Quantifying metric approximations of discrete groups

Arzhantseva, Goulnara; Cherix, Pierre-Alain

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 33 (2024) no. 3, pp. 831-877

We introduce and systematically study a profile function whose asymptotic behaviour quantifies the dimension or the size of a metric approximation of a finitely generated group $G$ by a family of groups $ℱ={\left\{(G_{\alpha },d_{\alpha },k_{\alpha },{\epsilon }_{\alpha })\right\}}_{\alpha \in I,}$ where each group $G_{\alpha }$ is equipped with a bi-invariant metric $d_{\alpha }$ and a dimension $k_{\alpha }$, for strictly positive real numbers ${\epsilon }_{\alpha }$ such that ${inf}_{\alpha }{\epsilon }_{\alpha }>0$. Through the notion of a residually amenable profile that we introduce, our approach generalises classical isoperimetric (aka Følner) profiles of amenable groups and recently introduced functions quantifying residually finite groups. Our viewpoint is much more general and covers hyperlinear and sofic approximations as well as many other metric approximations such as weakly sofic, weakly hyperlinear, and linear sofic approximations.

Classification : 20E26, 20F69, 20C99, 03C20
Keywords: Residually finite groups, sofic and hyperlinear groups, metric ultraproducts, amenable groups, full residual finiteness growth, Følner function.

[1] Arzhantseva, Goulnara Asymptotic approximations of finitely generated groups, Research Perspectives CRM Barcelona-Fall 2012 (Trends in Mathematics), Volume 1, Birkhäuser, 2014, pp. 7-16 | MR

[2] Arzhantseva, Goulnara; Păunescu, Liviu Almost commuting permutations are near commuting permutations, J. Funct. Anal., Volume 269 (2015) no. 3, pp. 745-757 (Accessed 2016-01-23) | MR | DOI | Zbl

[3] Arzhantseva, Goulnara; Păunescu, Liviu Linear sofic groups and algebras, Trans. Am. Math. Soc., Volume 369 (2017), pp. 2285-2310 | MR | DOI | Zbl

[4] Arzhantseva, Goulnara; Păunescu, Liviu Constraint metric approximations and equations in groups, J. Algebra, Volume 516 (2018), pp. 329-351 | MR | DOI | Zbl

[5] Bass, Hyman The degree of polynomial growth of finitely generated nilpotent groups, Proc. Lond. Math. Soc., Volume 25 (1972), pp. 603-614 | MR | DOI | Zbl

[6] Becker, Oren; Lubotzky, Alexander; Thom, Andreas Stability and invariant random subgroups, Duke Math. J., Volume 168 (2019) no. 12, pp. 2207-2234 | DOI | MR | Zbl

[7] Ben Yaacov, Itaï; Berenstein, Alexander; Henson, C. Ward; Usvyatsov, Alexander Model theory for metric structures, Model theory with applications to algebra and analysis. Vol. 2 (London Mathematical Society Lecture Note Series), Volume 350, Cambridge University Press, 2008, pp. 315-427 (Accessed 2014-04-07) | Zbl

[8] Berlai, Federico New residually amenable groups, permanence properties, and metric approximations, Ph. D. Thesis, University of Vienna (2016)

[9] Bou-Rabee, Khalid Quantifying residual finiteness, J. Algebra, Volume 323 (2010) no. 3, pp. 729-737 (Accessed 2015-08-27) | MR | DOI | Zbl

[10] Bou-Rabee, Khalid; de Cornulier, Yves Systolic growth of linear groups, Proc. Am. Math. Soc., Volume 144 (2016) no. 2, pp. 529-533 | DOI | MR | Zbl

[11] Bou-Rabee, Khalid; Hagen, Mark F.; Patel, Priyam Residual finiteness growths of virtually special groups, Math. Z., Volume 279 (2014) no. 1-2, pp. 297-310 (Accessed 2015-08-27) | MR | DOI | Zbl

[12] Bou-Rabee, Khalid; McReynolds, David B. Asymptotic growth and least common multiples in groups, Bull. Lond. Math. Soc., Volume 43 (2011) no. 6, pp. 1059-1068 (Accessed 2015-08-27) | MR | Zbl | DOI

[13] Bou-Rabee, Khalid; Seward, Brandon Arbitrarily large residual finiteness growth, J. Reine Angew. Math., Volume 710 (2016), pp. 199-204 | DOI | MR | Zbl

[14] Bou-Rabee, Khalid; Studenmund, Daniel Full residual finiteness growths of nilpotent groups, Isr. J. Math., Volume 214 (2016) no. 1, pp. 209-233 | DOI | MR | Zbl

[15] Brown, Nathanial P.; Dykema, Kenneth J.; Jung, Kenley Free entropy dimension in amalgamated free products, Proc. Lond. Math. Soc., Volume 97 (2008) no. 2, pp. 339-367 (with an appendix by Wolfgang Lück) | DOI | MR | Zbl

[16] Capraro, Valerio; Lupini, Martino Introduction to sofic and hyperlinear groups and Connes’ embedding conjecture, Lecture Notes in Mathematics, 2136, Springer, 2015 (Accessed 2016-01-06 with an appendix by Vladimir Pestov) | MR | DOI

[17] Cavaleri, Matteo Algorithms and quantifications in amenable and sofic groups, Ph. D. Thesis, Universita degli studi di Roma La Sapienza (2016)

[18] Ceccherini-Silberstein, Tullio; Coornaert, Michel Cellular automata and groups, Springer Monographs in Mathematics, Springer, 2010 (Accessed 2014-11-07) | MR | DOI

[19] Cornulier, Yves Sofic profile and computability of Cremona groups, Mich. Math. J., Volume 62 (2013) no. 4, pp. 823-841 | MR | Zbl

