Hostname: page-component-74656f9884-54qdv Total loading time: 0 Render date: 2026-11-10T20:44:48.077Z Has data issue: false hasContentIssue false
  • English
  • Français

Hyperlinear and Sofic Groups: A Brief Guide

Published online by Cambridge University Press:  15 January 2014

Vladimir G. Pestov
Vladimir G. Pestov*
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave. Ottawa, Ontario K1N 6N5, Canada E-mail: vpest283@uottawa.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

This is an introductory survey of the emerging theory of two new classes of (discrete, countable) groups, called hyperlinear and sofic groups. They can be characterized as subgroups of metric ultraproducts of families of, respectively, unitary groups U(n) and symmetric groups Sn , n ∈ ℕ. Hyperlinear groups come from theory of operator algebras (Connes' Embedding Problem), while sofic groups, introduced by Gromov, are motivated by a problem of symbolic dynamics (Gottschalk's Surjunctivity Conjecture). Open questions are numerous, in particular it is still unknown if every group is hyperlinear and/or sofic.

Information

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 14 , Issue 4 , December 2008 , pp. 449 - 480
Copyright
Copyright © Association for Symbolic Logic 2008

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.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1] Adams, S., Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups, Topology, vol. 33 (1994), pp. 765–783.Google Scholar
[2] Adian, S. I., Random walks on free periodic groups, Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya, vol. 46 (1982), pp. 1139–1149.Google Scholar
[3] Aldous, D. and Lyons, R., Processes on unimodular random networks, Electronic Journal of Probability, vol. 12 (2007), pp. 1454–1508.CrossRefGoogle Scholar
[4] Anantharaman-Delaroche, C., Amenability and exactness for dynamical systems and their C*-algebras, Transactions of the American Mathematical Society, vol. 354 (2002), pp. 4153–4178.CrossRefGoogle Scholar
[5] Anantharaman-Delaroche, C. and Renault, J., Amenable groupoids, with a foreword by Skandalis, Georges and Appendix B by Germain, E., Monographies de L'Enseignement Mathematique, vol. 36, L'Enseignement Mathématique, Geneva, 2000.Google Scholar
[6] Bekka, M. B., de la Harpe, P. de, and Valette, A., Kazhdan's property (T), New Mathematical Monographs, vol. 11, Cambridge University Press, 2008.Google Scholar
[7] Bekka, M. E. B., Amenable unitary representations of locally compact groups, Inventiones Mathematicae, vol. 100 (1990), pp. 383–401.CrossRefGoogle Scholar
[8] Bell, J. L. and Slomson, A. B., Models and ultraproducts: An introduction, North-Holland, Amsterdam–London, 1971, 2nd revised printing.Google Scholar
[9] Yaacov, I. Ben, Berenstein, A., Henson, C. W., and Usvyatsov, A., Model theory for metric structures, 112 pages, to be published in a Newton Institute volume in the Lecture Notes series of the London Mathematical Society, current version available from http://www.math.uiuc.edu/%7Ehenson/cfo/mtfms.pdf.Google Scholar
[10] Bergman, G. M., Generating infinite symmetric groups, Bulletin of the London Mathematical Society, vol. 38 (2006), pp. 429–440.Google Scholar
[11] Blattner, R. J., Automorphic group representations, Pacific Journal of Mathematics, vol. 8 (1958), pp. 665–677.Google Scholar
[12] Bowen, L., Isomorphism invariants for actions of sofic groups, arXiv:0804.3582v1 [math.DS], 22 pages.Google Scholar
[13] Brown, N. P. and Ozawa, N., C* -algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, R.I., 2008.Google Scholar
[14] Burger, M. and Mozes, S., Finitely presented simple groups and products of trees, Comptes Rendus Mathématique. Académie des Sciences. Paris Series I, vol. 324 (1997), pp. 747–752.CrossRefGoogle Scholar
