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Iterated identities and iterational depth of groups
1. | Département de Mathématiques et Applications, École Normale Supérieure, 45 Rue d’Ulm, 75005 Paris, France |
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. |
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. |
[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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