Abstract.
We prove that Connes’ Embedding Conjecture holds for the von Neumann algebras of sofic groups, that is sofic groups are hyperlinear. Hence we provide some new examples of hyperlinearity. We also show that the Determinant Conjecture holds for sofic groups as well. We introduce the notion of essentially free actions and amenable actions and study their properties.
Similar content being viewed by others
References
Bedos, E., de la Harpe, P.: Moyennabilité intérieure des groupes: définitions et exemples. Enseign. Math. (2) 32, 139–157 (1986)
Brenner, J.L.: Covering theorems for FINASIGs. VIII. Almost all conjugacy classes in A n have exponent ≤ 4. J. Austral. Math. Soc. Ser. A. 25, 210–214 (1978)
Champetier, C.: L’espace des groupes de type fini. Topology 39, 657–680 (2000)
Clair, B.: Residual amenability and the approximation of L2-invariants. Michigan Math. J. 46, 331–346 (1999)
Deuber, W.A., Simonovits, M., Sós, V.T.: A note on paradoxical metric spaces. Studia Sci. Hung. Math. 30, 17–23 (1995)
Elek, G., Szabó, E.: On sofic groups, to appear in the Journal of Group Theory (http://arXiv.org/abs/math/0305352).
Elek, G., Szabó, E.: Sofic groups and direct finiteness. Journal of Algebra 280, 426–434 (2004)
Gromov, M.: Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. 1, 109–197 (1999)
Hall, J.I.: Locally finite simple groups of finitary linear transformations. Finite and locally finite groups NATO Adv. Sci. Inst. Ser. C. Math. Phys 471, 147–188
Kirchberg, E.: On nonsemisplit extensions, tensor products and exactness of group C★-algebras. Invent. Math. 112, 449–489 (1993)
Lück, W.: Approximating L2-invariants by their finite-dimensional analogues. Geom. Funct. Analysis 4, 455–481 (1994)
Lück, W.: L2-invariants: theory and applications to geometry and K-theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. 44, Springer, Berlin Heidelberg (2002)
Ozawa, N.: About the QWEP conjecture. International Journal of Math. 15, 501–530 (2004)
Paterson, A.L.T.: Amenability. Mathematical Surveys and Monographs 29 American Mathematical Society, Providence (1988)
Radulescu, F.: The von Neumann algebra of the non-residually finite Baumslag group < a,b | a b3 a-1 = b2 > embeds into Rω, preprint 2002 (http://www.arxiv.org/abs/math.OA/0004172).
Schick, T.: L2-determinant class and approximation of L2-Betti numbers. Trans. Amer. Math. Soc. 353, 3247–3265 (2001)
Schick, T.: Integrality of L2-Betti numbers. Math. Ann. 317, 727–750 (2000)
Vershik, A.M., Gordon, E.I.: Groups that are locally embeddable in the class of finite groups. Algebra i Analiz 9, 71–97 (1997)
Weiss, B.: Sofic groups and dynamical systems, (Ergodic theory and harmonic analysis, Mumbai, 1999) Sankhya Ser. A. 62, 350–359 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000): 43A07, 55N25
Rights and permissions
About this article
Cite this article
Elek, G., Szabó, E. Hyperlinearity, essentially free actions and L2-invariants. The sofic property. Math. Ann. 332, 421–441 (2005). https://doi.org/10.1007/s00208-005-0640-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-005-0640-8