[15] Ceccherini-Silberstein, T. and Coornaert, M., Injective linear cellular automata and sofic groups, Israel Journal of Mathematics, vol. 161 (2007), pp. 1–15.CrossRefGoogle Scholar
[16] Cherix, P.-A., Cowling, M., Jolissaint, P., Julg, P., and Valette, A., Groups with the Haagerup property. Gromov's a-T-menability, Progress in Mathematics, vol. 197, Birkhäuser Verlag, Basel, 2001.Google Scholar
[17] Connes, A., Classification of injective factors, Annals of Mathematics, vol. 104 (1976), pp. 73–115.CrossRefGoogle Scholar
[18] Dacunha-Castelle, D. and Krivine, J.-L., Applications des ultraproduits à l'étude des espaces et des algèbres de Banach, Studia Mathematicae, vol. 41 (1972), pp. 315–334.Google Scholar
[19] Day, M. M., Amenable semigroups, Illinois Journal of Mathematics, vol. 1 (1957), pp. 509–544.Google Scholar
[20] de la Harpe, P. de, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000.Google Scholar
[21] de la Harpe, P. de and Plymen, R. J., Automorphic group representations: a new proof of Blattner's theorem, Journal of the London Mathematical Society, vol. 19 (1979), no. 2, pp. 509–522.CrossRefGoogle Scholar
[22] de la Harpe, P. de and Valette, A., La propriété (T) de Kazhdan pour les groupes localements compacts , Astérisque, vol. 175 (1989), 158 pp.Google Scholar
[23] Dye, H. A., On groups of measure preserving transformations, I, II, American Journal of Mathematics, vol. 81 (1959), pp. 119–159; vol. 85 (1963), pp. 551-576.Google Scholar
[24] Edjvetand, M. and Juhász, A., On equations over groups, International Journal of Algebra and Computation, vol. 4 (1994), pp. 451–468.Google Scholar
[25] Elek, G. and Szabó, E., Sofic groups and direct finiteness, Journal of Algebra, vol. 280 (2004), pp. 426–434.Google Scholar
[26] Elek, G. and Szabó, E., Hyperlinearity, essentially free actions and L2-invariants. The sofic property, Mathematische Annalen, vol. 332 (2005), no. 2, pp. 421–441.CrossRefGoogle Scholar
[27] Elek, G. and Szabó, E., On sofic groups, Journal of Group Theory, vol. 9 (2006), pp. 161–171.Google Scholar
[28] Farah, I., The commutant of B(H) in its ultrapower, abstract of talk at the Second Canada–France Congress (1-5 June 2008, UQAM, Montréal), http://www.cms.math.ca/Events/summer08/abs/set.html#if.Google Scholar
[29] Farah, I., Incompleteness, independence, and absoluteness or: When to call a set theorist?., Fields Institute Workshop around Connes' Embedding Problem (University of Ottawa), May 16-18 2008, invited lecture, http://www.fields.utoronto.ca/programs/scientific/07-08/embedding/abstracts.html.Google Scholar
[30] Freiling, C., Axioms of symmetry: throwing darts at the real number line, The Journal of Symbolic Logic, vol. 51 (1986), pp. 190–200.Google Scholar
[31] Ge, L. and Hadwin, D., Ultraproducts of C*-algebras, Recent advances in operator theory and related topics ( Szeged , 1999), Operator Theory: Advances and Applications, vol. 127, Birkhäuser, Basel, 2001, pp. 305–326.Google Scholar
[32] Gerstenhaber, M. and Rothaus, O. S., The solution of sets of equations in groups, Proceedings of the National Academy of Sciences of the United States of America, vol. 48 (1962), pp. 1531–1533.CrossRefGoogle ScholarPubMed
[33] Glebsky, L. and Martínez, L. M. Rivera, Sofic groups and profinite topology on free groups, arXiv:math0709.0026, Feb. 2008, 5 pp.Google Scholar
[34] Glebsky, L. Yu. and Gordon, E. I., On surjunctivity of the transition functions of cellular automata on groups, Taiwanese Journal of Mathematics, vol. 9 (2005), pp. 511–520.CrossRefGoogle Scholar
[35] Gottschalk, W., Some general dynamical notions, Recent advances in topological dynamics, Lecture Notes in Mathematics, vol. 318, Springer, Berlin, 1973, pp. 120–125.CrossRefGoogle Scholar
[36] Greenleaf, F. P., Invariant means on topological groups, Van Nostrand Mathematical Studies, vol. 16, Van Nostrand–Reinhold Co., NY–Toronto–London–Melbourne, 1969.Google Scholar
[37] Gromov, M., Hyperbolic groups, Essays in group theory (Gersten, S. M., editor), MSRI Publications, vol. 8, Springer, Berlin, 1987, pp. 75–263.CrossRefGoogle Scholar
[38] Gromov, M., Endomorphisms of symbolic algebraic varieties, Journal of European Mathematical Society, vol. 1 (1999), no. 2, pp. 109–197.Google Scholar
[39] Gromov, M., Random walk in random groups, Geometric and Functional Analysis, vol. 13 (2003), pp. 73–146.Google Scholar
[40] Guentner, E., Exactness of the one relator groups, Proceedings of the American Mathematical Society, vol. 130 (2002), pp. 1087–1093.Google Scholar
[41] Henson, C. W. and Iovino, J., Ultraproducts in analysis, Analysis and logic ( Mons , 1997), London Mathematical Society Lecture Note Series, vol. 262, Cambridge University Press, Cambridge, 2002, pp. 1–110.Google Scholar
[42] Herwig, B. and Lascar, D., Extending partial automorphisms and the profinite topology on free groups, Transactions of the American Mathematical Society, vol. 352 (2000), pp. 1985–2021.Google Scholar
[43] Hewitt, E., Rings of real-valued continuous functions. I, Transactions of the American Mathematical Society, vol. 64 (1948), pp. 45–99.CrossRefGoogle Scholar
[44] Higson, N. and Roe, J., Amenable group actions and the Novikov conjecture, Journal fur die Reine und Angewandte Mathematik, vol. 519 (2000), pp. 143–153.Google Scholar
[45] Hjorth, G., An oscillation theorem for groups of isometries, Geometric and Functional Analysis, vol. 18 (2008), no. 2, pp. 489–521.Google Scholar
[46] Hjorth, G. and Molberg, M., Free continuous actions on zero-dimensional spaces, Topology and its Applications, vol. 153 (2006), pp. 1116–1131.CrossRefGoogle Scholar
[47] Janssen, G., Restricted ultraproducts of finite von Neumann algebras, Contributions to non-standard analysis (Sympos., Oberwolfach, 1970), Studies in Logic and the Foundations of Mathematics, vol. 69, North-Holland, Amsterdam, 1972, pp. 101–114.CrossRefGoogle Scholar
[48] Jones, V. F. R., Von Neumann algebras, lecture notes, version of 13th May 2003, 121 pp., available from http://www.imsc.res.in/~vpgupta/hnotes/notes-jones.pdf.Google Scholar
[49] Kanamori, A., The higher infinite: Large cardinals in set theory from their beginnings, Perspectives in Mathematical Logic, Springer, Berlin, 1995.Google Scholar
[50] Kechris, A. S. and Miller, B. D., Topics in orbit equivalence, Lecture Notes in Mathematics, 1852, Springer, Berlin, 2004.Google Scholar
[51] Kechris, A. S. and Rosendal, C., Turbulence, amalgamation, and generic automorphisms of homogeneous structures, Proceedings of the London Mathematical Society, vol. 94 (2007), no. 3, pp. 302–350.Google Scholar
[52] Kirchberg, E., On nonsemisplit extensions, tensor products and exactness of group C*-algebras, Inventiones Mathematicae, vol. 112 (1993), pp. 449–489.Google Scholar
[53] Kirchberg, E., Discrete groups with Kazhdan's property T and factorization property are residually finite , Mathematische Annalen, vol. 299 (1994), pp. 551–563.CrossRefGoogle Scholar
[54] Kirchberg, E., Central sequences in C*-algebras and strongly purely infinite algebras, Operator algebras: The Abel symposium 2004, Abel Symposium, vol. 1, Springer, Berlin, 2006, pp. 175–231.Google Scholar
[55] Kropholler, P., Baumslag–Solitar groups and some other groups of cohomological dimension two, Commentarii Mathematici Helvetici, vol. 65 (1990), pp. 547–558.Google Scholar
[56] Łoś, J., Quelques remarques, théorèmes et problèmes sur les classes définissables d'algèbres, Mathematical interpretation of formal systems, North-Holland Publishing Co., Amsterdam, 1955, pp. 98–113.CrossRefGoogle Scholar
