Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Measure conjugacy invariants for actions of countable sofic groups

Author: Lewis Bowen
Journal: J. Amer. Math. Soc. 23 (2010), 217-245
MSC (2000): Primary 37A35
Published electronically: April 29, 2009
MathSciNet review: 2552252
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Sofic groups were defined implicitly by Gromov and explicitly by Weiss. All residually finite groups (and hence all linear groups) are sofic. The purpose of this paper is to introduce, for every countable sofic group $ G$, a family of measure-conjugacy invariants for measure-preserving $ G$-actions on probability spaces. These invariants generalize Kolmogorov-Sinai entropy for actions of amenable groups. They are computed exactly for Bernoulli shifts over $ G$, leading to a complete classification of Bernoulli systems up to measure-conjugacy for many groups, including all countable linear groups. Recent rigidity results of Y. Kida and S. Popa are utilized to classify Bernoulli shifts over mapping class groups and property (T) groups up to orbit equivalence and von Neumann equivalence, respectively.

References [Enhancements On Off] (What's this?)

  • [Bo03] L. Bowen. Periodicity and circle packings of the hyperbolic plane. Geom. Dedicata 102 (2003), 213-236. MR 2026846 (2005a:52014)
  • [Bo08a] L. Bowen. A measure-conjugacy invariant for free group actions. To appear in the Annals of Mathematics.
  • [Bo08b] L. Bowen. Weak isomorphisms between Bernoulli shifts. preprint. arXiv:0812.2718
  • [Bo09] L. Bowen. The ergodic theory of free group actions: entropy and the $ f$-invariant. preprint. arXiv:0902.0174
  • [CFW81] A. Connes, J. Feldman and B. Weiss. An amenable equivalence relation is generated by a single transformation. Ergodic Theory Dynamical Systems 1 (1981), no. 4, 431-450. MR 662736 (84h:46090)
  • [Co75] A. Connes. Sur la classification des facteurs de type $ {\rm II}$. (French) C. R. Acad. Sci. Paris Sr. A-B 281 (1975), no. 1, Aii, A13-A15. MR 0377534 (51:13706)
  • [Co76] A. Connes. Classification of injective factors. Cases $ II\sb{1},$ $ II\sb{\infty },$ $ III\sb{\lambda },$ $ \lambda \not=1$. Ann. of Math. (2) 104 (1976), no. 1, 73-115. MR 0454659 (56:12908)
  • [dHV89] P. de la Harpe, A. Valette. La propriété $ (T)$ de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger). (French) [Kazhdan's property $ (T)$ for locally compact groups (with an appendix by Marc Burger)]. With an appendix by M. Burger. Astérisque No. 175 (1989), 158 pp. MR 1023471 (90m:22001)
  • [Dy59] H. A. Dye. On groups of measure preserving transformation. I. Amer. J. Math. 81 (1959), 119-159. MR 0131516 (24:A1366)
  • [Dy63] H. A. Dye. On groups of measure preserving transformations. II. Amer. J. Math. 85 (1963), 551-576. MR 0158048 (28:1275)
  • [ES05] G. Elek and E. Szabó. Hyperlinearity, essentially free actions and $ L\sp 2$-invariants. The sofic property. Math. Ann. 332 (2005), no. 2, 421-441. MR 2178069 (2007i:43002)
  • [ES06] G. Elek and E. Szabó. On sofic groups. J. Group Theory 9 (2006), no. 2, 161-171. MR 2220572 (2007a:20037)
  • [Gl03] E. Glasner. Ergodic theory via joinings. Mathematical Surveys and Monographs, 101. American Mathematical Society, Providence, RI, 2003. xii+384 pp. MR 1958753 (2004c:37011)
  • [Gr74] E. K. Grossman. On the residual finiteness of certain mapping class groups. J. London Math. Soc. (2) 9 (1974/75), 160-164. MR 0405423 (53:9216)
  • [Gr99] M. Gromov. Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. 1 (1999), no. 2, 109-197. MR 1694588 (2000f:14003)
  • [Iv86] N. V. Ivanov. Algebraic properties of mapping class groups of surfaces. Geometric and algebraic topology, 15-35, Banach Center Publ., 18, PWN, Warsaw, 1986. MR 925854 (89a:57009)
  • [Jo91] V. F. R. Jones. von Neumann algebras in mathematics and physics. Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 121-138, Math. Soc. Japan, Tokyo, 1991. MR 1159209 (94c:46109a)
  • [Ki08] Y. Kida. Orbit equivalence rigidity for ergodic actions of the mapping class group. Geom. Dedicata 131 (2008), 99-109. MR 2369194 (2008k:37011)
  • [Ki75] J. C. Kieffer. A generalized Shannon-McMillan theorem for the action of an amenable group on a probability space. Ann. Probability 3 (1975), no. 6, 1031-1037. MR 0393422 (52:14232)
