Ordering the space of finitely generated groups
Annales de l'Institut Fourier, Volume 65 (2015) no. 5, pp. 2091-2144.

We consider the oriented graph whose vertices are isomorphism classes of finitely generated groups, with an edge from G to H if, for some generating set T in H and some sequence of generating sets S i in G, the marked balls of radius i in (G,S i ) and (H,T) coincide. We show that if a connected component of this graph contains at least one torsion-free nilpotent group G, then it consists of those groups which generate the same variety of groups as G. We show on the other hand that the first Grigorchuk group has infinite girth, and hence belongs to the same connected component as free groups.

The arrows in the graph define a preorder on the set of isomorphism classes of finitely generated groups. We show that a partial order can be imbedded in this preorder if and only if it is realizable by subsets of a countable set under inclusion.

We show that every countable group imbeds in a group of non-uniform exponential growth. In particular, there exist groups of non-uniform exponential growth that are not residually of subexponential growth and do not admit a uniform imbedding into Hilbert space.

Nous considérons le graphe orienté dont les sommets sont les classes d’isomorphisme de groupes de type fini, avec une arête de G à H si, pour une partie génératrice de H et une suite de parties génératrices de G, les boules marquées de rayon de plus en plus grand coincident dans G et H. Nous montrons que les composantes connexes de groupes nilpotents sans torsion sont leurs variétés, et qu’il y a une arête du premier groupe de Grigorchuk vers un groupe libre.

Les flèches dans ce graphe définissent un préordre sur l’ensemble des classes d’isomorphisme de groupes de type fini. Nous montrons qu’un ordre partiel se plonge dans ce préordre si et seulement s’il est réalisable par des ensembles d’un ensemble dénombrable pour l’inclusion.

Nous montrons que tout groupe dénombrable se plonge dans un groupe de croissance exponentielle non-uniforme. En particulier, il existe des groupes de croissance exponentielle non-uniforme qui ne sont pas résiduellement de croissance subexponentielle.

DOI: 10.5802/aif.2984
Classification: 20E10, 20E34, 20F65
Keywords: Topological space of marked groups, limit groups, varieties of groups, non-uniform exponential growth, universal statements and identities
Mot clés : Espace topologique des groupes marqués, groupes limites, variétés de groupes, croissance exponentielle non-uniforme, énoncés universels et identités
Bartholdi, Laurent 1; Erschler, Anna 2

1 Mathematisches Institut Georg-August Universität Bunsenstraße 3-5 D-37073 Göttingen (Germany)
2 C.N.R.S. UMR 8553 Département de mathématiques et applications École Normale Supérieure 45 rue d’Ulm 75005 Paris (France)
     author = {Bartholdi, Laurent and Erschler, Anna},
     title = {Ordering the space of finitely generated groups},
     journal = {Annales de l'Institut Fourier},
     pages = {2091--2144},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {5},
     year = {2015},
     doi = {10.5802/aif.2984},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2984/}
AU  - Bartholdi, Laurent
AU  - Erschler, Anna
TI  - Ordering the space of finitely generated groups
JO  - Annales de l'Institut Fourier
PY  - 2015
SP  - 2091
EP  - 2144
VL  - 65
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2984/
DO  - 10.5802/aif.2984
LA  - en
ID  - AIF_2015__65_5_2091_0
ER  - 
%0 Journal Article
%A Bartholdi, Laurent
%A Erschler, Anna
%T Ordering the space of finitely generated groups
%J Annales de l'Institut Fourier
%D 2015
%P 2091-2144
%V 65
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2984/
%R 10.5802/aif.2984
%G en
%F AIF_2015__65_5_2091_0
Bartholdi, Laurent; Erschler, Anna. Ordering the space of finitely generated groups. Annales de l'Institut Fourier, Volume 65 (2015) no. 5, pp. 2091-2144. doi : 10.5802/aif.2984. https://aif.centre-mersenne.org/articles/10.5802/aif.2984/

[1] Abért, Miklós Group laws and free subgroups in topological groups, Bull. London Math. Soc., Volume 37 (2005) no. 4, pp. 525-534 | DOI | Zbl

[2] Adyan, S. I. Problema Bernsaida i tozhdestva v gruppakh, Izdat. “Nauka”, Moscow, 1975, pp. 335

