## Stable decompositions and rigidity for products of countable equivalence relations

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- by Pieter Spaas;
- Trans. Amer. Math. Soc.
**376**(2023), 1867-1894 - DOI: https://doi.org/10.1090/tran/8800
- Published electronically: December 15, 2022
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## Abstract:

We show that the “stabilization” of any countable ergodic probability measure preserving (p.m.p.) equivalence relation which is not Schmidt, i.e. admits no central sequences in its full group, always gives rise to a stable equivalence relation with a unique stable decomposition, providing the first non-strongly ergodic such examples. In the proof, we moreover establish a new local characterization of the Schmidt property. We also prove some new structural results for product equivalence relations and orbit equivalence relations of diagonal product actions.## References

- Lewis Bowen, Daniel Hoff, and Adrian Ioana,
*von Neumann’s problem and extensions of non-amenable equivalence relations*, Groups Geom. Dyn.**12**(2018), no. 2, 399–448. MR**3813200**, DOI 10.4171/GGD/456 - Pierre-Alain Cherix, Michael Cowling, Paul Jolissaint, Pierre Julg, and Alain Valette,
*Groups with the Haagerup property*, Progress in Mathematics, vol. 197, Birkhäuser Verlag, Basel, 2001. Gromov’s a-T-menability. MR**1852148**, DOI 10.1007/978-3-0348-8237-8 - Marie Choda,
*Inner amenability and fullness*, Proc. Amer. Math. Soc.**86**(1982), no. 4, 663–666. MR**674101**, DOI 10.1090/S0002-9939-1982-0674101-6 - A. Connes,
*Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not =1$*, Ann. of Math. (2)**104**(1976), no. 1, 73–115. MR**454659**, DOI 10.2307/1971057 - A. Connes, J. Feldman, and B. Weiss,
*An amenable equivalence relation is generated by a single transformation*, Ergodic Theory Dynam. Systems**1**(1981), no. 4, 431–450 (1982). MR**662736**, DOI 10.1017/s014338570000136x - Daniel Drimbe, Daniel Hoff, and Adrian Ioana,
*Prime $\rm II_1$ factors arising from irreducible lattices in products of rank one simple Lie groups*, J. Reine Angew. Math.**757**(2019), 197–246. MR**4036574**, DOI 10.1515/crelle-2017-0039 - Daniel Drimbe,
*Solid ergodicity and orbit equivalence rigidity for coinduced actions*, Int. Math. Res. Not. IMRN**11**(2022), 8251–8279. MR**4425836**, DOI 10.1093/imrn/rnaa325 - D. Drimbe,
*Solid ergodicity and orbit equivalence rigidity for coinduced actions*, preprint arXiv:2003.03708v1, 2020. - H. A. Dye,
*On groups of measure preserving transformations. II*, Amer. J. Math.**85**(1963), 551–576. MR**158048**, DOI 10.2307/2373108 - I. Epstein,
*Orbit inequivalent actions of non-amenable groups*, Preprint, arXiv:0707.4215, 2007. - Pierre Eymard,
*Moyennes invariantes et représentations unitaires*, Lecture Notes in Mathematics, Vol. 300, Springer-Verlag, Berlin-New York, 1972 (French). MR**447969**, DOI 10.1007/BFb0060750 - Jacob Feldman and Calvin C. Moore,
*Ergodic equivalence relations, cohomology, and von Neumann algebras. II*, Trans. Amer. Math. Soc.**234**(1977), no. 2, 325–359. MR**578730**, DOI 10.1090/S0002-9947-1977-0578730-2 - Eusebio Gardella and Martino Lupini,
*On the classification problem of free ergodic actions of nonamenable groups*, C. R. Math. Acad. Sci. Paris**355**(2017), no. 10, 1037–1040 (English, with English and French summaries). MR**3716478**, DOI 10.1016/j.crma.2017.10.004 - Adrian Ioana,
*Non-orbit equivalent actions of $\Bbb F_n$*, Ann. Sci. Éc. Norm. Supér. (4)**42**(2009), no. 4, 675–696 (English, with English and French summaries). MR**2568879**, DOI 10.24033/asens.2106 - Adrian Ioana,
*Orbit inequivalent actions for groups containing a copy of $\Bbb F_2$*, Invent. Math.**185**(2011), no. 1, 55–73. MR**2810796**, DOI 10.1007/s00222-010-0301-8 - Adrian Ioana,
*Uniqueness of the group measure space decomposition for Popa’s $\scr {HT}$ factors*, Geom. Funct. Anal.**22**(2012), no. 3, 699–732. MR**2972606**, DOI 10.1007/s00039-012-0178-3 - Adrian Ioana, Alexander S. Kechris, and Todor Tsankov,
*Subequivalence relations and positive-definite functions*, Groups Geom. Dyn.**3**(2009), no. 4, 579–625. MR**2529949**, DOI 10.4171/GGD/71 - Adrian Ioana and Pieter Spaas,
*A class of $\textrm {II}_1$ factors with a unique McDuff decomposition*, Math. Ann.**375**(2019), no. 1-2, 177–212. MR**4000239**, DOI 10.1007/s00208-019-01862-z - Vaughan F. R. Jones and Klaus Schmidt,
