## Tohoku Mathematical Journal

### Invariants of orbit equivalence relations and Baumslag-Solitar groups

Yoshikata Kida

#### Abstract

To an ergodic, essentially free and measure-preserving action of a non-amenable Baumslag-Solitar group on a standard probability space, a flow is associated. The isomorphism class of the flow is shown to be an invariant of such actions of Baumslag-Solitar groups under weak orbit equivalence. Results on groups which are measure equivalent to Baumslag-Solitar groups are also provided.

#### Article information

Source
Tohoku Math. J. (2), Volume 66, Number 2 (2014), 205-258.

Dates
First available in Project Euclid: 9 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1404911861

Digital Object Identifier
doi:10.2748/tmj/1404911861

Mathematical Reviews number (MathSciNet)
MR3229595

Zentralblatt MATH identifier
1351.37016

#### Citation

Kida, Yoshikata. Invariants of orbit equivalence relations and Baumslag-Solitar groups. Tohoku Math. J. (2) 66 (2014), no. 2, 205--258. doi:10.2748/tmj/1404911861. https://projecteuclid.org/euclid.tmj/1404911861

#### References

• S. Adams, Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups, Topology 33 (1994), 765–783.
• S. Adams, Indecomposability of equivalence relations generated by word hyperbolic groups, Topology 33 (1994), 785–798.
• C. Anantharaman-Delaroche and J. Renault, Amenable groupoids, Monogr. Enseign. Math. 36, Enseignement Math., Geneva, 2000.
• H. Aoi, A remark on the commensurability for inclusions of ergodic measured equivalence relations, Hokkaido Math. J. 37 (2008), 545–560.
• H. Aoi and T. Yamanouchi, On the normalizing groupoids and the commensurability groupoids for inclusions of factors associated to ergodic equivalence relations-subrelations, J. Funct. Anal. 240 (2006), 297–333.
• U. Bader, A. Furman and R. Sauer, Integrable measure equivalence and rigidity of hyperbolic lattices, Invent. Math. 194 (2013), 213–379.
• G. Baumslag and D. Solitar, Some two-generator one-relator non-Hopfian groups, Bull. Amer. Math. Soc. 68 (1962), 199–201.
• M. B. Bekka and M. Mayer, Ergodic theory and topological dynamics of group actions on homogeneous spaces, London Math. Soc. Lecture Note Ser. 269, Cambridge Univ. Press, Cambridge, 2000.
• L. Bowen, Measure conjugacy invariants for actions of countable sofic groups, J. Amer. Math. Soc. 23 (2010), 217–245.
• A. Connes, J. Feldman and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynam. Systems 1 (1981), 431–450.
• H. A. Dye, On groups of measure preserving transformation. I, Amer. J. Math. 81 (1959), 119–159.
• H. A. Dye, On groups of measure preserving transformation. II, Amer. J. Math. 85 (1963), 551–576.
• J. Feldman, C. E. Sutherland and R. J. Zimmer, Subrelations of ergodic equivalence relations, Ergodic Theory Dynam. Systems 9 (1989), 239–269.
• D. Fisher, D. W. Morris and K. Whyte, Nonergodic actions, cocycles and superrigidity, New York J. Math. 10 (2004), 249–269.
• A. Furman, Gromov's measure equivalence and rigidity of higher rank lattices, Ann. of Math. (2) 150 (1999), 1059–1081.
• A. Furman, Orbit equivalence rigidity, Ann. of Math. (2) 150 (1999), 1083–1108.
• A. Furman, A survey of measured group theory, in Geometry, rigidity, and group actions, 296–374, Univ. Chicago Press, Chicago and London, 2011.
• D. Gaboriau, Orbit equivalence and measured group theory, in Proceedings of the International Congress of Mathematicians (Hyderabad, India, 2010), Vol. III, 1501–1527, Hindustan Book Agency, New Delhi, 2010.
• N. D. Gilbert, J. Howie, V. Metaftsis and E. Raptis, Tree actions of automorphism groups, J. Group Theory 3 (2000), 213–223.
• M. Gromov, Asymptotic invariants of infinite groups, in Geometric group theory, Vol. 2 (Sussex, 1991), 1–295, London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press, Cambridge, 1993.
