Abstract
We give a brief survey of some classification results on orbit equivalence of probability measure preserving actions of countable groups. The notion of ℓ2 Betti numbers for groups is gently introduced. An account of orbit equivalence invariance for ℓ2 Betti numbers is presented together with a description of the theory of equivalence relation actions on simplicial complexes. We relate orbit equivalence to a measure theoretic analogue of commensurability and quasi-isometry of groups: measure equivalence. Rather than a complete description of these subjects, a lot of examples are provided.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
S. Adams. Trees and amenable equivalence relations. Ergodic Theory Dynam. Systems, 10(1):1–14, 1990.
S. Adams and R. Spatzier. Kazhdan groups, cocycles and trees. Amer. J. Math., 112(2):271–287, 1990.
M. Atiyah. Elliptic operators, discrete groups and von Neumann algebras. In Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974), pages 43-72. Astérisque, No. 32-33. Soc. Math. Prance, Paris, 1976.
A. Connes, J. Feldman, and B. Weiss. An amenable equivalence relation is generated by a single transformation. Ergodic Theory Dynam. Systems, 1(4):431–450 (1982), 1981.
J. Cheeger and M. Gromov. L 2-cohomology and group cohomology. Topology, 25(2):189–215, 1986.
A. Connes. A survey of foliations and operator algebras. In Operator algebras and applications, Part I (Kingston, Ont., 1980), pages 521-628. Amer. Math. Soc, Providence, R.I., 1982.
H. Dye. On groups of measure preserving transformation. I. Amer. J. Math., 81:119–159, 1959.
H. Dye. On groups of measure preserving transformations. II. Amer. J. Math., 85:551–576, 1963.
B. Eckmann. Introduction to l 2-methods in topology: reduced l 2 -homology, harmonic chains, l 2-Betti numbers. Israel J. Math., 117:183–219, 2000. Notes prepared by Guido Mislin.
R. Feres. An introduction to Cocycle Super-Rigidity. In Rigidity in Dynamics and Geometry (Cambridge, UK, 2000). Springer-Verlag, 2002.
J. Feldman and C. Moore. Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Trans. Amer. Math. Soc., 234(2):289–324, 1977.
J. Feldman, C. Moore. Ergodic equivalence relations, cohomology, and von Neumann algebras. II. Trans. Amer. Math. Soc., 234(2):325–359, 1977.
A. Furman. Gromov’s measure equivalence and rigidity of higher rank lattices. Ann. of Math. (2), 150(3):1059–1081, 1999.
A. Furman. Orbit equivalence rigidity. Ann. of Math. (2), 150(3):1083–1108, 1999.
D. Gaboriau. Mercuriale de groupes et de relations. C. R. Acad. Sci. Paris Sér. I Math., 326(2):219–222, 1998.
D. Gaboriau. Coût des relations d’équivalence et des groupes. Invent. Math., 139(1):41–98, 2000.
D. Gaboriau. Sur la (co-)homologie L 2 des actions préservant une mesure. C. R. Acad. Sci. Paris Sér. I Math., 330(5):365–370, 2000.
D. Gaboriau. Invariants ℓ2 de relations d’équivalence et de groupes. preprint.
S. Gefter and V. Golodets. Fundamental groups for ergodic actions and actions with unit fundamental groups. Publ. Res. Inst. Math. Sci., 24(6):821–847 (1989), 1988.
M. Gromov. Asymptotic invariants of infinite groups. In Geometrie group theory, Vol. 2 (Sussex, 1991), pages 1–295. Cambridge Univ. Press, Cambridge, 1993.
P. de la Harpe. Topics in geometric group theory. University of Chicago Press, Chicago, IL, 2000.
G. Levitt. On the cost of generating an equivalence relation. Ergodic Theory Dynam. Systems, 15(6):1173–1181, 1995.
J.-S. Li. Nonvanishing theorems for the cohomology of certain arithmetic quotients. J. Reine Angew. Math., 428:177–217, 1992.
W. Lück. L 2-torsion and 3-manifolds. In Low-dimensional topology (Knoxville, TN, 1992), pages 75–107. Internat. Press, Cambridge, MA, 1994.
W. Lück. L 2-invariants of regular coverings of compact manifolds and CW-complexes to appear in “Handbook of Geometric topology”, Elsevier, 1998.
W. Lück. Dimension theory of arbitrary modules over finite von Neumann algebras and L 2-Betti numbers. I. Foundations. J. Reine Angew. Math., 495:135–162, 1998.
W. Lück. Dimension theory of arbitrary modules over finite von Neumann algebras and L 2-Betti numbers. II. Applications to Grothendieck groups, L 2-Euler characteristics and Burnside groups. J. Reine Angew. Math., 496:213–236, 1998.
W. Lück. L 2-Invariants and K-Theory. Mathematics-Monograph, Springer-Verlag, in preparation.
C. Moore. Ergodic theory and von Neumann algebras. In Operator algebras and applications, Part 2 (Kingston, Ont., 1980), pages 179-226. Amer. Math. Soc, Providence, R.I., 1982.
F. Murray and J. von Neumann. On rings of operators. Ann. of Math., II. Ser., 37:116–229, 1936.
D. Ornstein and B. Weiss. Ergodic theory of amenable group actions. I. The Rohlin lemma. Bull. Amer. Math. Soc. (N.S.), 2(1):161–164, 1980.
K. Schmidt. Some solved and unsolved problems concerning orbit equivalence of countable group actions. In Proceedings of the conference on ergodic theory and related topics, II (Georgenthal, 1986), pages 171-184, Leipzig, 1987. Teubner.
R. Zimmer. Amenable ergodic group actions and an application to Poisson boundaries of random walks. J. Functional Analysis, 27(3):350–372, 1978.
R. Zimmer. Ergodic theory and semisimple groups. Birkhäuser Verlag, Basel, 1984.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Gaboriau, D. (2002). On Orbit Equivalence of Measure Preserving Actions. In: Burger, M., Iozzi, A. (eds) Rigidity in Dynamics and Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04743-9_8
Download citation
DOI: https://doi.org/10.1007/978-3-662-04743-9_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-07751-7
Online ISBN: 978-3-662-04743-9
eBook Packages: Springer Book Archive