Some remarks on finitarily approximable groups
Journal de l’École polytechnique — Mathématiques, Volume 5 (2018), pp. 239-258.

The concept of a 𝒞-approximable group, for a class of finite groups 𝒞, is a common generalization of the concepts of a sofic, weakly sofic, and linear sofic group. Glebsky raised the question whether all groups are approximable by finite solvable groups with arbitrary invariant length function. We answer this question by showing that any non-trivial finitely generated perfect group does not have this property, generalizing a counterexample of Howie. In a related note, we prove that any non-trivial group which can be approximated by finite groups has a non-trivial quotient that can be approximated by finite projective special linear groups. Moreover, we discuss the question which connected Lie groups can be embedded into a metric ultraproduct of finite groups with invariant length function. We prove that these are precisely the abelian ones, providing a negative answer to a question of Doucha. Referring to a problem of Zilber, we show that the identity component of a Lie group, whose topology is generated by an invariant length function and which is an abstract quotient of a product of finite groups, has to be abelian. Both of these last two facts give an alternative proof of a result of Turing. Finally, we solve a conjecture of Pillay by proving that the identity component of a compactification of a pseudofinite group must be abelian as well. All results of this article are applications of theorems on generators and commutators in finite groups by the first author and Segal. In Section 4 we also use results of Liebeck and Shalev on bounded generation in finite simple groups.

Le concept de groupe 𝒞-approximable, pour une classe de groupes finis 𝒞, généralise à la fois les notions de groupe sofique, faiblement sofique et linéairement sofique. Glebsky a soulevé la question de savoir si tous les groupes sont approximables par des groupes finis résolubles avec une fonction longueur invariante arbitraire. Nous résolvons cette question en montrant que tout groupe parfait non trivial de type fini n’a pas cette propriété, en généralisant un contre-exemple dû à Howie. Dans le même esprit, nous montrons que tout groupe non trivial qui peut être approximé par des groupes finis a un quotient non trivial qui peut être approximé par des groupes projectifs spéciaux linéaires finis. De plus, nous discutons la question de savoir quels groupes de Lie peuvent être plongés dans un ultraproduit métrique de groupes finis avec fonction longueur invariante. Nous montrons que ce sont exactement les groupes abéliens, fournissant ainsi une réponse négative à une question de Doucha. En relation avec un problème de Zilber, nous montrons que la composante neutre d’un groupe de Lie dont la topologie est engendrée par une fonction longueur invariante et qui est un quotient abstrait d’un produit de groupes finis, doit être abélienne. Ces deux derniers résultats permettent de donner une nouvelle preuve d’un résultat de Turing. Enfin, nous résolvons une conjecture de Pillay en montrant que la composante neutre d’une compactification d’un groupe pseudo-fini doit aussi être abélienne. Tous les résultats de cet article sont des applications de théorèmes, dus au premier auteur et à Segal, sur les générateurs et les commutateurs dans un groupe fini, ainsi que de résultats de Liebeck et Shalev.

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DOI: 10.5802/jep.69
Classification: 20E26,  20E18,  03C20,  20F69
Keywords: Sofic, C-approximable group, metric ultraproduct
Nikolay Nikolov 1; Jakob Schneider 2; Andreas Thom 2

1 Mathematical Institute, University of Oxford Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
2 Technische Universität Dresden, Institut für Geometrie 01062 Dresden, Germany
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Nikolay Nikolov; Jakob Schneider; Andreas Thom. Some remarks on finitarily approximable groups. Journal de l’École polytechnique — Mathématiques, Volume 5 (2018), pp. 239-258. doi : 10.5802/jep.69. https://jep.centre-mersenne.org/articles/10.5802/jep.69/

[1] G. Arzhantseva & L. Păunescu - “Linear sofic groups and algebras”, Trans. Amer. Math. Soc. 369 (2017) no. 4, p. 2285-2310 | DOI | MR | Zbl

[2] Y. Cornulier - “A sofic group away from amenable groups”, Math. Ann. 350 (2011) no. 2, p. 269-275 | DOI | MR | Zbl

[3] P. Deligne - “Extensions centrales non résiduellement finies de groupes arithmétiques”, C. R. Acad. Sci. Paris Sér. A-B 287 (1978) no. 4, p. A203-A208 | Zbl

[4] M. Doucha - “Metric topological groups: their metric approximation and metric ultraproducts”, Groups Geom. Dyn. (to appear), arXiv:1601.07449 | MR | Zbl

[5] G. Elek & E. Szabó - “Sofic groups and direct finiteness”, J. Algebra 280 (2004) no. 2, p. 426-434 | DOI | MR | Zbl

[6] G. Elek & E. Szabó - “Hyperlinearity, essentially free actions and L 2 -invariants. The sofic property”, Math. Ann. 332 (2005) no. 2, p. 421-441 | DOI | MR | Zbl

