93 (2) (1971), 579-585. MR 44:956

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