Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T14:07:30.921Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak containment of measure-preserving group actions

Part of: Methods of category theory in functional analysis Measure-theoretic ergodic theory Abstract harmonic analysis Noncompact transformation groups Special aspects of infinite or finite groups Ergodic theory

Published online by Cambridge University Press:  17 April 2019

PETER J. BURTON  and
ALEXANDER S. KECHRIS
PETER J. BURTON
Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, TX78712-1202, USA email [email protected]
ALEXANDER S. KECHRIS
Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, CA91125, USA email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper concerns the study of the global structure of measure-preserving actions of countable groups on standard probability spaces. Weak containment is a hierarchical notion of complexity of such actions, motivated by an analogous concept in the theory of unitary representations. This concept gives rise to an associated notion of equivalence of actions, called weak equivalence, which is much coarser than the notion of isomorphism (conjugacy). It is well understood now that, in general, isomorphism is a very complex notion, a fact which manifests itself, for example, in the lack of any reasonable structure in the space of actions modulo isomorphism. On the other hand, the space of weak equivalence classes is quite well behaved. Another interesting fact that relates to the study of weak containment is that many important parameters associated with actions, such as the type, cost, and combinatorial parameters, turn out to be invariants of weak equivalence and in fact exhibit desirable monotonicity properties with respect to the pre-order of weak containment, a fact that can be useful in certain applications. There has been quite a lot of activity in this area in the last few years, and our goal in this paper is to provide a survey of this work.

