Browse issues
no. 3
Quantifying metric approximations of discrete groups
Goulnara Arzhantseva1; Pierre-Alain Cherix2
1 Universität Wien, Fakultät für Mathematik, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria.
2 Université de Genève, Section de Mathématiques, Uni Dufour, 24 rue du Général Dufour, Case postale 64, 1211 Genève 4, Switzerland.
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 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={(G α ,d α ,k α ,ε α )} α∈I, where each group G α is equipped with a bi-invariant metric d α and a dimension k α , for strictly positive real numbers ε α such that inf α ε α >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.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1788
Classification: 20E26, 20F69, 20C99, 03C20
Keywords: Residually finite groups, sofic and hyperlinear groups, metric ultraproducts, amenable groups, full residual finiteness growth, Følner function.

Goulnara Arzhantseva 1; Pierre-Alain Cherix 2

1 Universität Wien, Fakultät für Mathematik, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria.
2 Université de Genève, Section de Mathématiques, Uni Dufour, 24 rue du Général Dufour, Case postale 64, 1211 Genève 4, Switzerland.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{AFST_2024_6_33_3_831_0,
     author = {Goulnara Arzhantseva and Pierre-Alain Cherix},
     title = {Quantifying metric approximations of discrete groups},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {831--877},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 33},
     number = {3},
     year = {2024},
     doi = {10.5802/afst.1788},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1788/}
}
TY  - JOUR
AU  - Goulnara Arzhantseva
AU  - Pierre-Alain Cherix
TI  - Quantifying metric approximations of discrete groups
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2024
SP  - 831
EP  - 877
VL  - 33
IS  - 3
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1788/
DO  - 10.5802/afst.1788
LA  - en
ID  - AFST_2024_6_33_3_831_0
ER  - 
%0 Journal Article
%A Goulnara Arzhantseva
%A Pierre-Alain Cherix
%T Quantifying metric approximations of discrete groups
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2024
%P 831-877
%V 33
%N 3
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1788/
%R 10.5802/afst.1788
%G en
%F AFST_2024_6_33_3_831_0
Goulnara Arzhantseva; Pierre-Alain Cherix. Quantifying metric approximations of discrete groups. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 3, pp. 831-877. doi : 10.5802/afst.1788. https://afst.centre-mersenne.org/articles/10.5802/afst.1788/

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

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

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

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

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

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

[7] Itaï Ben Yaacov; Alexander Berenstein; C. Ward Henson; Alexander Usvyatsov 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] Federico Berlai New residually amenable groups, permanence properties, and metric approximations, Ph. D. Thesis, University of Vienna (2016)

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

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

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

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

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

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

[15] Nathanial P. Brown; Kenneth J. Dykema; Kenley Jung 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] Valerio Capraro; Martino Lupini 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) | DOI

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

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

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

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

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

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

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

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

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

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

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

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

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

[30] Mikhael Gromov 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) | Zbl

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

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

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

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

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

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

[37] Justin Tatch Moore 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) | DOI | Zbl

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

[39] Vladimir Pestov; Aleksandra Kwiatkowska 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 | DOI | Zbl

[40] Sorin Popa Free-independent sequences in type 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] Florin Rădulescu The von Neumann algebra of the non-residually finite Baumslag group 〈a,b|ab 3 a -1 =b 2 〉 embeds into R ω , Hot topics in operator theory (Theta Series in Advanced Mathematics), Volume 9, Theta, 2008, pp. 173-185 (Accessed 2016-09-04) | Zbl

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

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

[44] William Slofstra; Thomas Vidick 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] Abel Stolz Properties of Linearly Sofic Groups (2013) | arXiv

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

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

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

[49] Dan Voiculescu 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] Benjamin Weiss 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) | Zbl

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

Cited by Sources: