no. 1
Metric Approximations of Wreath Products
[Approximation métrique de produits en couronne]
Hayes, Ben1 ; Sale, Andrew W.2
1 University of Virginia Charlottesville, VA 22904 (USA)
2 Cornell University Ithaca, NY 14853 (USA)
Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 423-455.

On connait aujourd’hui de nombreux groupes sofiques. Néanmoins il existe peu de résultats concernant la stabilité de la propriété de soficité. Ce travail s’intéresse au produit en couronne de groupes sofiques mais aussi de groupes vérifiant des propriétés d’approximations métriques plus générales.

Considérons un groupe sofique H et un groupe dénombrable discret G. Notre résultat principal démontre que si G est sofique, hyperlinéaire, faiblement sofique ou linéairement sofique, alors G≀H est respectivement sofique, hyperlinéaire, faiblement sofique ou linéairement sofique. Grâce à la soficité de H nous construisons explicitement dans chacun des cas ci-dessus une approximation métrique pour G≀H.

Given the large class of groups already known to be sofic, there is seemingly a shortfall in results concerning their permanence properties. We address this problem for wreath products, and in particular investigate the behaviour of more general metric approximations of groups under wreath products.

Our main result is the following. Suppose that H is a sofic group and G is a countable, discrete group. If G is sofic, hyperlinear, weakly sofic, or linear sofic, then G≀H is also sofic, hyperlinear, weakly sofic, or linear sofic respectively. In each case we construct relevant metric approximations, extending a general construction of metric approximations for G≀H that uses soficity of H.

Reçu le :
Révisé le :
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DOI : 10.5802/aif.3166
Classification : 20E26, 20F65, 43A07
Keywords: sofic groups, wreath products, hyperlinear groups, linear sofic groups, weakly sofic groups
Mot clés : groupes sofiques, produits en couronne, groupes hyperlinéaires, groupes linéairement sofiques, groupes faiblement sofiques

Hayes, Ben 1 ; Sale, Andrew W. 2

1 University of Virginia Charlottesville, VA 22904 (USA)
2 Cornell University Ithaca, NY 14853 (USA)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Metric {Approximations} of {Wreath} {Products}},
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Hayes, Ben; Sale, Andrew W. Metric Approximations of Wreath Products. Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 423-455. doi : 10.5802/aif.3166. https://aif.centre-mersenne.org/articles/10.5802/aif.3166/

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