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Orbit equivalence and actions of

Published online by Cambridge University Press:  12 March 2014

Asge Törnquist
Asge Törnquist*
Affiliation:
Department of Mathematics, University of Copenhagen, Universitetsparken 5, Dk-2100 Copenhagen, Denmark. E-mail: [email protected]
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Abstract

In this paper we show that there are “E0 many” orbit inequivalent free actions of the free groups , 2 ≤ n ≤ ∞ by measure preserving transformations on a standard Borel probability space. In particular, there are uncountably many such actions.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 1 , March 2006 , pp. 265 - 282
Copyright
Copyright © Association for Symbolic Logic 2006

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References

REFERENCES

[1]Becker, H. and Kechris, A., The descriptive set theory of Polish group actions, London Mathematical Society Lecture Notes, vol. 232, Cambridge University Press, 1996.Google Scholar
[2]Bekka, B., de la Harpe, P., and Valette, A., Kazhdan's property (T), to appear, 2003.Google Scholar
[3]Burger, M., Kazhdan constants for SL3(ℤ), Journal für die reine und angewandte Mathematik, vol. 413 (1991), pp. 36–67.Google Scholar
[4]Connes, A., Feldman, J., and Weiss, B., An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynamical Systems, vol. 1 (1981), pp. 431–450.CrossRefGoogle Scholar
[5]Connes, A. and Weiss, B., Property T and asymptotically invariant sequences, Israel Journal of Mathematics, vol. 37 (1980), pp. 209–210.CrossRefGoogle Scholar
[6]de la Harpe, P. and Valette, A., La propriété (T) de Kazhdan pour les groupes localement compacts, Astérisque, vol. 175 (1989).Google Scholar
[7]Dye, H. A., On groups of measure preserving transformation. I, American Journal of Mathematics, vol.81 (1959), pp. 119–159.CrossRefGoogle Scholar
[8]Dye, H. A., On groups of measure preserving transformation. II, American Journal of Mathematics, vol. 85 (1963), pp. 551–576.CrossRefGoogle Scholar
[9]Gaboriau, D., On orbit equivalence of measure preserving actions, Rigidity in dynamics and geometry (Cambridge 2000), Springer, 2002, pp. 167–186.CrossRefGoogle Scholar
[10]Gaboriau, D. and Popa, S., An uncountable family ofnon orbit equivalent actions of Fn, preprint, 2003.Google Scholar
[11]Halmos, P., Lectures on ergodic theory, Chelsea Publishing Co., New York, 1956.Google Scholar
[12]Harrington, L., Kechris, A., and Louveau, A., A Glimm-Efross dichotomy for Borel equivalence relations, Journal of the American Mathematical Society, vol. 3 (1990), no. 4, pp. 903–928.CrossRefGoogle Scholar
[13]Hjorth, G., Notes from a course at Notre Dame, 2000, lecture notes.Google Scholar
[14]Hjorth, G., The isomorphism relation on countable torsion free abelian groups, Fundamenta Mathematical vol. 175 (2002), pp. 241–257.CrossRefGoogle Scholar
[15]Hjorth, G., A converse to Dye's theorem, preprint, 2004.CrossRefGoogle Scholar
[16]Jackson, S., Kechris, A., and Louveau, A., Countable Borel equivalence relations, Journal of Mathematical Logic, vol. 2 (2002), no. 1, pp. 1–80.CrossRefGoogle Scholar
[17]Kazhdan, D., Connection of the dual space of a group with the structure of its closed subgroups, Functional Analysis audits Applications, vol. 1 (1967), pp. 63–65.CrossRefGoogle Scholar
[18]Kechris, A., Classical descriptive set theory, Springer, 1995.CrossRefGoogle Scholar
[19]Kechris, A., Unitary representations and modular actions, preprint, 2003.Google Scholar
[20]Kechris, A. and Miller, B., Topics in orbit equivalence, Lecture Notes in Mathematics, vol. 1852, Springer, 2004.Google Scholar
[21]Ornstein, D. and Weiss, B., Ergodic theory of amenable group actions, Bulletin of the American Mathematical Society, vol. 2 (1980), pp. 161–164.CrossRefGoogle Scholar
[22]Popa, S., On a class of type II1 factors with Betti numbers invariants, MSRI preprint, math.OA/0209130, 2001.Google Scholar
[23]Reiter, H. and Stegeman, J., Classical harmonic analysis and locally compact groups, London Mathematical Society Monographs, new series, vol. 22, Oxford University Press, 2001.Google Scholar
[24]Schmidt, K., Asymptotically invariant sequences and an action of SL(2, ℤ) on the sphere, Israel Journal of Mathematics, vol. 37 (1980), pp. 193–208.CrossRefGoogle Scholar
[25]Shalom, Y., Bounded generation and Kazhdan's property (T), Publications Mathématiques Institut de Hautes Études Scientifiques, (2001).Google Scholar
[26]Törnquist, A., The Borel complexity of orbit equivalence, Ph.D. thesis, UCLA, 2005.Google Scholar