Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Fuchsian manifolds


Author: Su Shing Chen
Journal: Trans. Amer. Math. Soc. 203 (1975), 247-256
MSC: Primary 53C20
DOI: https://doi.org/10.1090/S0002-9947-1975-0362135-1
MathSciNet review: 0362135
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Recently Eberlein and O'Neill have investigated Riemannian manifolds of negative sectional curvature. For visibility manifolds, they have obtained a classification into three types: parabolic, axial and fuchsian. Fundamental groups of fuchsian manifolds of finite type will be investigated. The main theorem is that isometry groups of certain (not necessarily compact) fuchsian manifolds are finite. Fundamental groups of fuchsian manifolds of finite type are not amenable. The spectral radius of the random matrix of the fundamental group of a compact Riemannian manifold of negative sectional curvature is less than one.


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 40 #4891. MR 0251664 (40:4891)
  • [2] S. Chen, On Lobatchewsky manifolds. Bull. Amer. Math. Soc. 80 (1974), 244-247. MR 0339014 (49:3777)
  • [3] -, Amenability of isometry groups of Riemannian manifolds, Chinese J. Math. 2 (1974), 31-38. MR 0380666 (52:1563)
  • [4] P. Eberlein, Geodesic flows on negatively curved manifolds. I, Ann. of Math. (2) 95 (1972), 492-510. MR 46 #10024. MR 0310926 (46:10024)
  • [5] -, Some properties of the fundamental group of a fuchsian manifold, Invent. Math. 19 (1973), 5-13. MR 0400250 (53:4085)
  • [6] P. Eberlein and B. O'Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45-109. MR 0336648 (49:1421)
  • [7] L. Greenberg, Discrete subgroups of the Lorentz group, Math. Scand. 10 (1962), 85-107. MR 25 #5128. MR 0141731 (25:5128)
  • [8] -, Conformal transformations of Riemann surfaces, Amer. J. Math. 82 (1960), 749-760. MR 23 #A319. MR 0122988 (23:A319)
  • [9] -, Discrete groups of motions, Canad. J. Math. 12 (1960), 415-426. MR 22 #5932. MR 0115130 (22:5932)
  • [10] -, Commensurable groups of Moebius transformations, discontinuous groups and Riemann surfaces, Ann. of Math. Studies, no. 79, Princeton Univ. Press, Princeton, N. J., 1974. MR 0379689 (52:594)
  • [11] H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959), 336-354. MR 22 #253. MR 0109367 (22:253)
  • [12] J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1-7. MR 38 #636. MR 0232311 (38:636)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C20

Retrieve articles in all journals with MSC: 53C20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0362135-1
Keywords: Fuchsian manifold, negative sectional curvature, limit set, properly discontinuous group
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society