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A note on the fundamental group of a manifold of negative curvature

Published online by Cambridge University Press:  24 October 2008

J. C. Wood
J. C. Wood
Affiliation:
Department of Pure Mathematics, University of Leeds
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Extract

Let Y be a compact connected C∞ Riemannian manifold with negative sectional curvatures. Let G be a non-trivial subgroup of the fundamental group π1(Y). G is known to be cyclic if it is abelian (Preissmann (6)) or contains a subnormal abelian (hence cyclic) subgroup (Yau(9)). These results may be generalized as follows: Say that a group G is of type (α) if ∃a ∈ G, a ≠ e, such that for all b belonging to a set of generators for G we have ambn = bqap for some integers m, n, p, q with either m = p or n = q.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 83 , Issue 3 , May 1978 , pp. 415 - 417
Copyright
Copyright © Cambridge Philosophical Society 1978

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References

REFERENCES

(1)Al'ber, S. I.The topology of functional manifolds and the calculus of variations in the large. Russian Math. Surveys 25 (1970), 51–117.CrossRefGoogle Scholar
(2)Bishop, R. L. & O'neill, B.Manifolds of negative curvature. Trans. Amer. Math. Soc. 145 (1969), 1–49.CrossRefGoogle Scholar
(3)Byers, W. P.On a theorem of Preissmann. Proc. Amer. Math. Soc. 24 (1970), 50–51.CrossRefGoogle Scholar
(4)Eells, J. & Sampson, J. H.Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86 (1964), 109–160.CrossRefGoogle Scholar
(5)Hartman, P.On homotopic harmonic maps. Canad. J. Math. 19 (1967), 673–687.CrossRefGoogle Scholar
(6)Preissmann, A.Quelques propriétés globales des espaces de Riemann. Comm. Math. Helv. 15 (1942), 175–216.CrossRefGoogle Scholar
(7)Sampson, J. Some properties and applications of harmonic mappings. (To be published.)Google Scholar
(8)Schoen, R. & Yau, S. T.Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature. Comm. Math. Helv. 39 (51) (1976), 333–341.CrossRefGoogle Scholar
(9)Yau, S. T.On the fundamental group of compact manifolds of non-positive curvature. Ann. of Math. 93 (1971), 579–585.CrossRefGoogle Scholar