[Remarques sur les groupes finitairement approximables]
Le concept de groupe -approximable, pour une classe de groupes finis , 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.
The concept of a -approximable group, for a class of finite groups , 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.
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DOI : 10.5802/jep.69
Keywords: Sofic, C-approximable group, metric ultraproduct
Mot clés : Sofique, groupe C-approximable, ultraproduit métrique
Nikolay Nikolov 1 ; Jakob Schneider 2 ; Andreas Thom 2
CC-BY-ND 4.0
@article{JEP_2018__5__239_0,
author = {Nikolay Nikolov and Jakob Schneider and Andreas Thom},
title = {Some remarks on finitarily approximable groups},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {239--258},
publisher = {\'Ecole polytechnique},
volume = {5},
year = {2018},
doi = {10.5802/jep.69},
zbl = {06988579},
mrnumber = {3749196},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.69/}
}
TY - JOUR AU - Nikolay Nikolov AU - Jakob Schneider AU - Andreas Thom TI - Some remarks on finitarily approximable groups JO - Journal de l’École polytechnique — Mathématiques PY - 2018 SP - 239 EP - 258 VL - 5 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.69/ DO - 10.5802/jep.69 LA - en ID - JEP_2018__5__239_0 ER -
%0 Journal Article %A Nikolay Nikolov %A Jakob Schneider %A Andreas Thom %T Some remarks on finitarily approximable groups %J Journal de l’École polytechnique — Mathématiques %D 2018 %P 239-258 %V 5 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.69/ %R 10.5802/jep.69 %G en %F JEP_2018__5__239_0
Nikolay Nikolov; Jakob Schneider; Andreas Thom. Some remarks on finitarily approximable groups. Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 239-258. doi : 10.5802/jep.69. https://jep.centre-mersenne.org/articles/10.5802/jep.69/
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