• P. Hahn, The regular representations of measure groupoids, Trans. Amer. Math. Soc. 242 (1978), 35–72.
• T. Hamachi, Y. Oka and M. Osikawa, Flows associated with ergodic non-singular transformation groups, Publ. Res. Inst. Math. Sci. 11 (1975), 31–50.
• T. Hamachi and M. Osikawa, Ergodic groups of automorphisms and Krieger's theorems, Sem. Math. Sci., 3, Keio University, Department of Mathematics, Yokohama, 1981.
• G. Hjorth and A. S. Kechris, Rigidity theorems for actions of product groups and countable Borel equivalence relations, Mem. Amer. Math. Soc. 177 (2005), no. 833.
• V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), 1–25.
• A. S. Kechris, Classical descriptive set theory, Grad. Texts in Math. 156, Springer-Verlag, New York, 1995.
• Y. Kida, Orbit equivalence rigidity for ergodic actions of the mapping class group, Geom. Dedicata 131 (2008), 99–109.
• Y. Kida, Introduction to measurable rigidity of mapping class groups, in Handbook of Teichmüller theory, Vol. II, 297–367, IRMA Lect. Math. Theor. Phys. 13, Eur. Math. Soc., Zürich, 2009.
• Y. Kida, Rigidity of amalgamated free products in measure equivalence, J. Topol. 4 (2011), 687–735.
• Y. Kida, Examples of amalgamated free products and coupling rigidity, Ergodic Theory Dynam. Systems 33 (2013), 499–528.
• W. Krieger, On non-singular transformations of a measure space. II, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 11 (1969), 98–119.
• W. Krieger, On ergodic flows and the isomorphism of factors, Math. Ann. 223 (1976), 19–70.
• G. Levitt, On the automorphism group of generalized Baumslag-Solitar groups, Geom. Topol. 11 (2007), 473–515.
• R. C. Lyndon and P. E. Schupp, Combinatorial group theory, Classics Math., Springer-Verlag, Berlin, 2001.
• G. W. Mackey, Ergodic theory and virtual groups, Math. Ann. 166 (1966), 187–207.
• D. I. Moldavanskiǐ, On the isomorphisms of Baumslag-Solitar groups, Ukrainian Math. J. 43 (1991), 1569–1571.
• N. Monod and Y. Shalom, Orbit equivalence rigidity and bounded cohomology, Ann. of Math. (2) 164 (2006), 825–878.
• D. S. Ornstein and B. Weiss, Ergodic theory of amenable group actions. I, The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.) 2 (1980), 161–164.
• V. G. Pestov, Hyperlinear and sofic groups: a brief guide, Bull. Symbolic Logic 14 (2008), 449–480.
• S. Popa, Some computations of 1-cohomology groups and construction of non-orbit-equivalent actions, J. Inst. Math. Jussieu 5 (2006), 309–332.
• S. Popa, Deformation and rigidity for group actions and von Neumann algebras, in International Congress of Mathematicians. Vol. I, 445–477, Eur. Math. Soc., Zürich, 2007.
• S. Popa, On the superrigidity of malleable actions with spectral gap, J. Amer. Math. Soc. 21 (2008), 981–1000.
• A. Ramsay, Virtual groups and group actions, Adv. Math. 6 (1971), 253–322.
• V. A. Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation 1952 (1952), no. 71, 1–54.
• R. Sauer and A. Thom, A spectral sequence to compute $L^2$-Betti numbers of groups and groupoids, J. Lond. Math. Soc. (2) 81 (2010), 747–773.
• K. Schmidt, Asymptotic properties of unitary representations and mixing, Proc. Lond. Math. Soc. (3) 48 (1984), 445–460.
• J.-P. Serre, Trees, Springer Monogr. Math., Springer-Verlag, Berlin, 2003.
• Y. Shalom, Measurable group theory, in European Congress of Mathematics, 391–423, Eur. Math. Soc., Zürich, 2005.
• M. Takesaki, Theory of operator algebras. III, Encyclopaedia Math. Sci., 127. Operator Algebras and Non-commutative Geometry, 8. Springer-Verlag, Berlin, 2003.
• S. Vaes, Rigidity for von Neumann algebras and their invariants, in Proceedings of the International Congress of Mathematicians (Hyderabad, India, 2010), Vol. III, 1624–1650, Hindustan Book Agency, New Delhi, 2010.
• R. J. Zimmer, Ergodic theory and semisimple groups, Monogr. Math. 81, Birkhäuser Verlag, Basel, 1984.