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2015, 9: 257-284. doi: 10.3934/jmd.2015.9.257

Iterated identities and iterational depth of groups

Anna Erschler 1,

1. 

Département de Mathématiques et Applications, École Normale Supérieure, 45 Rue d’Ulm, 75005 Paris, France

Received  September 2014 Revised  July 2015 Published  September 2015

Given a word $w$ on $n$ letters, we study groups which satisfy ``iterated identity'' $w$, meaning that for all $x_1, \dots, x_n$ there exists $N$ such that the $N-th$ iteration of $w$ of Engel type, applied to $x_1, \dots, x_n$, is equal to the identity. We define bounded groups and groups which are multiscale with respect to identities. This notion of being multiscale can be viewed as a self-similarity conditions for the set of identities, satisfied by a group. In contrast with torsion groups and Engel groups, groups which are multiscale with respect to identities appear among finitely generated elementary amenable groups. We prove that any polycyclic, as well as any metabelian group is bounded, and we compute the iterational depth for various wreath products. We study the set of iterated identities satisfied by a given group, which is not necessarily a subgroup of a free group and not necessarily invariant under conjugation, in contrast with usual identities. Finally, we discuss another notion of iterated identities of groups, which we call solvability type iterated identities, and its relation to elementary classes of varieties of groups.
Keywords: nilpotent groups, group identity, amenable groups, metabelian groups, Engel groups, elementary amenable groups., verbal dynamics, wreath products, Universal law, solvable groups, polycyclic groups, Grigorchuk group, verbal map, torsion groups.
Mathematics Subject Classification: Primary: 20E10, 20F69 ; Secondary: 20E22, 43A07, 20F16, 20F18, 20F19, 20F45, 20F5.
Citation: Anna Erschler. Iterated identities and iterational depth of groups. Journal of Modern Dynamics, 2015, 9: 257-284. doi: 10.3934/jmd.2015.9.257
References:
[1]

M. Abért, Group laws and free subgroups in topological groups, Bull. London Math. Soc., 37 (2005), 525-534. doi: 10.1112/S002460930500425X.  Google Scholar

[2]

S. I. Adjan, Infinite irreducible systems of group identities, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 34 (1970), 715-734.  Google Scholar

[3]

S. I. Adyan, Problema Bernsaĭda i tozhdestva v gruppakh, (Russian) Izdat. "Nauka'', Moscow, 1975.  Google Scholar

[4]

S. V. Alešin, Finite automata and the Burnside problem for periodic groups, (Russian) Mat. Zametki, 11 (1972), 319-328.  Google Scholar

[5]

T. Bandman, G.-M. Greuel, F. Grunewald, B. Kunyavskiĭ, G. Pfister and E. Plotkin, Identities for finite solvable groups and equations in finite simple groups, Compos. Math., 142 (2006), 734-764. doi: 10.1112/S0010437X0500179X.  Google Scholar

[6]

T. Bandman, F. Grunewald and B. Kunyavskiĭ, Geometry and arithmetic of verbal dynamical systems on simple groups, With an appendix by Nathan Jones, Groups Geom. Dyn., 4 (2010), 607-655. doi: 10.4171/GGD/98.  Google Scholar

[7]

T. Bandman, S. Garion and F. Grunewald, On the surjectivity of Engel words on $PSL(2,q)$, Groups Geom. Dyn., 6 (2012), 409-439. doi: 10.4171/GGD/162.  Google Scholar

[8]

L. Bartholdi, R. Grigorchuk and V. Nekrashevych, From fractal groups to fractal sets, in Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, 25-118.  Google Scholar

[9]

R. Brandl and J. S. Wilson, Characterization of finite soluble groups by laws in a small number of variables, J. Algebra, 116 (1988), 334-341. doi: 10.1016/0021-8693(88)90221-9.  Google Scholar

[10]

J. N. Bray, J. S. Wilson and R. A. Wilson, A characterization of finite soluble groups by laws in two variables, Bull. London Math. Soc., 37 (2005), 179-186. doi: 10.1112/S0024609304003959.  Google Scholar

[11]

C. Chou, Elementary amenable groups, Illinois J. Math., 24 (1980), 396-407.  Google Scholar

[12]

E. S. Golod, Some problems of Burnside type, (Russian) in Proc. Internat. Congr. Math. (Moscow, 1966), Izdat. "Mir'', Moscow, 1968, 284-289.  Google Scholar

[13]

R. I. Grigorčuk, On Burnside's problem on periodic groups, (Russian) Funktsional. Anal. i Prilozhen., 14 (1980), 53-54.  Google Scholar

[14]

R. I. Grigorchuk, Branch groups, (Russian) Mat. Zametki, 67 (2000), 852-858; translation in Math. Notes, 67 (2000), 718-723. doi: 10.1007/BF02675625.  Google Scholar

