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Diagonal actions and Borel equivalence relations

Published online by Cambridge University Press:  12 March 2014

Longyun Ding  and
Su Gao
Longyun Ding
Affiliation:
School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, P. R. China, E-mail: [email protected]
Su Gao
Affiliation:
Department of Mathematics, Po Box 311430, University of North Texas, Denton, TX 76203, USA, E-mail: [email protected]
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Abstract

We investigate diagonal actions of Polish groups and the related intersection operator on closed subgroups of the acting group. The Borelness of the diagonal orbit equivalence relation is characterized and is shown to be connected with the Borelness of the intersection operator. We also consider relatively tame Polish groups and give a characterization of them in the class of countable products of countable abelian groups. Finally an example of a logic action is considered and its complexity in the Borel reducbility hierarchy determined.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 4 , December 2006 , pp. 1081 - 1096
Copyright
Copyright © Association for Symbolic Logic 2006

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References

REFERENCES

[1]Becker, H. and Kechris, A. S., The descriptive set theory of Polish group actions, London Mathematical Society Lecture Notes Series, vol. 232, Cambridge University Press, 1996.CrossRefGoogle Scholar
[2]Hjorth, G. and Kechris, A. S., Borel equivalence relations and classifications of countable models, Annals of Pure and Applied Logic, vol. 82 (1996), pp. 221–272.CrossRefGoogle Scholar
[3]Hjorth, G., Kechris, A. S., and Louveau, A., Borel equivalence relations induced by actions of the symmetric group, Annals of Pure and Applied Logic, vol. 92 (1998), pp. 63–112.CrossRefGoogle Scholar
[4]Kechris, A. S., Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer Verlag, 1995.CrossRefGoogle Scholar
[5]Robinson, D. J. S., A course in the theory of groups, Graduate Texts in Mathematics, vol. 80, Springer, 1996.CrossRefGoogle Scholar
[6]Sami, R., Polish group actions and the Vaught conjecture, Transactions of the American Mathematical Society, vol. 341 (1994), pp. 335–353.CrossRefGoogle Scholar
[7]Samoilenko, Y. S., Spectral theory of families of self-adjoint operators, Mathematics and Its Applications (Soviet Series), vol. 57, Kluwer Academic Publishers, 1991.CrossRefGoogle Scholar
[8]Solecki, S., Equivalence relations induced by actions of Polish groups, Transactions of the American Mathematical Society, vol. 347 (1995), pp. 4765–4777.CrossRefGoogle Scholar