[3] Akhmedov, Azer On the girth of finitely generated groups, J. Algebra, Volume 268 (2003) no. 1, pp. 198-208 | DOI | Zbl

[4] Akhmedov, Azer The girth of groups satisfying Tits alternative, J. Algebra, Volume 287 (2005) no. 2, pp. 275-282 | DOI | Zbl

[5] Akhmedov, Azer; Stein, Melanie; Taback, Jennifer Free limits of Thompson’s group F, Geom. Dedicata, Volume 155 (2011), pp. 163-176 | DOI | Zbl

[6] Alešin, S. V. Finite automata and the Burnside problem for periodic groups, Mat. Zametki, Volume 11 (1972), pp. 319-328

[7] Arzhantseva, G. N.; Burillo, J.; Lustig, M.; Reeves, L.; Short, H.; Ventura, E. Uniform non-amenability, Adv. Math., Volume 197 (2005) no. 2, pp. 499-522 | DOI | Zbl

[8] Bartholdi, Laurent A Wilson group of non-uniformly exponential growth, C. R. Math. Acad. Sci. Paris, Volume 336 (2003) no. 7, pp. 549-554 | DOI | Zbl

[9] Bartholdi, Laurent; Erschler, Anna Growth of permutational extensions, Invent. Math., Volume 189 (2012) no. 2, pp. 431-455 | DOI | Zbl

[10] Bass, H. The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. (3), Volume 25 (1972), pp. 603-614 | Zbl

[11] Baumslag, Benjamin Residually free groups, Proc. London Math. Soc. (3), Volume 17 (1967), pp. 402-418 | Zbl

[12] Baumslag, Gilbert Subgroups of finitely presented metabelian groups, J. Austral. Math. Soc., Volume 16 (1973), pp. 98-110 (Collection of articles dedicated to the memory of Hanna Neumann, I) | Zbl

[13] Baumslag, Gilbert; Myasnikov, Alexei; Remeslennikov, Vladimir Algebraic geometry over groups. I. Algebraic sets and ideal theory, J. Algebra, Volume 219 (1999) no. 1, pp. 16-79 | DOI | Zbl

[14] Baumslag, Gilbert; Neumann, B. H.; Neumann, Hanna; Neumann, Peter M. On varieties generated by a finitely generated group, Math. Z., Volume 86 (1964), pp. 93-122 | Zbl

[15] Bestvina, Mladen; Feighn, Mark Notes on Sela’s work: limit groups and Makanin-Razborov diagrams, Geometric and cohomological methods in group theory (London Math. Soc. Lecture Note Ser.), Volume 358, Cambridge Univ. Press, Cambridge, 2009, pp. 1-29 | Zbl

[16] Brieussel, Jérémie Behaviors of entropy on finitely generated groups, Ann. Probab., Volume 41 (2013) no. 6, pp. 4116-4161 | DOI | Zbl

[17] Brin, Matthew G. The free group of rank 2 is a limit of Thompson’s group F, Groups Geom. Dyn., Volume 4 (2010) no. 3, pp. 433-454 | DOI | Zbl

[18] Burger, Marc; Mozes, Shahar Finitely presented simple groups and products of trees, C. R. Acad. Sci. Paris Sér. I Math., Volume 324 (1997) no. 7, pp. 747-752 | DOI | Zbl

[19] Carlitz, L.; Wilansky, A.; Milnor, John; Struble, R. A.; Felsinger, Neal; Simoes, J. M. S.; Power, E. A.; Shafer, R. E.; Maas, R. E. Problems and Solutions: Advanced Problems: 5600-5609, Amer. Math. Monthly, Volume 75 (1968) no. 6, pp. 685-687 | DOI

[20] Chabauty, Claude Limite d’ensembles et géométrie des nombres, Bull. Soc. Math. France, Volume 78 (1950), pp. 143-151 | Numdam | Zbl

[21] Champetier, Christophe; Guirardel, Vincent Limit groups as limits of free groups, Israel J. Math., Volume 146 (2005), pp. 1-75 | DOI | Zbl

[22] Chapuis, Olivier Universal theory of certain solvable groups and bounded Ore group rings, J. Algebra, Volume 176 (1995) no. 2, pp. 368-391 | DOI | Zbl

