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On the fundamental groups of negatively curved manifolds with finite volume


Author: Midori S. Goto
Journal: Proc. Amer. Math. Soc. 75 (1979), 99-103
MSC: Primary 53C20; Secondary 22E40
DOI: https://doi.org/10.1090/S0002-9939-1979-0529223-7
MathSciNet review: 529223
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Abstract: We will prove that if M is a complete, simply connected riemannian manifold with sectional curvature K, $ a \leqslant K < 0$, for $ a > 0$ and $ \Gamma $ a properly discontinuous group of isometries of M acting freely on M with volume $ (M/\Gamma )$ finite, then $ M/\Gamma $ is compact if and only if $ \Gamma $ consists of hyperbolic elements.


References [Enhancements On Off] (What's this?)

  • [1] R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1-49. MR 0251664 (40:4891)
  • [2] P. Eberlein and B. O'Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45-109. MR 0336648 (49:1421)
  • [3] D. Gromall and J. A. Wolf, Some relations between the metric structure and the algebraic structure of the fundamental groups in manifolds of nonpositive curvature, Bull. Amer. Math. Soc. 77 (1971), 545-552. MR 0281122 (43:6841)
  • [4] M. Gromov, Manifolds of negative curvature, preprint.
  • [5] -, Almost flat manifolds, preprint.
  • [6] E. Heintze, Mannigfaltigkeiten negativer Krümmung, preprint, 1976. MR 1940403 (2003m:53056)
  • [7] B. Lawson and S. T. Yau, Compact manifolds of nonpositive curvature, J. Differential Geometry 7 (1972), 211-228. MR 0334083 (48:12402)
  • [8] W. P. Thurston, The geometry and topology of 3-manifolds, preprint, 1978.
  • [9] S. T. Yau, On the fundamental group of compact manifolds of non-positive curvature, Ann. of Math. 93 (1971), 579-585. MR 0283726 (44:956)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0529223-7
Keywords: Fundamental group, negatively curved manifold, hyperbolic isometry, Dirichlet region
Article copyright: © Copyright 1979 American Mathematical Society

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