Keywords

measure-preserving group actionsweak containmentweak equivalence

MSC classification

Primary: 37A15: General groups of measure-preserving transformations 22F10: Measurable group actions 28D15: General groups of measure-preserving transformations
Secondary: 37A20: Orbit equivalence, cocycles, ergodic equivalence relations 20F65: Geometric group theory 43A07: Means on groups, semigroups, etc.; amenable groups 46M07: Ultraproducts
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 10 , October 2020 , pp. 2681 - 2733
Check for updates
Copyright
© Cambridge University Press, 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abért, M. and Elek, G.. Dynamical properties of profinite actions. Ergod. Th. & Dynamic. Sys. 32 (2012), 1805–1835.Google Scholar
Abért, M. and Elek, G.. The space of actions, partition metric and combinatorial rigidity. Preprint, 2011, arXiv:1108.21471v1.Google Scholar
Adams, S., Elliott, G.A. and Giordano, T.. Amenable actions of groups. Trans. Amer. Math. Soc. 344(2) (1994), 803–822.Google Scholar
Abért, M., Glasner, Y. and Virág, B.. Kesten’s theorem for invariant random subgroups. Duke Math. J. 163(3) (2014), 465–488.Google Scholar
Abért, M. and Nikolov, N.. Rank gradient, cost of groups and the rank versus Heegard genus problem. J. Eur. Math. Soc. (JEMS) 14(5) (2012), 1657–1677.Google Scholar
Aaserud, A. and Popa, S.. Approximate equivalence of group actions. Ergod. Th. & Dynam. Sys. 38(4) (2018), 1201–1237.Google Scholar
Abért, M. and Weiss, B.. Bernoulli actions are weakly contained in any free action. Ergod. Th. & Dynam. Sys. 33 (2013), 323–333.Google Scholar
Bader, U., Duchesne, B. and Lécureux, J.. Amenable invariant normal subgroups. Israel J. Math. 213(1) (2016), 399–422.CrossRefGoogle Scholar
Bekka, B., de la Harpe, P. and Valette, A.. Kazhdan’s Property (T). Cambridge University Press, Cambridge, 2008.CrossRefGoogle Scholar
Ben Yaacov, I., Berenstein, A., Henson, C. W. and Usvyatsov, A.. Model theory for metric structures. Model Theory with Applications to Algebra and Analysis. Vol. 2 (London Math. Soc. Lecture Note Series, 350) . Cambridge University Press, Cambridge, 2008, pp. 315–427.CrossRefGoogle Scholar
Bernshteyn, A.. Multiplication of weak equivalence classes may be discontinuous. Preprint, 2018,arXiv:1803.09307v1.Google Scholar
Bernshteyn, A.. Ergodic theorems for the shift action and pointwise versions of the Abért-Weiss theorem. Preprint, 2018, arXiv:1808.00596v1.Google Scholar
Bordenave, C. and Collins, B.. Eigenvalues of random lifts and polynomial of random permutations matrices. Preprint, 2018, arXiv:1801.00876v1.Google Scholar
Bowen, L.. Periodicity and packings of the hyperbolic plane. Geom. Dedicata 102 (2004), 213–236.Google Scholar
Bowen, L.. Weak density of orbit equivalence classes of free group actions. Groups Geom. Dyn. 9(3) (2015), 811–830.Google Scholar
Bowen, L.. Measure conjugacy invariants for actions of countable sofic groups. J. Amer. Math. Soc. 23(1) (2010), 217–245.Google Scholar
Bowen, L., Grigorchuk, R. and Kravchenko, R.. Invariant random subgroups of lamplighter groups. Israel J. Math. 207(2) (2015), 763–782.Google Scholar
Bowen, L. and Tucker-Drob, R. D.. On a co-induction question of Kechris. Israel. J. Math. 194 (2013), 209–224.Google Scholar
Bowen, L. and Tucker-Drob, R. D.. The space of stable weak equivalence classes of measure-preserving actions. J. Funct. Anal. 274(11) (2018), 3170–3196.Google Scholar
Burton, P. J.. Topology and convexity in the space of actions modulo weak equivalence. Ergod. Th. & Dynam. Sys. 38(7) (2018), 2508–2536.Google Scholar
Burton, P. J.. A topological semigroup structure on the space of actions modulo weak equivalence. Preprint, 2015, arXiv:1501.04373v1.Google Scholar
Burton, P. J. and Kechris, A. S.. Invariant random subgroups and action versus representation maximality. Proc. Amer. Math. Soc. 145(9) (2017), 3961–3971.Google Scholar
Burton, P., Lupini, M. and Tamuz, O.. Weak equivalence of stationary actions and the entropy realization problem. Preprint, 2016, arXiv:1603.05013v2.Google Scholar
Carderi, A.. Ultraproducts, weak equivalence and sofic entropy. Preprint, 2015, arXiv:1509.03189v1.Google Scholar
Carderi, A., Gaboriau, D. and de la Salle, M.. Non-standard limits of graphs and some orbit equivalence invariants. Preprint, 2018, arXiv:1812.00704v2.Google Scholar
Conley, C. T. and Kechris, A. S.. Measurable chromatic and independence numbers for ergodic graphs and group actions. Groups Geom. Dyn. 7 (2013), 127–180.CrossRefGoogle Scholar
Conley, C. T., Kechris, A. S. and Tucker-Drob, R. D.. Ultraproducts of measure preserving actions and graph combinatorics. Ergod. Th. & Dynam. Sys. 33 (2013), 334–374.Google Scholar
Csóka, E. and Lippner, G.. Invariant random perfect matchings in Cayley graphs. Groups Geom. Dyn. 11(1) (2017), 211–243.Google Scholar
Creutz, D. and Peterson, J.. Stabilizers of ergodic actions of lattices and commensurators. Trans. Amer. Math. Soc. 369(6) (2017), 4119–4166.Google Scholar
Connes, A. and Weiss, B.. Property T and almost invariant sequences. Israel J. Math. 37 (1980), 209–2010.Google Scholar
Dixmier, J.. C*-algebras. North Holland, Amsterdam, 1977.Google Scholar
Dudko, A. and Grigorchuk, R.. On spectra of Koopman, groupoid and quasi-regular representations. J. Mod. Dyn. 11 (2017), 99–123.Google Scholar
Elek, G.. Finite graphs and amenability. J. Funct. Anal. 263 (2012), 2593–2614.Google Scholar
Elek, G. and Królicki, K.. Invariant subsets of the space of subgroups, equational compactness and the weak equivalence of actions. Preprint, 2016, arXiv:1608.05332.Google Scholar
Elek, G. and Lippner, G.. Sofic equivalence relations. J. Funct. Anal. 258 (2010), 1692–1708.Google Scholar
Gaboriau, D.. Coût des relations d’equivalence et des groupes. Invent. Math. 139 (2000), 41–98.Google Scholar