[57] Luxemburg, W. A. J., A general theory of monads, Applications of model theory to algebra, analysis, and probability, Holt, Rinehart and Winston, New York, 1969, pp. 18–86.Google Scholar
[58] Magnus, W., Karrass, A., and Solitar, D., Combinatorial group theory, Dover Publications, Mineola, NY, 2004, reprint of the 1976 second edition.Google Scholar
[59] Malcev, A. I., On isomorphic matrix representations of infinite groups, Matematicheskii Sbornik (N.S.), vol. 8 (1940), no. 50, pp. 405–422.Google Scholar
[60] McDuff, D., Central sequences and the hyperfinite factor, Proceedings of the London Mathematical Society, vol. 21 (1970), no. 3, pp. 443–461.Google Scholar
[61] Van Thé, L. Nguyen, Structural Ramsey theory of metric spaces and topological dynamics of isometry groups, Memoirs of the American Mathematical Society, to appear.Google Scholar
[62] Ol'shanskiĭ, A. Yu., Geometry of defining relations in groups, Mathematics and its Applications (Soviet Series), vol. 70, Kluwer, Dordrecht, 1991.Google Scholar
[63] Ozawa, N., Amenable actions and exactness for discrete groups, Comptes Rendus de l'Académie des Sciences. Série I. Mathématiques. Paris, vol. 324 (1997), pp. 747–752.Google Scholar
[64] Ozawa, N., About the QWEP conjecture, International Journal of Mathematics, vol. 15 (2004), pp. 501–530.Google Scholar
[65] Paterson, A. T., Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988.CrossRefGoogle Scholar
[66] Pedersen, G. K., C*-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc., London-New York, 1979.Google Scholar
[67] Pisier, G., Introduction to operator space theory, London Mathematical Society Lecture Note Series, vol. 294, Cambridge University Press, Cambridge, 2003.CrossRefGoogle Scholar
[68] Radulescu, F., The von Neumann algebra of the non-residually finite Baumslag group 〈a, b ∣ ab3a−1 = b2〉 embeds into Rω , arXiv:math/0004172v3, 2000, 16 pp.Google Scholar
[69] Robinson, A., Non-standard analysis, Studies in Logic and Foundations of Mathematics, North-Holland, Amsterdam–London, 1966, (second printing, 1970).Google Scholar
[70] Rosendal, C. and Solecki, S., Automatic continuity of homomorphisms and fixed points on metric compacta, Israel Journal of Mathematics, vol. 162 (2007), pp. 349–371.CrossRefGoogle Scholar
[71] Sakai, S., C*-algebras and W*-algebras, Springer, Berlin, 1971; reprinted, Springer, 1998.Google Scholar
[72] Sanov, I. N., A property of a representation of a free group, Rossiĭskaya Akademiya Nauk. Doklady Akademii Nauk, vol. 57 (1947), pp. 657–659, in Russian.Google Scholar
[73] Thom, A., Sofic groups and diophantine approximation, arXiv:math/0701294v3 [math.FA], 15 pages, Jan. 2007.Google Scholar
[74] Valette, A., Amenable representations and finite injective von Neumann algebras, Proceedings of the American Mathematical Society, vol. 125 (1997), pp. 1841–1843.CrossRefGoogle Scholar
[75] Vershik, A. M. and Gordon, E. I., Groups that are locally embeddable in the class of finite groups, St. Petersburg Mathematical Journal, vol. 9 (1998), pp. 49–67.Google Scholar
[76] Wagon, S., The Banach–Tarski paradox, Cambridge University Press, 1985.Google Scholar
[77] Wassermann, S., On tensor products of certain group C*-algebras, Journal of Functional Analysis, vol. 23 (1976), pp. 239–254.CrossRefGoogle Scholar
[78] Weaver, N., Set theory and C*-algebras, this Bulletin, vol. 13 (2007), pp. 1–20.Google Scholar
[79] Weiss, B., Sofic groups and dynamical systems, Sankhyā Series A, vol. 62 (2000), no. 3, pp. 350–359, available at: http://202.54.54.147/search/62a3/eh06fnl.pdf.Google Scholar
[80] Zimmer, R. J., Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984.Google Scholar