[23] Chapuis, Olivier -free metabelian groups, J. Symbolic Logic, Volume 62 (1997) no. 1, pp. 159-174 | DOI | Zbl

[24] Cherix, Pierre-Alain; Cowling, Michael; Jolissaint, Paul; Julg, Pierre; Valette, Alain Groups with the Haagerup property, Progress in Mathematics, 197, Birkhäuser Verlag, Basel, 2001, pp. viii+126 (Gromov’s a-T-menability) | DOI | Zbl

[25] de Cornulier, Yves; Guyot, Luc; Pitsch, Wolfgang On the isolated points in the space of groups, J. Algebra, Volume 307 (2007) no. 1, pp. 254-277 | DOI | Zbl

[26] de Cornulier, Yves; Mann, Avinoam Some residually finite groups satisfying laws, Geometric group theory (Trends Math.), Birkhäuser, Basel, 2007, pp. 45-50 | DOI | Zbl

[27] Greendlinger, Martin Dehn’s algorithm for the word problem, Comm. Pure Appl. Math., Volume 13 (1960), pp. 67-83 | Zbl

[28] Grigorchuk, R. I. Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat., Volume 48 (1984) no. 5, pp. 939-985 | Zbl

[29] Grigorčuk, R. I. On Burnside’s problem on periodic groups, Funktsional. Anal. i Prilozhen., Volume 14 (1980) no. 1, pp. 53-54 | Zbl

[30] Gromov, Mikhael Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. (1981) no. 53, pp. 53-73 | Numdam | Zbl

[31] Gromov, Mikhael Structures métriques pour les variétés riemanniennes, Textes Mathématiques [Mathematical Texts], 1, CEDIC, Paris, 1981, pp. iv+152 (Edited by J. Lafontaine and P. Pansu)

[32] Gromov, Mikhael Random walk in random groups, Geom. Funct. Anal., Volume 13 (2003) no. 1, pp. 73-146 | DOI | Zbl

[33] Guyot, Luc Limits of dihedral groups, Geom. Dedicata, Volume 147 (2010), pp. 159-171 | DOI | Zbl

[34] Guyot, Luc Limits of metabelian groups, Internat. J. Algebra Comput., Volume 22 (2012) no. 4, pp. 1250031, 30 | DOI | Zbl

[35] Hall, P. Finiteness conditions for soluble groups, Proc. London Math. Soc. (3), Volume 4 (1954), pp. 419-436 | Zbl

[36] Hall, Philip The Edmonton notes on nilpotent groups, Queen Mary College Mathematics Notes, Mathematics Department, Queen Mary College, London, 1969, pp. iii+76 | Zbl

[37] de la Harpe, Pierre Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000, pp. vi+310 | Zbl

[38] de la Harpe, Pierre Uniform growth in groups of exponential growth, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part II (Haifa, 2000), Volume 95 (2002), pp. 1-17 | DOI | Zbl

[39] Hirsch, K. A. On infinite soluble groups. III, Proc. London Math. Soc. (2), Volume 49 (1946), pp. 184-194 | Zbl

[40] Kharlampovich, Olga; Myasnikov, Alexei Irreducible affine varieties over a free group. I. Irreducibility of quadratic equations and Nullstellensatz, J. Algebra, Volume 200 (1998) no. 2, pp. 472-516 | DOI | Zbl

[41] Kharlampovich, Olga; Myasnikov, Alexei Irreducible affine varieties over a free group. II. Systems in triangular quasi-quadratic form and description of residually free groups, J. Algebra, Volume 200 (1998) no. 2, pp. 517-570 | DOI | Zbl

[42] Kharlampovich, Olga; Myasnikov, Alexei Equations and fully residually free groups, Combinatorial and geometric group theory (Trends Math.), Birkhäuser/Springer Basel AG, Basel, 2010, pp. 203-242 | DOI | Zbl

[43] Kharlampovich, Olga; Myasnikov, Alexei Limits of relatively hyperbolic groups and Lyndon’s completions, J. Eur. Math. Soc. (JEMS), Volume 14 (2012) no. 3, pp. 659-680 | DOI | Zbl