Glasner, E.. Ergodic Theory via Joinings. American Mathematical Society, Providence, RI, 2003.Google Scholar
Glasner, E. and King, J.. A zero-one law for dynamical properties. Contemp. Math. 215 (1998), 231–242.Google Scholar
Gromov, M.. Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. (JEMS) 2 (1999), 109–197.Google Scholar
Gaboriau, D. and Lyons, R.. A measurable-group-theoretic solution to von Neumann’s problem. Invent. Math. 177(3) (2009), 533–540.Google Scholar
Gaboriau, D. and Seward, B.. Cost, $l^{2}$ -Betti numbers and the sofic entropy of some algebraic actions. Preprint, 2015, arXiv:1509.02482.Google Scholar
Gamarnik, D. and Sudan, M.. Limits of local algorithms and sparse random graphs. Ann. Probab. 45(4) (2017), 2353–2376.Google Scholar
Glasner, E., Thouvenot, J.-P. and Weiss, B.. Every countable group has the weak Rohlin property. Bull. Lond. Math. Soc. 138(6) (2006), 932–936.Google Scholar
Glasner, E. and Weiss, B.. Kazhdan’s property T and the geometry of the collection of invariant measures. Geom. Funct. Anal. 7 (1997), 917–935.Google Scholar
Hayes, B.. Weak equivalence to Bernoulli shifts for some algebraic actions. Preprint, 2017,arXiv:1709.05372.Google Scholar
Hayes, B.. Max-min theorems for weak containment, square summable homoclinic points, and completely positive entropy. Preprint, 2019, arXiv:1902.06600.Google Scholar
Hjorth, G. and Kechris, A. S.. Rigidity theorems for actions of product groups and countable Borel equivalence relations. Mem. Amer. Math. Soc 177(833) (2005).Google Scholar
Hatami, H., Lovász, L. and Szegedy, B.. Limits of locally-globally convergent graph sequences. Geom. Funct. Anal. 24(1) (2014), 269–296.Google Scholar
Ibarlucía, T. and Tsankov, T.. A model-theoretic approach to rigidity of strongly ergodic distal actions. Preprint, 2018, arXiv:1808.00341.Google Scholar
Ioana, A. and Tucker-Drob, R.. Weak containment rigidity for distal actions. Adv. Math. 302 (2016), 309–322.Google Scholar
Jacod, J. and Protter, P.. Probability Essentials. Springer, Berlin, 2004.Google Scholar
Jones, V. and Schmidt, K.. Asymptotically invariant sequences and approximate finiteness. Amer. J. Math. 109 (1987), 91–114.Google Scholar
Kaimanovich, V. A.. Amenability, hyperfiniteness, and isoperimetric inequalities. C.R. Acad. Sci. Paris, Sér. I 235 (1997), 999–1004.Google Scholar
Kechris, A. S.. Global Aspects of Ergodic Group Actions. American Mathematical Society, Providence, RI, 2010.Google Scholar
Kechris, A. S.. Weak containment in the space of actions of a free group. Israel J. Math. 189 (2012), 461–507.CrossRefGoogle Scholar
Kechris, A. S.. Classical Descriptive Set Theory. New York, Providence, RI, 1995.Google Scholar
Kechris, A. S.. The space of measure preserving equivalence relations and graphs. Preprint, 2018 (posted at http://math.caltech.edu/∼kechris/).Google Scholar
Kechris, A. S.. Unitary representations and modular actions. J. Math. Sci. 140(3) (2007), 398–425.Google Scholar
Kechris, A. S. and Marks, A. S.. Descriptive graph combinatorics. Preprint, 2018 (posted at http://math.caltech.edu/∼kechris/).Google Scholar
Kerr, D.. Sofic measure entropy via finite partitions. Groups Geom. Dyn. 7(3) (2013), 617–632.Google Scholar
Kerr, D.. Bernoulli actions of sofic groups have completely positive entropy. Israel J. Math. 202(1) (2014), 461–474.Google Scholar
Kerr, D. and Li, H.. Ergodic Theory. Springer, Cham, 2016.Google Scholar
Kerr, D. and Pichot, M.. Asymptotic abelianess, weak mixing, and property T. J. Reine Angew. Math. 623 (2008), 213–235.Google Scholar
Kechris, A. S. and Miller, B. D.. Topics in Orbit Equivalence. Springer, Berlin, 2004.Google Scholar
Kechris, A. S. and Quorning, V.. Co-induction and invariant random subgroups. Preprint, 2019 (posted at http://math.caltech.edu/∼kechris/).Google Scholar
Kechris, A. S. and Tsankov, T.. Amenable actions and almost invariant sets. Proc. Amer. Math. Soc. 136(2) (2007), 687–697.Google Scholar
Le Maître, F.. Highly faithful actions and dense free subgroups in full groups. Groups Geom. Dyn. 12(1) (2018), 207–230.Google Scholar
Lyons, R. and Nazarov, F.. Perfect matchings as IID factors of non-amenable groups. Eur. J. Combin. 32 (2011), 1115–1125.Google Scholar
Losert, V. and Rindler, H.. Almost invariant sets. Bull. Lond. Math. Soc. 13 (1981), 145–148.Google Scholar
Lubotzky, A. and Shalom, Y.. Finite representations in the unitary dual and Ramanujan groups. Contemp. Math. 347 (2004), 173–189.Google Scholar
Lubotzky, A. and Zuk, Y.. On property $(\unicode[STIX]{x1D70F})$ . Preprint, 2003 (posted at http://www.ma.huji.ac.il/∼alexlub/).Google Scholar
Popa, S.. Some computations of 1-cohomology groups and construction of non-orbit equivalent actions. J. Inst. Math. Jussieu 15(2) (2007), 309–332.Google Scholar
Pestov, V. G. and Uspenskij, V. V.. Representations of residually finite groups by isometries of the Urysohn space. J. Ramanujan Math. Soc. 21 (2006), 189–203.Google Scholar
Schmidt, K.. Amenability, Kazhdan’s property T, strong ergodicity and invariant means for ergodic group actions. Ergod. Th. & Dynam. Sys. 1 (1981), 223–236.Google Scholar
Seward, B.. Weak containment and sofic entropy. Preprint, 2016, arXiv:1602.06680.Google Scholar
Tucker-Drob, R. D.. Shift-minimal groups, fixed price 1, and the unique trace property. Preprint, 2012, arXiv:1211.6395v3.Google Scholar
Tucker-Drob, R. D.. Weak equivalence and non-classifiability of measure preserving actions. Ergod. Th. & Dynam. Sys. 35 (2015), 293–336.CrossRefGoogle Scholar
Zimmer, R. J.. On the von Neumann algebra of an ergodic group action. Proc. Amer. Math. Soc. 41 (2015), 23–31.Google Scholar
Zimmer, R. J.. Ergodic Theory and Semisimple Groups. Birkhäuser, Boston, 1984.Google Scholar