Metric Approximations of Wreath Products

Hayes, Ben; Sale, Andrew W.

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 $G\wr H$ 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 $G\wr H$.

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 $G\wr H$ 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 $G\wr H$ that uses soficity of $H$.

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

[1] Arzhantseva, Goulnara; Păunescu, Liviu Linear sofic groups and algebras, Trans. Am. Math. Soc., Volume 369 (2017) no. 4, pp. 2285-2310 | DOI | MR | Zbl

[2] Bowen, Lewis 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] Capraro, Valerio; Lupini, Martino 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] Ciobanu, Laura; Holt, Derek F.; Rees, Sarah Sofic groups: graph products and graphs of groups, Pacific J. Math., Volume 271 (2014) no. 1, pp. 53-64 | DOI | MR | Zbl

[5] Connes, Alain Classification of injective factors. Cases $I{I}_1,$ $I{I}_{\infty },$ $II{I}_{\lambda },$ $\lambda \ne 1$, Ann. Math., Volume 104 (1976) no. 1, pp. 73-115 | DOI | MR | Zbl

[6] Dykema, Ken; Kerr, David; Pichot, Mikaël Sofic dimension for discrete measured groupoids, Trans. Am. Math. Soc., Volume 366 (2014) no. 2, pp. 707-748 | DOI | MR | Zbl

[7] Elek, Gábor; Lippner, Gábor Sofic equivalence relations, J. Funct. Anal., Volume 258 (2010) no. 5, pp. 1692-1708 | DOI | MR | Zbl

[8] Elek, Gábor; Szabó, Endre Sofic groups and direct finiteness, J. Algebra, Volume 280 (2004) no. 2, pp. 426-434 | DOI | MR | Zbl

[9] Elek, Gábor; Szabó, Endre Hyperlinearity, essentially free actions and $L^2$-invariants. The sofic property, Math. Ann., Volume 332 (2005) no. 2, pp. 421-441 | DOI | MR | Zbl

[10] Elek, Gábor; Szabó, Endre On sofic groups, J. Group Theory, Volume 9 (2006) no. 2, pp. 161-171 | DOI | MR | Zbl

[11] Elek, Gábor; Szabó, Endre Sofic representations of amenable groups, Proc. Amer. Math. Soc., Volume 139 (2011) no. 12, pp. 4285-4291 | DOI | MR | Zbl

[12] Glebsky, Lev; Rivera, Luis Manuel Sofic groups and profinite topology on free groups, J. Algebra, Volume 320 (2008) no. 9, pp. 3512-3518 | DOI | MR | Zbl

[13] Gromov, Mikhael Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. (JEMS), Volume 1 (1999) no. 2, pp. 109-197 | DOI | MR | Zbl

[14] Holt, Derek F.; Rees, Sarah Some closure results for $𝒞$-approximable groups, Pac. J. Math., Volume 287 (2017) no. 2, pp. 393-409 | DOI | MR | Zbl

[15] Iima, Kei-ichiro; Iwamatsu, Ryo On the Jordan decomposition of tensored matrices of Jordan canonical forms, Math. J. Okayama Univ., Volume 51 (2009), pp. 133-148 | MR | Zbl

[16] Kerr, David; Li, Hanfeng Entropy and the variational principle for actions of sofic groups, Invent. Math., Volume 186 (2011) no. 3, pp. 501-558 | DOI | MR | Zbl

[17] Lück, Wolfgang $L^2$-invariants: theory and applications to geometry and $K$-theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 44, Springer, 2002, xvi+595 pages | DOI | MR | Zbl

[18] Martsinkovksky, Alex; Vlassov, Anatoly The representation rings of $k\left[x\right]$ (http://mathserver.neu.edu/~martsinkovsky/GreenExcerpt.pdf)

[19] Păunescu, Liviu On sofic actions and equivalence relations, J. Funct. Anal., Volume 261 (2011) no. 9, pp. 2461-2485 | DOI | MR | Zbl

[20] Pestov, Vladimir G. Hyperlinear and sofic groups: a brief guide, Bull. Symb. Logic, Volume 14 (2008) no. 4, pp. 449-480 | DOI | MR | Zbl

[21] Popa, Sorin Independence properties in subalgebras of ultraproduct ${\mathrm{II}}_1$ factors, J. Funct. Anal., Volume 266 (2014) no. 9, pp. 5818-5846 | DOI | MR | Zbl

[22] Rădulescu, Florin The von Neumann algebra of the non-residually finite Baumslag group $\langle a,b|a{b}^3{a}^{-1}=b^2\rangle$ embeds into $R^{\omega }$, Hot topics in operator theory (Theta Series in Advanced Mathematics), Volume 9, Theta, 2008, pp. 173-185 | MR | Zbl

[23] Sale, Andrew W. Metric behaviour of the Magnus embedding, Geom. Dedicata, Volume 176 (2015), pp. 305-313 | DOI | MR | Zbl

[24] Vershik, Anatoli M.; Gordon, Evgeniĭ I. Groups that are locally embeddable in the class of finite groups, Algebra Anal., Volume 9 (1997) no. 1, pp. 71-97 | MR | Zbl

[25] Weiss, Benjamin 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