Fuchsian manifolds
HTML articles powered by AMS MathViewer
- by Su Shing Chen
- Trans. Amer. Math. Soc. 203 (1975), 247-256
- DOI: https://doi.org/10.1090/S0002-9947-1975-0362135-1
- PDF | Request permission
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
- R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1–49. MR 251664, DOI 10.1090/S0002-9947-1969-0251664-4
- Su Shing Chen, On Lobatchewsky manifolds, Bull. Amer. Math. Soc. 80 (1974), 244–247. MR 339014, DOI 10.1090/S0002-9904-1974-13444-0
- Su Shing Chen, Amenability of isometry groups of Riemannian manifolds, Chinese J. Math. 2 (1974), no. 1, 31–38. MR 380666
- Patrick Eberlein, Geodesic flows on negatively curved manifolds. I, Ann. of Math. (2) 95 (1972), 492–510. MR 310926, DOI 10.2307/1970869
- Patrick Eberlein, Some properties of the fundamental group of a Fuchsian manifold, Invent. Math. 19 (1973), 5–13. MR 400250, DOI 10.1007/BF01418848
- P. Eberlein and B. O’Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45–109. MR 336648, DOI 10.2140/pjm.1973.46.45
- Leon Greenberg, Discrete subgroups of the Lorentz group, Math. Scand. 10 (1962), 85–107. MR 141731, DOI 10.7146/math.scand.a-10515
- Leon Greenberg, Conformal transformations of Riemann surfaces, Amer. J. Math. 82 (1960), 749–760. MR 122988, DOI 10.2307/2372937
- Leon Greenberg, Discrete groups of motions, Canadian J. Math. 12 (1960), 415–426. MR 115130, DOI 10.4153/CJM-1960-036-8
- Leon Greenberg, Commensurable groups of Moebius transformations, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973) Ann. of Math. Studies, No. 79, Princeton Univ. Press, Princeton, N.J., 1974, pp. 227–237. MR 0379689
- Harry Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959), 336–354. MR 109367, DOI 10.1090/S0002-9947-1959-0109367-6
- J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1–7. MR 232311, DOI 10.4310/jdg/1214501132
Bibliographic Information
- © Copyright 1975 American Mathematical Society
- 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