Lipschitz precompactness for closed negatively curved manifolds
HTML articles powered by AMS MathViewer
- by Igor Belegradek PDF
- Proc. Amer. Math. Soc. 127 (1999), 1201-1208 Request permission
Abstract:
We prove that, given a integer $n\ge 3$ and a group $\pi$, the class of closed Riemannian $n$-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
- Werner Ballmann, Mikhael Gromov, and Viktor Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics, vol. 61, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 823981, DOI 10.1007/978-1-4684-9159-3
- I. Belegradek, Intersections in hyperbolic manifolds, Geometry & Topology 2 (1998), 117–144, electronic: http://www.maths.warvick.ac.uk/gt/.
- Mladen Bestvina, Degenerations of the hyperbolic space, Duke Math. J. 56 (1988), no. 1, 143–161. MR 932860, DOI 10.1215/S0012-7094-88-05607-4
- Mladen Bestvina and Mark Feighn, Stable actions of groups on real trees, Invent. Math. 121 (1995), no. 2, 287–321. MR 1346208, DOI 10.1007/BF01884300
- M. Burger and V. Schroeder, Amenable groups and stabilizers of measures on the boundary of a Hadamard manifold, Math. Ann. 276 (1987), no. 3, 505–514. MR 875344, DOI 10.1007/BF01450845
- B. H. Bowditch, Discrete parabolic groups, J. Differential Geom. 38 (1993), no. 3, 559–583. MR 1243787, DOI 10.4310/jdg/1214454483
- F. T. Farrell and L. E. Jones, Negatively curved manifolds with exotic smooth structures, J. Amer. Math. Soc. 2 (1989), no. 4, 899–908. MR 1002632, DOI 10.1090/S0894-0347-1989-1002632-2
- F. T. Farrell and L. E. Jones, Complex hyperbolic manifolds and exotic smooth structures, Invent. Math. 117 (1994), no. 1, 57–74. MR 1269425, DOI 10.1007/BF01232234
- F. T. Farrell and L. E. Jones, Topological rigidity for compact non-positively curved manifolds, Differential geometry: Riemannian geometry (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 229–274. MR 1216623
- Marco Rigoli, The conformal Gauss map of submanifolds of the Möbius space, Ann. Global Anal. Geom. 5 (1987), no. 2, 97–116. MR 944775, DOI 10.1007/BF00127853
- David Gabai, On the geometric and topological rigidity of hyperbolic $3$-manifolds, J. Amer. Math. Soc. 10 (1997), no. 1, 37–74. MR 1354958, DOI 10.1090/S0894-0347-97-00206-3
- D. Gabai, G. R. Meyerhoff, and N. Thurston, Homotopy Hyperbolic 3-Manifolds are Hyperbolic, MSRI Preprint No. 1996-058 (1996).
- R. E. Greene and H. Wu, Lipschitz convergence of Riemannian manifolds, Pacific J. Math. 131 (1988), no. 1, 119–141. MR 917868, DOI 10.2140/pjm.1988.131.119
- M. Kapovich and B. Leeb, On asymptotic cones and quasi-isometry classes of fundamental groups of $3$-manifolds, Geom. Funct. Anal. 5 (1995), no. 3, 582–603. MR 1339818, DOI 10.1007/BF01895833
- Robion C. Kirby and Laurence C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Annals of Mathematics Studies, No. 88, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. With notes by John Milnor and Michael Atiyah. MR 0645390, DOI 10.1515/9781400881505
- Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1996. Reprint of the 1963 original; A Wiley-Interscience Publication. MR 1393940
- Frédéric Paulin, Topologie de Gromov équivariante, structures hyperboliques et arbres réels, Invent. Math. 94 (1988), no. 1, 53–80 (French). MR 958589, DOI 10.1007/BF01394344
- Frédéric Paulin, Outer automorphisms of hyperbolic groups and small actions on $\textbf {R}$-trees, Arboreal group theory (Berkeley, CA, 1988) Math. Sci. Res. Inst. Publ., vol. 19, Springer, New York, 1991, pp. 331–343. MR 1105339, DOI 10.1007/978-1-4612-3142-4_{1}2
- F. Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. Math. 87 (1968), 56–88.
- Shing-tung Yau, On the fundamental group of compact manifolds of non-positive curvature, Ann. of Math. (2) 93 (1971), 579–585. MR 283726, DOI 10.2307/1970888
Additional Information
- Igor Belegradek
- Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
- Address at time of publication: Department of Mathematics and Statistics, McMaster University, 1280 Main St. West, Hamilton, Ontario, Canada L8S 4K1
- MR Author ID: 340900
- Email: igorb@math.umd.edu, belegi@icarus.math.mcmaster.ca
- Received by editor(s): July 30, 1997
- Communicated by: Christopher Croke
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 1201-1208
- MSC (1991): Primary 53C20, 53C23; Secondary 20F32, 57R55
- DOI: https://doi.org/10.1090/S0002-9939-99-04654-7
- MathSciNet review: 1476116