[20] de Cornulier, Yves; Tessera, Romain; Valette, Alain Isometric group actions on Hilbert spaces: growth of cocycles, Geom. Funct. Anal., Volume 17 (2007) no. 3, pp. 770-792 (Accessed 2016-08-28) | MR | DOI | Zbl

[21] Dykema, Kenneth J.; 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

[22] Elek, Gábor; Grabowski, Łukasz Almost commuting matrices with respect to the rank metric, Groups Geom. Dyn., Volume 15 (2021) no. 3, pp. 1059-1083 | DOI | MR | Zbl

[23] Elek, Gábor; Szabó, Endre Sofic groups and direct finiteness, J. Algebra, Volume 280 (2004) no. 2, pp. 426-434 (Accessed 2014-03-09) | MR | DOI | Zbl

[24] Elek, Gábor; Szabó, Endre On sofic groups, J. Group Theory, Volume 9 (2006) no. 2, pp. 161-171 (Accessed 2014-03-09) | MR | Zbl

[25] Gismatullin, Jakub On approximation of groups: weak soficity and weak hyperlinearity (2013) (in preparation)

[26] Glebsky, Lev Almost commuting matrices with respect to normalized Hilbert-Schmidt norm (2010) | arXiv

[27] Glebsky, Lev Approximations of groups, characterizations of sofic groups, and equations over groups, J. Algebra, Volume 477 (2017), pp. 147-162 | DOI | MR | Zbl

[28] 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

[29] Gromov, Mikhael Groups of polynomial growth and expanding maps, Publ. Math., Inst. Hautes Étud. Sci., Volume 53 (1981), pp. 53-73 (Accessed 2016-01-23) | MR | DOI | Numdam | Zbl

[30] Gromov, Mikhael Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) (London Mathematical Society Lecture Note Series), Volume 182, Cambridge University Press, 1993, pp. 1-295 (Accessed 2015-12-12) | MR | Zbl

[31] Gromov, Mikhael Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc., Volume 1 (1999) no. 2, pp. 109-197 (Accessed 2013-11-03) | MR | Zbl | DOI

[32] Guivarc’h, Yves Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. Fr., Volume 101 (1973), pp. 333-379 | MR | DOI | Numdam | Zbl

[33] Hadwin, Don; Shulman, Tatiana Stability of group relations under small Hilbert-Schmidt perturbations, J. Funct. Anal., Volume 275 (2018) no. 4, pp. 761-792 | DOI | MR | Zbl

[34] Hayes, Ben; Sale, Andrew W. Metric approximations of wreath products, Ann. Inst. Fourier, Volume 68 (2018) no. 1, pp. 423-455 | MR | DOI | Numdam | Zbl

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

[36] Malcev, Anatoliĭ I. On isomorphic matrix representations of infinite groups, Mat. Sb., N. Ser., Volume 8 (50) (1940), pp. 405-422 | MR | Zbl

[37] Moore, Justin Tatch Fast growth in the Følner function for Thompson’s group $F$, Groups Geom. Dyn., Volume 7 (2013) no. 3, pp. 633-651 (Accessed 2016-01-23) | MR | DOI | Zbl

[38] Pestov, Vladimir Hyperlinear and sofic groups: A brief guide, Bull. Symb. Log., Volume 14 (2008) no. 04, pp. 449-480 | MR | DOI | Zbl

[39] Pestov, Vladimir; Kwiatkowska, Aleksandra An introduction to hyperlinear and sofic groups, Appalachian Set Theory 2006-2012 (London Mathematical Society Lecture Note Series), Volume 406, Cambridge University Press, 2012, pp. 145-185 | MR | Zbl | DOI

[40] Popa, Sorin Free-independent sequences in type ${\mathrm{II}}_1$ factors and related problems, Recent advances in operator algebras (Orléans, 1992) (Astérisque), Volume 232, Société Mathématique de France, 1995, pp. 187-202 | MR | Zbl

[41] 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 (Accessed 2016-09-04) | Zbl

[42] Remeslennikov, Vladimir N. $\exists$-free groups, Sib. Mat. Zh., Volume 30 (1989) no. 6, pp. 193-197

[43] Segal, Daniel Polycyclic groups, Cambridge Tracts in Mathematics, 82, Cambridge University Press, 1983, xiv+289 pages | DOI | MR

[44] Slofstra, William; Vidick, Thomas Entanglement in non-local games and the hyperlinear profile of groups, Ann. Henri Poincaré, Volume 19 (2018) no. 10, pp. 2979-3005 | DOI | MR | Zbl

[45] Stolz, Abel Properties of Linearly Sofic Groups (2013) | arXiv

[46] Thom, Andreas About the metric approximation of Higman’s group, J. Group Theory, Volume 15 (2012) no. 2, pp. 301-310 (Accessed 2014-11-07) | MR | DOI | Zbl

[47] Vershik, Anatoliĭ M. Amenability and approximation of infinite groups, Sel. Math. Sov., Volume 2 (1982) no. 4, pp. 311-330 | MR | Zbl

[48] Vershik, Anatoliĭ 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

[49] Voiculescu, Dan A strengthened asymptotic freeness result for random matrices with applications to free entropy, Int. Math. Res. Not., Volume 1998 (1998) no. 1, pp. 41-63 | DOI | MR | Zbl

[50] Weiss, Benjamin Sofic groups and dynamical systems, Sankhyā, Ser. A, Volume 62 (2000) no. 3, pp. 350-359 Accessed 2013-11-03 Ergodic theory and harmonic analysis (Mumbai, 1999) | MR | Zbl

[51] Wolf, Joseph A. Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Differ. Geom., Volume 2 (1968), pp. 421-446 | MR | Zbl