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 is a sofic group and is a countable, discrete group. If is sofic, hyperlinear, weakly sofic, or linear sofic, then 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 that uses soficity of .
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 et un groupe dénombrable discret . Notre résultat principal démontre que si est sofique, hyperlinéaire, faiblement sofique ou linéairement sofique, alors est respectivement sofique, hyperlinéaire, faiblement sofique ou linéairement sofique. Grâce à la soficité de nous construisons explicitement dans chacun des cas ci-dessus une approximation métrique pour .
Revised:
Accepted:
Published online:
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
@article{AIF_2018__68_1_423_0, author = {Hayes, Ben and Sale, Andrew W.}, title = {Metric {Approximations} of {Wreath} {Products}}, journal = {Annales de l'Institut Fourier}, pages = {423--455}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {68}, number = {1}, year = {2018}, doi = {10.5802/aif.3166}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3166/} }
TY - JOUR AU - Hayes, Ben AU - Sale, Andrew W. TI - Metric Approximations of Wreath Products JO - Annales de l'Institut Fourier PY - 2018 SP - 423 EP - 455 VL - 68 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3166/ DO - 10.5802/aif.3166 LA - en ID - AIF_2018__68_1_423_0 ER -
%0 Journal Article %A Hayes, Ben %A Sale, Andrew W. %T Metric Approximations of Wreath Products %J Annales de l'Institut Fourier %D 2018 %P 423-455 %V 68 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3166/ %R 10.5802/aif.3166 %G en %F AIF_2018__68_1_423_0
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/
[1] Linear sofic groups and algebras, Trans. Am. Math. Soc., Volume 369 (2017) no. 4, pp. 2285-2310 | DOI | MR | Zbl
[2] Measure conjugacy invariants for actions of countable sofic groups, J. Am. Math. Soc., Volume 23 (2010) no. 1, pp. 217-245 | DOI | MR | Zbl
[3] Introduction to sofic and hyperlinear groups and Connes’ embedding conjecture, Lecture Notes in Mathematics, 2136, Springer, 2015, viii+151 pages (With an appendix by Vladimir Pestov) | DOI | MR | Zbl
[4] Sofic groups: graph products and graphs of groups, Pacific J. Math., Volume 271 (2014) no. 1, pp. 53-64 | DOI | MR | Zbl
[5] Classification of injective factors. Cases , Ann. Math., Volume 104 (1976) no. 1, pp. 73-115 | DOI | MR | Zbl
[6] Sofic dimension for discrete measured groupoids, Trans. Am. Math. Soc., Volume 366 (2014) no. 2, pp. 707-748 | DOI | MR | Zbl
[7] Sofic equivalence relations, J. Funct. Anal., Volume 258 (2010) no. 5, pp. 1692-1708 | DOI | MR | Zbl
[8] Sofic groups and direct finiteness, J. Algebra, Volume 280 (2004) no. 2, pp. 426-434 | DOI | MR | Zbl
[9] Hyperlinearity, essentially free actions and -invariants. The sofic property, Math. Ann., Volume 332 (2005) no. 2, pp. 421-441 | DOI | MR | Zbl
[10] On sofic groups, J. Group Theory, Volume 9 (2006) no. 2, pp. 161-171 | DOI | MR | Zbl
[11] Sofic representations of amenable groups, Proc. Amer. Math. Soc., Volume 139 (2011) no. 12, pp. 4285-4291 | DOI | MR | Zbl
[12] Sofic groups and profinite topology on free groups, J. Algebra, Volume 320 (2008) no. 9, pp. 3512-3518 | DOI | MR | Zbl
[13] Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. (JEMS), Volume 1 (1999) no. 2, pp. 109-197 | DOI | MR | Zbl
[14] Some closure results for -approximable groups, Pac. J. Math., Volume 287 (2017) no. 2, pp. 393-409 | DOI | MR | Zbl
[15] On the Jordan decomposition of tensored matrices of Jordan canonical forms, Math. J. Okayama Univ., Volume 51 (2009), pp. 133-148 | MR | Zbl
[16] Entropy and the variational principle for actions of sofic groups, Invent. Math., Volume 186 (2011) no. 3, pp. 501-558 | DOI | MR | Zbl
[17] -invariants: theory and applications to geometry and -theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 44, Springer, 2002, xvi+595 pages | DOI | MR | Zbl
[18] The representation rings of (http://mathserver.neu.edu/~martsinkovsky/GreenExcerpt.pdf)
[19] On sofic actions and equivalence relations, J. Funct. Anal., Volume 261 (2011) no. 9, pp. 2461-2485 | DOI | MR | Zbl
[20] Hyperlinear and sofic groups: a brief guide, Bull. Symb. Logic, Volume 14 (2008) no. 4, pp. 449-480 | DOI | MR | Zbl
[21] Independence properties in subalgebras of ultraproduct factors, J. Funct. Anal., Volume 266 (2014) no. 9, pp. 5818-5846 | DOI | MR | Zbl
[22] The von Neumann algebra of the non-residually finite Baumslag group embeds into , Hot topics in operator theory (Theta Series in Advanced Mathematics), Volume 9, Theta, 2008, pp. 173-185 | MR | Zbl
[23] Metric behaviour of the Magnus embedding, Geom. Dedicata, Volume 176 (2015), pp. 305-313 | DOI | MR | Zbl
[24] Groups that are locally embeddable in the class of finite groups, Algebra Anal., Volume 9 (1997) no. 1, pp. 71-97 | MR | Zbl
[25] Sofic groups and dynamical systems, Sankhyā Ser. A, Volume 62 (2000) no. 3, pp. 350-359 Ergodic theory and harmonic analysis (Mumbai, 1999) | MR | Zbl
Cited by Sources: