# American Institute of Mathematical Sciences

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

## Iterated identities and iterational depth of groups

 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.
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. [2] S. I. Adjan, Infinite irreducible systems of group identities, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 34 (1970), 715-734. [3] S. I. Adyan, Problema Bernsaĭda i tozhdestva v gruppakh, (Russian) Izdat. "Nauka'', Moscow, 1975. [4] S. V. Alešin, Finite automata and the Burnside problem for periodic groups, (Russian) Mat. Zametki, 11 (1972), 319-328. [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. [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. [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. [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. [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. [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. [11] C. Chou, Elementary amenable groups, Illinois J. Math., 24 (1980), 396-407. [12] E. S. Golod, Some problems of Burnside type, (Russian) in Proc. Internat. Congr. Math. (Moscow, 1966), Izdat. "Mir'', Moscow, 1968, 284-289. [13] R. I. Grigorčuk, On Burnside's problem on periodic groups, (Russian) Funktsional. Anal. i Prilozhen., 14 (1980), 53-54. [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. [15] L. Bartholdi and R. I. Grigorchuk, On parabolic subgroups and Hecke algebras of some fractal groups, Serdica Math. J., 28 (2002), 47-90. [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. [17] N. Gupta and S. Sidki, On the Burnside problem for periodic groups, Math. Z., 182 (1983), 385-388. doi: 10.1007/BF01179757. [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. [19] P. Hall, Finiteness conditions for soluble groups, Proc. London Math. Soc. (3), 4 (1954), 419-436. [20] P. Hall, The Edmonton notes on nilpotent groups, Queen Mary College Mathematics Notes, Mathematics Department, Queen Mary College, London, 1969. [21] W. Magnus, Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring, (German) Math. Ann., 111 (1935), 259-280. doi: 10.1007/BF01472217. [22] V. Nekrashevych, Self-Similar Groups, Mathematical Surveys and Monographs, 117, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/surv/117. [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. [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. [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. [26] E. Ribnere, Sequences of words characterizing finite solvable groups, Monatsh. Math., 157 (2009), 387-401. doi: 10.1007/s00605-008-0034-6. [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. [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. [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. [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.

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##### References:
 [1] M. Abért, Group laws and free subgroups in topological groups, Bull. London Math. Soc., 37 (2005), 525-534. doi: 10.1112/S002460930500425X. [2] S. I. Adjan, Infinite irreducible systems of group identities, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 34 (1970), 715-734. [3] S. I. Adyan, Problema Bernsaĭda i tozhdestva v gruppakh, (Russian) Izdat. "Nauka'', Moscow, 1975. [4] S. V. Alešin, Finite automata and the Burnside problem for periodic groups, (Russian) Mat. Zametki, 11 (1972), 319-328. [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. [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. [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. [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. [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. [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. [11] C. Chou, Elementary amenable groups, Illinois J. Math., 24 (1980), 396-407. [12] E. S. Golod, Some problems of Burnside type, (Russian) in Proc. Internat. Congr. Math. (Moscow, 1966), Izdat. "Mir'', Moscow, 1968, 284-289. [13] R. I. Grigorčuk, On Burnside's problem on periodic groups, (Russian) Funktsional. Anal. i Prilozhen., 14 (1980), 53-54. [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. [15] L. Bartholdi and R. I. Grigorchuk, On parabolic subgroups and Hecke algebras of some fractal groups, Serdica Math. J., 28 (2002), 47-90. [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. [17] N. Gupta and S. Sidki, On the Burnside problem for periodic groups, Math. Z., 182 (1983), 385-388. doi: 10.1007/BF01179757. [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. [19] P. Hall, Finiteness conditions for soluble groups, Proc. London Math. Soc. (3), 4 (1954), 419-436. [20] P. Hall, The Edmonton notes on nilpotent groups, Queen Mary College Mathematics Notes, Mathematics Department, Queen Mary College, London, 1969. [21] W. Magnus, Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring, (German) Math. Ann., 111 (1935), 259-280. doi: 10.1007/BF01472217. [22] V. Nekrashevych, Self-Similar Groups, Mathematical Surveys and Monographs, 117, American Mathematical Society, Providence, RI, 2005. doi: 10.1090/surv/117. [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. [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. [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. [26] E. Ribnere, Sequences of words characterizing finite solvable groups, Monatsh. Math., 157 (2009), 387-401. doi: 10.1007/s00605-008-0034-6. [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. [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. [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. [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.
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