[7] T. Gelander - “Limits of finite homogeneous metric spaces”, Enseign. Math. (2) 59 (2013) no. 1-2, p. 195-206 | DOI | MR | Zbl

[8] L. Glebsky - “Approximations of groups, characterizations of sofic groups, and equations over groups”, J. Algebra 477 (2017), p. 147-162 | DOI | MR | Zbl

[9] L. Glebsky & L. M. Rivera - “Sofic groups and profinite topology on free groups”, J. Algebra 320 (2008) no. 9, p. 3512-3518 | DOI | MR | Zbl

[10] E. Goursat - “Sur les substitutions orthogonales et les divisions régulières de l’espace”, Ann. Sci. École Norm. Sup. 6 (1889), p. 9-102 | DOI | Zbl

[11] M. Gromov - “Endomorphisms of symbolic algebraic varieties”, J. Eur. Math. Soc. (JEMS) 1 (1999) no. 2, p. 109-197 | DOI | MR | Zbl

[12] D. F. Holt & S. Rees - “Some closure results for 𝒞-approximable groups” (2016), arXiv:1601.01836 | Zbl

[13] J. Howie - “The p-adic topology on a free group: A counterexample”, Math. Z. 187 (1984) no. 1, p. 25-27 | MR | Zbl

[14] A. Kar & N. Nikolov - “A non-LEA sofic group”, Proc. Indian Acad. Sci. Math. Sci. 127 (2017) no. 2, p. 289-293 | DOI | MR | Zbl

[15] A. Klyachko & A. Thom - “New topological methods to solve equations over groups”, Algebraic Geom. Topol. 17 (2017) no. 1, p. 331-353 | MR | Zbl

[16] M. W. Liebeck & A. Shalev - “Diameters of finite simple groups: sharp bounds and applications”, Ann. of Math. (2) (2001), p. 383-406 | DOI | MR | Zbl

[17] D. H. McLain - “A characteristically simple group”, Math. Proc. Cambridge Philos. Soc. 50 (1954) no. 4, p. 641-642 | DOI | MR | Zbl

[18] N. Nikolov & D. Segal - “Generators and commutators in finite groups; abstract quotients of compact groups”, Invent. Math. 190 (2012) no. 3, p. 513-602 | DOI | MR | Zbl

[19] A. Ould Houcine & F. Point - “Alternatives for pseudofinite groups”, J. Group Theory 16 (2013) no. 4, p. 461-495 | MR | Zbl

[20] A. Pillay - “Remarks on compactifications of pseudofinite groups”, Fund. Math. 236 (2017) no. 2, p. 193-200 | DOI | MR | Zbl

[21] D. J. S. Robinson - Finiteness conditions and generalized soluble groups, Ergeb. Math. Grenzgeb. (3), Springer-Verlag, New York-Berlin, 1972 | Zbl

[22] D. Segal - “Closed subgroups of profinite groups”, Proc. London Math. Soc. (3) 81 (2000) no. 1, p. 29-54 | DOI | MR | Zbl

[23] D. Segal - Words: notes on verbal width in groups, London Mathematical Society Lect. Note Series, vol. 361, Cambridge University Press, Cambridge, 2009 | MR | Zbl

[24] S. Shelah - “On the cardinality of ultraproduct of finite sets”, J. Symbolic Logic 35 (1970) no. 1, p. 83-84 | DOI | MR | Zbl

[25] A. Stolz & A. Thom - “On the lattice of normal subgroups in ultraproducts of compact simple groups”, Proc. London Math. Soc. (3) 108 (2014) no. 1, p. 73-102 | DOI | MR | Zbl

[26] A. Thom - “Examples of hyperlinear groups without factorization property”, Groups Geom. Dyn. 4 (2010) no. 1, p. 195-208 | DOI | MR | Zbl

[27] A. Thom - “About the metric approximation of Higman’s group”, J. Group Theory 15 (2012) no. 2, p. 301-310 | MR | Zbl

[28] A. Thom & J. S. Wilson - “Metric ultraproducts of finite simple groups”, Comptes Rendus Mathématique 352 (2014) no. 6, p. 463-466 | DOI | MR | Zbl

[29] A. Thom & J. S. Wilson - “Some geometric properties of metric ultraproducts of finite simple groups” (2016), arXiv:1606.03863

[30] A. M. Turing - “Finite approximations to Lie groups”, Ann. of Math. (1938), p. 105-111 | DOI | MR | Zbl

[31] J. S. Wilson - Profinite groups, London Mathematical Society Monographs, vol. 19, The Clarendon Press, Oxford University Press, New York, 1998 | MR | Zbl

[32] B. Zilber - “Perfect infinities and finite approximation”, in Infinity and truth, Lect. Notes Series Inst. Math. Sci. Natl. Univ. Singapore, vol. 25, World Sci. Publ., Hackensack, NJ, 2014, p. 199-223 | DOI | MR | Zbl

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