Kida, Yoshikata Splitting in orbit equivalence, treeable groups, and the Haagerup property. (English) Zbl 1446.20075 Fujiwara, Koji (ed.) et al., Hyperbolic geometry and geometric group theory. Proceedings of the 7th Seasonal Institute of the Mathematical Society of Japan (MSJ-SI), Tokyo, Japan, July 30 – August 5, 2014. Tokyo: Mathematical Society of Japan (MSJ). Adv. Stud. Pure Math. 73, 167-214 (2017). Summary: Let \(G\) be a discrete countable group and \(C\) its central subgroup with \(G/C\) treeable. We show that for any treeable action of \(G/C\) on a standard probability space \(X\), the groupoid \(G\ltimes X\) is isomorphic to the direct product of \(C\) and \((G/C)\ltimes X\), through cohomology of groupoids. We apply this to show that any group in the minimal class of groups containing treeable groups and closed under taking direct products, commensurable groups and central extensions has the Haagerup property.For the entire collection see [Zbl 1380.20002]. Cited in 4 Documents MSC: 20J06 Cohomology of groups 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20N02 Sets with a single binary operation (groupoids) 22D55 Kazhdan’s property (T), the Haagerup property, and generalizations 28D05 Measure-preserving transformations PDFBibTeX XMLCite \textit{Y. Kida}, Adv. Stud. Pure Math. 73, 167--214 (2017; Zbl 1446.20075) Full Text: DOI arXiv Euclid References: [1] M. Caspers, R. Okayasu, A. Skalski, and R. Tomatsu, Generalisations of the Haagerup approximation property to arbitrary von Neumann algebras, C. R. Acad. Sci. Paris352(2014), 507-510. · Zbl 1311.46055 [2] P.-A. Cherix, M. Cowling, P. Jolissaint, P. Julg, and A. Valette,Groups with the Haagerup property. Gromov’s a-T-menability, Progr. Math., 197, Birkh¨auser Verlag, Basel, 2001. · Zbl 1030.43002 [3] J. Feldman and C. C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I,Trans. Amer. Math. Soc.234(1977), 289- 324. · Zbl 0369.22009 [4] D. Fisher, D. W. Morris, and K. Whyte, Nonergodic actions, cocycles and superrigidity,New York J. Math.10(2004), 249-269. · Zbl 1074.37001 [5] D. Gaboriau, Coˆut des relations d’´equivalence et des groupes,Invent. Math. 139(2000), 41-98. · Zbl 0939.28012 [6] D. Gaboriau, Examples of groups that are measure equivalent to the free group,Ergodic Theory Dynam. Systems25(2005), 1809-1827. · Zbl 1130.37311 [7] U. Haagerup, An example of a nonnuclearC∗-algebra, which has the metric approximation property,Invent. Math.50(1979), 279-293. · Zbl 0408.46046 [8] G. Hjorth, A lemma for cost attained,Ann. Pure Appl. Logic143(2006), 87-102. · Zbl 1101.37004 [9] P. Jolissaint, Haagerup approximation property for finite von Neumann algebras,J. Operator Theory48(2002), 549-571. · Zbl 1029.46091 [10] P. Jolissaint, The Haagerup property for measure-preserving standard equivalence relations,Ergodic Theory Dynam. Systems25(2005), 161- 174. · Zbl 1070.37002 [11] A. S. Kechris,Classical descriptive set theory, Grad. Texts in Math., 156, Springer-Verlag, New York, 1995. [12] Y. Kida, Stable actions of central extensions and relative property (T), Israel J. Math.207(2015), 925-959. · Zbl 1373.37007 [13] R. Okayasu and R. Tomatsu, Haagerup approximation property for arbitrary von Neumann algebras,Publ. Res. Inst. Math. Sci.51(2015), 567-603. · Zbl 1335.46052 [14] S. Popa, On a class of type II1factors with Betti numbers invariants,Ann. of Math. (2)163(2006), 809-899. · Zbl 1120.46045 [15] A. Ramsay, Virtual groups and group actions,Adv. Math.6(1971), 253- 322. · Zbl 0216.14902 [16] C. Series, An application of groupoid cohomology,Pacific J. Math.92 (1981), 415-432. · Zbl 0471.28016 [17] Y. Ueda, Notes on treeability and costs for discrete groupoids in operator algebra framework, inOperator Algebras: The Abel Symposium 2004, 259-279, Abel Symp., 1, Springer, Berlin, 2006. [18] J. J. Westman, Cohomology for ergodic groupoids,Trans. Amer. Math. Soc.146(1969), 465-471. · Zbl 0271.18013 [19] J. J. Westman, Cohomology for the ergodic actions of countable groups, Proc. Amer. Math. Soc.30(1971), 318-320. · Zbl 0229.28012 [20] D. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.