Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T17:52:52.205Z Has data issue: false hasContentIssue false

Orbit equivalence and actions of

Published online by Cambridge University Press:  12 March 2014

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

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
Copyright
Copyright © Association for Symbolic Logic 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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. 3667.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. 431450.CrossRefGoogle Scholar
[5]Connes, A. and Weiss, B., Property T and asymptotically invariant sequences, Israel Journal of Mathematics, vol. 37 (1980), pp. 209210.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. 119159.CrossRefGoogle Scholar
[8]Dye, H. A., On groups of measure preserving transformation. II, American Journal of Mathematics, vol. 85 (1963), pp. 551576.CrossRefGoogle Scholar
[9]Gaboriau, D., On orbit equivalence of measure preserving actions, Rigidity in dynamics and geometry (Cambridge 2000), Springer, 2002, pp. 167186.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. 903928.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. 241257.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. 180.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. 6365.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. 161164.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. 193208.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