[15]

L. Bartholdi and R. I. Grigorchuk, On parabolic subgroups and Hecke algebras of some fractal groups, Serdica Math. J., 28 (2002), 47-90.  Google Scholar

[16]

M. Gromov, Hyperbolic groups, in Essays in Group Theory, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987, 75-263. doi: 10.1007/978-1-4613-9586-7_3.  Google Scholar

[17]

N. Gupta and S. Sidki, On the Burnside problem for periodic groups, Math. Z., 182 (1983), 385-388. doi: 10.1007/BF01179757.  Google Scholar

[18]

R. Guralnick, E. Plotkin and A. Shalev, Burnside-type problems related to solvability, Internat. J. Algebra Comput., 17 (2007), 1033-1048. doi: 10.1142/S0218196707003962.  Google Scholar

[19]

P. Hall, Finiteness conditions for soluble groups, Proc. London Math. Soc. (3), 4 (1954), 419-436.  Google Scholar

[20]

P. Hall, The Edmonton notes on nilpotent groups, Queen Mary College Mathematics Notes, Mathematics Department, Queen Mary College, London, 1969.  Google Scholar

[21]

W. Magnus, Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring, (German) Math. Ann., 111 (1935), 259-280. doi: 10.1007/BF01472217.  Google Scholar

[22]

V. Nekrashevych, Self-Similar Groups, Mathematical Surveys and Monographs, 117, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/surv/117.  Google Scholar

[23]

D. V. Osin, Elementary classes of groups, (Russian) Mat. Zametki, 72 (2002), 84-93; translation in Math. Notes, 72 (2002), 75-82. doi: 10.1023/A:1019869105364.  Google Scholar

[24]

E. L. Pervova, Everywhere dense subgroups of a group of tree automorphisms, (Russian) Tr. Mat. Inst. Steklova, 231 (2000), Din. Sist., Avtom. i Beskon. Gruppy, 356-367; translation in Proc. Steklov Inst. Math., (2000), 339-350.  Google Scholar

[25]

B. I. Plotkin, Notes on Engel groups and Engel elements in groups. Some generalizations, Izv. Ural. Gos. Univ. Mat. Mekh., 7(36) (2005), 153-166, 192-193.  Google Scholar

[26]

E. Ribnere, Sequences of words characterizing finite solvable groups, Monatsh. Math., 157 (2009), 387-401. doi: 10.1007/s00605-008-0034-6.  Google Scholar

[27]

J. G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc., 74 (1968), 383-437. doi: 10.1090/S0002-9904-1968-11953-6.  Google Scholar

[28]

J. G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable. IV, V, VI, Pacific J. Math., 48 (1973), 511-592, ibid. 50 (1974), 215-297, ibid. 51 (1974), 573-630. doi: 10.2140/pjm.1973.48.511.  Google Scholar

[29]

J. S. Wilson, Two-generator conditions for residually finite groups, Bull. London Math. Soc., 23 (1991), 239-248. doi: 10.1112/blms/23.3.239.  Google Scholar

[30]

E. I. Zel'manov, Solution of the restricted Burnside problem for $2$-groups, (Russian) Mat. Sb., 182 (1991), 568-592; translation in Math. USSR-Sb., 72 (1992), 543-565.  Google Scholar

show all references

References:
[1]

M. Abért, Group laws and free subgroups in topological groups, Bull. London Math. Soc., 37 (2005), 525-534. doi: 10.1112/S002460930500425X.  Google Scholar

[2]

S. I. Adjan, Infinite irreducible systems of group identities, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 34 (1970), 715-734.  Google Scholar

[3]

S. I. Adyan, Problema Bernsaĭda i tozhdestva v gruppakh, (Russian) Izdat. "Nauka'', Moscow, 1975.  Google Scholar

[4]

S. V. Alešin, Finite automata and the Burnside problem for periodic groups, (Russian) Mat. Zametki, 11 (1972), 319-328.  Google Scholar

[5]

T. Bandman, G.-M. Greuel, F. Grunewald, B. Kunyavskiĭ, G. Pfister and E. Plotkin, Identities for finite solvable groups and equations in finite simple groups, Compos. Math., 142 (2006), 734-764. doi: 10.1112/S0010437X0500179X.  Google Scholar

[6]

T. Bandman, F. Grunewald and B. Kunyavskiĭ, Geometry and arithmetic of verbal dynamical systems on simple groups, With an appendix by Nathan Jones, Groups Geom. Dyn., 4 (2010), 607-655. doi: 10.4171/GGD/98.  Google Scholar

[7]

T. Bandman, S. Garion and F. Grunewald, On the surjectivity of Engel words on $PSL(2,q)$, Groups Geom. Dyn., 6 (2012), 409-439. doi: 10.4171/GGD/162.  Google Scholar