*Asymptotically invariant sequences and approximate finiteness*, Amer. J. Math.**109**(1987), no. 1, 91–114. MR**878200**, DOI 10.2307/2374553 - Richard V. Kadison and John R. Ringrose,
*Fundamentals of the theory of operator algebras. Vol. I*, Graduate Studies in Mathematics, vol. 15, American Mathematical Society, Providence, RI, 1997. Elementary theory; Reprint of the 1983 original. MR**1468229**, DOI 10.1090/gsm/015 - Alexander S. Kechris,
*Global aspects of ergodic group actions*, Mathematical Surveys and Monographs, vol. 160, American Mathematical Society, Providence, RI, 2010. MR**2583950**, DOI 10.1090/surv/160 - Yoshikata Kida,
*Inner amenable groups having no stable action*, Geom. Dedicata**173**(2014), 185–192. MR**3275298**, DOI 10.1007/s10711-013-9936-0 - Yoshikata Kida,
*Stable actions and central extensions*, Math. Ann.**369**(2017), no. 1-2, 705–722. MR**3694658**, DOI 10.1007/s00208-017-1553-z - Yoshikata Kida and Robin Tucker-Drob,
*Inner amenable groupoids and central sequences*, Forum Math. Sigma**8**(2020), Paper No. e29, 84. MR**4108920**, DOI 10.1017/fms.2020.15 - Amine Marrakchi,
*Stability of products of equivalence relations*, Compos. Math.**154**(2018), no. 9, 2005–2019. MR**3867291**, DOI 10.1112/s0010437x18007388 - Narutaka Ozawa and Sorin Popa,
*Some prime factorization results for type $\textrm {II}_1$ factors*, Invent. Math.**156**(2004), no. 2, 223–234. MR**2052608**, DOI 10.1007/s00222-003-0338-z - Sorin Popa,
*Strong rigidity of $\rm II_1$ factors arising from malleable actions of $w$-rigid groups. I*, Invent. Math.**165**(2006), no. 2, 369–408. MR**2231961**, DOI 10.1007/s00222-006-0501-4 - Sorin Popa,
*On Ozawa’s property for free group factors*, Int. Math. Res. Not. IMRN**11**(2007), Art. ID rnm036, 10. MR**2344271**, DOI 10.1093/imrn/rnm036 - Sorin Popa and Stefaan Vaes,
*Unique Cartan decomposition for $\rm II_1$ factors arising from arbitrary actions of free groups*, Acta Math.**212**(2014), no. 1, 141–198. MR**3179609**, DOI 10.1007/s11511-014-0110-9 - Román Sasyk,
*A note on the classification of gamma factors*, Rev. Un. Mat. Argentina**57**(2016), no. 1, 1–7. MR**3519280** - Roman Sasyk and Asger Törnquist,
*The classification problem for von Neumann factors*, J. Funct. Anal.**256**(2009), no. 8, 2710–2724. MR**2503171**, DOI 10.1016/j.jfa.2008.11.010 - Klaus Schmidt,
*Asymptotically invariant sequences and an action of $\textrm {SL}(2,\,\textbf {Z})$ on the $2$-sphere*, Israel J. Math.**37**(1980), no. 3, 193–208. MR**599454**, DOI 10.1007/BF02760961 - Klaus Schmidt,
*Some solved and unsolved problems concerning orbit equivalence of countable group actions*, Proceedings of the conference on ergodic theory and related topics, II (Georgenthal, 1986) Teubner-Texte Math., vol. 94, Teubner, Leipzig, 1987, pp. 171–184. MR**931144** - I. M. Singer,
*Automorphisms of finite factors*, Amer. J. Math.**77**(1955), 117–133. MR**66567**, DOI 10.2307/2372424 - Pieter Spaas,
*Non-classification of Cartan subalgebras for a class of von Neumann algebras*, Adv. Math.**332**(2018), 510–552. MR**3810261**, DOI 10.1016/j.aim.2018.05.007 - M. Takesaki,
*Theory of operator algebras. I*, Encyclopaedia of Mathematical Sciences, vol. 124, Springer-Verlag, Berlin, 2002. Reprint of the first (1979) edition; Operator Algebras and Non-commutative Geometry, 5. MR**1873025** - Robin D. Tucker-Drob,
*Invariant means and the structure of inner amenable groups*, Duke Math. J.**169**(2020), no. 13, 2571–2628. MR**4142752**, DOI 10.1215/00127094-2019-0070 - Stefaan Vaes,
*Explicit computations of all finite index bimodules for a family of $\textrm {II}_1$ factors*, Ann. Sci. Éc. Norm. Supér. (4)**41**(2008), no. 5, 743–788 (English, with English and French summaries). MR**2504433**, DOI 10.24033/asens.2081

## Bibliographic Information

**Pieter Spaas**- Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
- MR Author ID: 1272392
- Email: pisp@math.ku.dk
- Received by editor(s): May 8, 2021
- Received by editor(s) in revised form: November 7, 2021, and August 17, 2022
- Published electronically: December 15, 2022
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**376**(2023), 1867-1894 - MSC (2020): Primary 37A20, 46L10; Secondary 03E15
- DOI: https://doi.org/10.1090/tran/8800
- MathSciNet review: 4549694