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Stable actions and central extensions

Abstract

A probability-measure-preserving action of a countable group is called stable if its transformation-groupoid absorbs the ergodic hyperfinite equivalence relation of type \({\text {II}}_1\) under direct product. We show that for a countable group G and its central subgroup C, if G / C has a stable action, then so does G. Combining a previous result of the author, we obtain a characterization of a central extension having a stable action.

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References

  1. 1.

    Connes, A.: Classification of injective factors. Cases \(\text{ II }_1\), \(\text{ II }_\infty \), \(\text{ III }_\lambda \), \(\lambda \ne 1\). Ann. Math. (2) 104, 73–115 (1976)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Deprez, T., Vaes, S.: Inner amenability, property Gamma, McDuff \(\text{ II }_1\) factors and stable equivalence relations, preprint, to appear in Ergodic Theory Dyn. Syst., arXiv:1604.02911

  3. 3.

    Dye, H.A.: On groups of measure preserving transformation. I. Am. J. Math. 81, 119–159 (1959)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Feldman, J., Sutherland, C.E., Zimmer, R.J.: Subrelations of ergodic equivalence relations. Ergod. Theory Dyn. Syst. 9, 239–269 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Folland, G.B.: A Course in Abstract Harmonic Analysis. Textbooks in Mathematics, 2nd edn. CRC Press, Boca Raton (2016)

    MATH  Google Scholar 

  6. 6.

    Gaboriau, D.: Examples of groups that are measure equivalent to the free group. Ergod. Theory Dyn. Syst. 25, 1809–1827 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Jones, V.F.R., Schmidt, K.: Asymptotically invariant sequences and approximate finiteness. Am. J. Math. 109, 91–114 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Kechris, A.S.: Global Aspects of Ergodic Group Actions. Mathematical Surveys and Monographs, vol. 160. American Mathematical Society, Providence (2010)

    Book  Google Scholar 

  9. 9.

    Kida, Y.: Inner amenable groups having no stable action. Geom. Dedic. 173, 185–192 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Kida, Y.: Stability in orbit equivalence for Baumslag–Solitar groups and Vaes groups. Groups Geom. Dyn. 9, 203–235 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Kida, Y.: Stable actions of central extensions and relative property (T). Isr. J. Math. 207, 925–959 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Kida, Y.: Splitting in orbit equivalence, treeable groups, and the Haagerup property, preprint, to appear in the Proceedings of the MSJ-SI Conference in 2014, arXiv:1403.0688

  13. 13.

    McDuff, D.: Central sequences and the hyperfinite factor. Proc. Lond. Math. Soc. (3) 21, 443–461 (1970)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Phelps, R.R.: Lectures on Choquet’s Theorem. Lecture Notes in Mathematics, vol. 1757, 2nd edn. Springer, Berlin (2001)

    MATH  Google Scholar 

  15. 15.

    Popa, S., Vaes, S.: On the fundamental group of \({\text{ II }}_1\) factors and equivalence relations arising from group actions. In: Quanta of Maths. Clay Mathematics Proceedings, vol. 11. American Mathematical Society, Providence, pp. 519–541 (2010)

  16. 16.

    Series, C.: An application of groupoid cohomology. Pac. J. Math. 92, 415–432 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Tucker-Drob, R.: Invariant means and the structure of inner amenable groups, preprint, arXiv:1407.7474

  18. 18.

    Varadarajan, V.S.: Groups of automorphisms of Borel spaces. Trans. Am. Math. Soc. 109, 191–220 (1963)

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Yoshikata Kida.

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The author was supported by JSPS Grant-in-Aid for Scientific Research, No. 25800063.

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Kida, Y. Stable actions and central extensions. Math. Ann. 369, 705–722 (2017). https://doi.org/10.1007/s00208-017-1553-z

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