Metric Approximations of Wreath Products
Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 423-455.

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 GH 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 GH that uses soficity of H.

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 GH 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 GH.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3166
Classification: 20E26,  20F65,  43A07
Keywords: sofic groups, wreath products, hyperlinear groups, linear sofic groups, weakly sofic groups
Hayes, Ben 1; Sale, Andrew W. 2

1 University of Virginia Charlottesville, VA 22904 (USA)
2 Cornell University Ithaca, NY 14853 (USA)
License: CC-BY-ND 4.0
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Hayes, Ben; Sale, Andrew W. Metric Approximations of Wreath Products. Annales de l'Institut Fourier, Volume 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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