  • [Ko58] A. N. Kolmogorov. A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces. (Russian) Dokl. Akad. Nauk SSSR (N.S.) 119 (1958), 861-864. MR 0103254 (21:2035a)
  • [Ko59] A. N. Kolmogorov. Entropy per unit time as a metric invariant of automorphisms. (Russian) Dokl. Akad. Nauk SSSR 124 (1959), 754-755. MR 0103255 (21:2035b)
  • [Ma40] A. I. Mal'cev. On isomorphic matrix representations of infinite groups of matrices. Mat. Sb. 8, 405-422 (1940). Amer. Math. Soc. Transl. (2) 45, 1-18 (1965). MR 0003420 (2:216d)
  • [Ma82] G. A. Margulis. Finitely-additive invariant measures on Euclidean spaces. Ergodic Theory Dynam. Systems 2 (1982), no. 3-4, 383-396 (1983). MR 721730 (85g:28004)
  • [Mc69] D. McDuff. A countable infinity of $ \Pi \sb{1}$ factors. Ann. of Math. (2) 90 (1969), 361-371. MR 0256183 (41:840)
  • [MvN36] F. J. Murray and J. von Neumann. On rings of operators. Ann. of Math. (2) 37 (1936), no. 1, 116-229. MR 1503275
  • [MvN43] F. J. Murray and J. von Neumann. On rings of operators. IV. Ann. of Math. (2) 44, (1943). 716-808. MR 0009096 (5:101a)
  • [Ol91] A. Yu. Ol'shanskii. Geometry of defining relations in groups. Translated from the 1989 Russian original by Yu. A. Bakhturin. Mathematics and its Applications (Soviet Series), 70. Kluwer Academic Publishers Group, Dordrecht, 1991. xxvi+505 pp. MR 1191619 (93g:20071)
  • [Or70a] D. Ornstein. Bernoulli shifts with the same entropy are isomorphic. Advances in Math. 4 (1970), 337-352. MR 0257322 (41:1973)
  • [Or70b] D. Ornstein. Two Bernoulli shifts with infinite entropy are isomorphic. Advances in Math. 5 (1970), 339-348. MR 0274716 (43:478a)
  • [OW80] D. Ornstein and B. Weiss. Ergodic theory of amenable group actions. I. The Rohlin lemma. Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 1, 161-164. MR 551753 (80j:28031)
  • [OW87] D. Ornstein and B. Weiss. Entropy and isomorphism theorems for actions of amenable groups. J. Analyse Math. 48 (1987), 1-141. MR 910005 (88j:28014)
  • [Pa69] W. Parry. Entropy and generators in ergodic theory. W. A. Benjamin, Inc., New York-Amsterdam 1969 xii+124 pp. MR 0262464 (41:7071)
  • [Pe08] V. Pestov. Hyperlinear and sofic groups: a brief guide. Bull. Symbolic Logic 14 (2008), no. 4, 449-480. MR 2460675
  • [Po06] S. Popa. Strong rigidity of $ \rm II\sb 1$ factors arising from malleable actions of $ w$-rigid groups. II. Invent. Math. 165 (2006), no. 2, 409-451. MR 2231962 (2007h:46084)
  • [Po07] S. Popa. Deformation and rigidity for group actions and von Neumann algebras. International Congress of Mathematicians. Vol. I, 445-477, Eur. Math. Soc., Zürich, 2007. MR 2334200 (2008k:46186)
  • [Po08] S. Popa. On the superrigidity of malleable actions with spectral gap. J. Amer. Math. Soc. 21 (2008), no. 4, 981-1000. MR 2425177
  • [Sc63] J. Schwartz. Two finite, non-hyperfinite, non-isomorphic factors. Comm. Pure Appl. Math. 16 (1963), 19-26. MR 0149322 (26:6812)
  • [Si55] I. M. Singer. Automorphisms of finite factors. Amer. J. Math. 77 (1955), 117-133. MR 0066567 (16:597f)
  • [Si59] Ya. G. Sinaĭ. On the concept of entropy for a dynamic system. (Russian) Dokl. Akad. Nauk SSSR 124 (1959), 768-771. MR 0103256 (21:2036a)
  • [St75] A. M. Stepin. Bernoulli shifts on groups. (Russian) Dokl. Akad. Nauk SSSR 223 (1975), no. 2, 300-302. MR 0409769 (53:13521)
  • [Ti72] J. Tits. Free subgroups in linear groups. J. Algebra 20 (1972), 250-270. MR 0286898 (44:4105)
  • [We00] B. Weiss. Sofic groups and dynamical systems. Ergodic theory and Harmonic Analysis, Mumbai, 1999. Sankhya Ser. A 62, (2000) no. 3, 350-359. MR 1803462 (2001j:37022)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 37A35

Retrieve articles in all journals with MSC (2000): 37A35

Additional Information

Lewis Bowen
Affiliation: Department of Mathematics, University of Hawaii, 2565 McCarthy Mall, Keller 409, Honolulu, HI 96822

Keywords: Entropy, Ornstein's isomorphism theorem, Bernoulli shifts, measure conjugacy, orbit equivalence, von Neumann equivalence, sofic groups, group measure space construction
Received by editor(s): August 20, 2008
Published electronically: April 29, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society