[8]

L. Bartholdi, R. Grigorchuk and V. Nekrashevych, From fractal groups to fractal sets, in Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, 25-118.  Google Scholar

[9]

R. Brandl and J. S. Wilson, Characterization of finite soluble groups by laws in a small number of variables, J. Algebra, 116 (1988), 334-341. doi: 10.1016/0021-8693(88)90221-9.  Google Scholar

[10]

J. N. Bray, J. S. Wilson and R. A. Wilson, A characterization of finite soluble groups by laws in two variables, Bull. London Math. Soc., 37 (2005), 179-186. doi: 10.1112/S0024609304003959.  Google Scholar

[11]

C. Chou, Elementary amenable groups, Illinois J. Math., 24 (1980), 396-407.  Google Scholar

[12]

E. S. Golod, Some problems of Burnside type, (Russian) in Proc. Internat. Congr. Math. (Moscow, 1966), Izdat. "Mir'', Moscow, 1968, 284-289.  Google Scholar

[13]

R. I. Grigorčuk, On Burnside's problem on periodic groups, (Russian) Funktsional. Anal. i Prilozhen., 14 (1980), 53-54.  Google Scholar

[14]

R. I. Grigorchuk, Branch groups, (Russian) Mat. Zametki, 67 (2000), 852-858; translation in Math. Notes, 67 (2000), 718-723. doi: 10.1007/BF02675625.  Google Scholar

[15]

L. Bartholdi and R. I. Grigorchuk, On parabolic subgroups and Hecke algebras of some fractal groups, Serdica Math. J., 28 (2002), 47-90.  Google Scholar

[16]

M. Gromov, Hyperbolic groups, in Essays in Group Theory, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987, 75-263. doi: 10.1007/978-1-4613-9586-7_3.  Google Scholar

[17]

N. Gupta and S. Sidki, On the Burnside problem for periodic groups, Math. Z., 182 (1983), 385-388. doi: 10.1007/BF01179757.  Google Scholar

[18]

R. Guralnick, E. Plotkin and A. Shalev, Burnside-type problems related to solvability, Internat. J. Algebra Comput., 17 (2007), 1033-1048. doi: 10.1142/S0218196707003962.  Google Scholar

[19]

P. Hall, Finiteness conditions for soluble groups, Proc. London Math. Soc. (3), 4 (1954), 419-436.  Google Scholar

[20]

P. Hall, The Edmonton notes on nilpotent groups, Queen Mary College Mathematics Notes, Mathematics Department, Queen Mary College, London, 1969.  Google Scholar

[21]

W. Magnus, Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring, (German) Math. Ann., 111 (1935), 259-280. doi: 10.1007/BF01472217.  Google Scholar

[22]

V. Nekrashevych, Self-Similar Groups, Mathematical Surveys and Monographs, 117, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/surv/117.  Google Scholar

[23]

D. V. Osin, Elementary classes of groups, (Russian) Mat. Zametki, 72 (2002), 84-93; translation in Math. Notes, 72 (2002), 75-82. doi: 10.1023/A:1019869105364.  Google Scholar

[24]

E. L. Pervova, Everywhere dense subgroups of a group of tree automorphisms, (Russian) Tr. Mat. Inst. Steklova, 231 (2000), Din. Sist., Avtom. i Beskon. Gruppy, 356-367; translation in Proc. Steklov Inst. Math., (2000), 339-350.  Google Scholar

[25]

B. I. Plotkin, Notes on Engel groups and Engel elements in groups. Some generalizations, Izv. Ural. Gos. Univ. Mat. Mekh., 7(36) (2005), 153-166, 192-193.  Google Scholar

[26]

E. Ribnere, Sequences of words characterizing finite solvable groups, Monatsh. Math., 157 (2009), 387-401. doi: 10.1007/s00605-008-0034-6.  Google Scholar

[27]

J. G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc., 74 (1968), 383-437. doi: 10.1090/S0002-9904-1968-11953-6.  Google Scholar

[28]

J. G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable. IV, V, VI, Pacific J. Math., 48 (1973), 511-592, ibid. 50 (1974), 215-297, ibid. 51 (1974), 573-630. doi: 10.2140/pjm.1973.48.511.  Google Scholar

[29]

J. S. Wilson, Two-generator conditions for residually finite groups, Bull. London Math. Soc., 23 (1991), 239-248. doi: 10.1112/blms/23.3.239.  Google Scholar

[30]

E. I. Zel'manov, Solution of the restricted Burnside problem for $2$-groups, (Russian) Mat. Sb., 182 (1991), 568-592; translation in Math. USSR-Sb., 72 (1992), 543-565.  Google Scholar

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