[44] Malcev, A. On isomorphic matrix representations of infinite groups, Rec. Math. [Mat. Sbornik] N.S., Volume 8 (50) (1940), pp. 405-422

[45] Mann, Avinoam The growth of free products, J. Algebra, Volume 326 (2011), pp. 208-217 | DOI | Zbl

[46] Nekrashevych, Volodymyr A group of non-uniform exponential growth locally isomorphic to IMG (z 2 +i), Trans. Amer. Math. Soc., Volume 362 (2010) no. 1, pp. 389-398 | DOI | Zbl

[47] Neumann, Hanna Varieties of groups, Springer-Verlag New York, Inc., New York, 1967, pp. x+192 | Zbl

[48] Olʼshanskiĭ, A. Yu.; Sapir, M. V. On F k -like groups, Algebra Logika, Volume 48 (2009) no. 2, p. 245-257, 284, 286–287 | DOI | Zbl

[49] Ould Houcine, Abderezak Limit groups of equationally Noetherian groups, Geometric group theory (Trends Math.), Birkhäuser, Basel, 2007, pp. 103-119 | DOI | Zbl

[50] Paulin, Frédéric Sur la théorie élémentaire des groupes libres (d’après Sela), Astérisque (2004) no. 294, pp. ix, 363-402 | Numdam | Zbl

[51] Pervova, E. L. Everywhere dense subgroups of a group of tree automorphisms, Tr. Mat. Inst. Steklova, Volume 231 (2000) no. Din. Sist., Avtom. i Beskon. Gruppy, pp. 356-367 | Zbl

[52] Pervova, E. L. Maximal subgroups of some non locally finite p-groups, Internat. J. Algebra Comput., Volume 15 (2005) no. 5-6, pp. 1129-1150 | DOI | Zbl

[53] Pride, Stephen J. The concept of “largeness” in group theory, Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976) (Stud. Logic Foundations Math.), Volume 95, North-Holland, Amsterdam-New York, 1980, pp. 299-335 | Zbl

[54] Remeslennikov, V. N. -free groups, Sibirsk. Mat. Zh., Volume 30 (1989) no. 6, pp. 193-197 | DOI | Zbl

[55] Schleimer, Saul On the girth of groups (2003) (http://homepages.warwick.ac.uk/~masgar/Maths/girth.pdf)

[56] Sela, Zlil Diophantine geometry over groups. I. Makanin-Razborov diagrams, Publ. Math. Inst. Hautes Études Sci. (2001) no. 93, pp. 31-105 | DOI | Numdam | Zbl

[57] Shalom, Yehuda Rigidity of commensurators and irreducible lattices, Invent. Math., Volume 141 (2000) no. 1, pp. 1-54 | DOI | Zbl

[58] Thomas, Simon On the concept of “largeness” in group theory, J. Algebra, Volume 322 (2009) no. 12, pp. 4181-4197 | DOI | Zbl

[59] Timošenko, E. I. The preservation of elementary and universal equivalence in the wreath product, Algebra i Logika, Volume 7 (1968) no. 4, pp. 114-119 | Zbl

[60] Timoshenko, E. I. On universally equivalent solvable groups, Algebra Log., Volume 39 (2000) no. 2, p. 227-240, 245 | DOI | Zbl

[61] Timoshenko, E. I. On universal theories of metabelian groups and the Shmel ' kin embedding, Sibirsk. Mat. Zh., Volume 42 (2001) no. 5, p. 1168-1175, iv | DOI | Zbl

[62] Timoshenko, E. I. Universal equivalence of partially commutative metabelian groups, Algebra Logika, Volume 49 (2010) no. 2, p. 263-290, 296, 299 | DOI | Zbl

[63] Wilson, John S. Further groups that do not have uniformly exponential growth, J. Algebra, Volume 279 (2004) no. 1, pp. 292-301 | DOI | Zbl

[64] Wilson, John S. On exponential growth and uniformly exponential growth for groups, Invent. Math., Volume 155 (2004) no. 2, pp. 287-303 | DOI | Zbl

[65] Zarzycki, Roland Limits of Thompson’s group F, Combinatorial and geometric group theory (Trends Math.), Birkhäuser/Springer Basel AG, Basel, 2010, pp. 307-315 | DOI | Zbl

Cited by Sources: