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An example of an amenable action from geometry

Published online by Cambridge University Press:  19 September 2008

R. J. Spatzier
R. J. Spatzier
Affiliation:
Department of Mathematics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA
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Abstract

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Let M be a compact manifold of not necessarily constant negative curvature. We observe that π1(M) acts amenably on the sphere at infinity of the universal cover of M with respect to a natural measure class. We also note that this action is of type III1.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 7 , Issue 2 , June 1987 , pp. 289 - 293
Copyright
Copyright © Cambridge University Press 1987

References

REFERENCES

[A]Anosov, D. V.. Geodesic Flows on Closed Riemannian Manifolds of Negative Curvature. Trudy Steklov, 90 (1967), Proceedings of the Steklov Institute of Mathematics, (Petrovskii, and Mikolskii, , ed.), Amer. Math. Soc. 1969.Google Scholar
[BGS]Ballmann, W., Gromov, M. & Schroeder, V.. Manifolds of Non-positive Curvature. Progress in Mathematics vol. 61, Birkhauser 1985.Google Scholar
[B1]Bowen, R.. Periodic orbits for hyperbolic flows. Amer. J. Math. 94 (1972), 1–30.Google Scholar
[B2]Bowen, R.. Anosov foliations are hyperfinite. Annals of Math. 106 (1977), 549–565.Google Scholar
[E]Eberlein, P.. Rigidity problems of manifolds of non-positive curvature. To appear in Proceedings of Conference on Differential Geometry and Global Analysis, Berlin, June 1984.Google Scholar
[Ma1]Margulis, G. A.. Applications of ergodic theory to the investigation of manifolds of negative curvature. Fund. Anal. & Appl. 3 (no. 4), (1969), 335–336.Google Scholar
[Ma2]Margulis, G. A.. Certain measures associated with U-flows on compact manifolds. Fund. Anal. Appl. 4 (no. 1), (1970), 55–67.CrossRefGoogle Scholar
[M]Moore, C. C.. Ergodic theory and von Neumann algebras. Operator Algebras and Applications, Proceedings of Symposia in Pure Mathematics, vol. 38, part 2, 179–226, Amer. Math. Soc. 1982.CrossRefGoogle Scholar
[PP]Parry, W. & Pollicott, M.. An analogue of the prime number theorem for closed orbits of Axiom A flows. Annals of Math. 118 (1983), 573–591.Google Scholar
[R]Ramsay, A.. Virtual groups and group actions. Adv. in Math. 6 (1971), 253–322.Google Scholar
[S]Series, C.. The Poincare flow of a foliation. Amer. J. Math. 102 (1980), 93–128.Google Scholar
[Su]Sullivan, D.. Discrete conformal groups and measurable dynamics. Bull. Amer. Math. Soc. 6 (1982), 57–73Google Scholar
[T]Toll, C. H.. A multiplicative asymptotic for the prime geodesic theorem. Thesis, University of Maryland (1984).Google Scholar
[Z1]Zimmer, R. J.. Hyperfinite factors and amenable ergodic actions. Invent. Math. 41 (1977), 23–31.Google Scholar
[Z2]Zimmer, R. J.. Amenable ergodic group actions and an application to Poisson boundaries of random walks. J. Fund. Anal. 27 (1978), 350–372.Google Scholar
[Z3]Zimmer, R. J.. Ergodic theory, group representations and rigidity. Bull. Amer. Math. Soc. 6 (1982), 383–416.Google Scholar