Mathematical Notes

, 86:457 | Cite as

Monomorphisms of free Burnside groups

  • V. S. AtabekyanEmail author


In the paper, it is proved that, for any odd n ≥ 1039, there are words u(x, y) and υ(x, y) over the group alphabet {x, y} such that, if a and b are any two noncommuting elements of the free Burnside group B(m, n), then, for some k, the elements u(a k , b) and υ(a k , b) freely generate a free Burnside subgroup of the group B(m, n). In particular, the facts proved in the paper imply the uniform nonamenability of the group B(m, n) for odd n, n ≥ 1039.

Key words

absolutely free group free Burnside group uniformly nonamenable group residually finite group 2-generated subgroup Tarski monster Hopfian group 


  1. 1.
    S. I. Adyan [Adjan, Adian], The Burnside Problem and Identities in Groups (Nauka, Moscow, 1975; Springer-Verlag, Berlin-New York, 1979).Google Scholar
  2. 2.
    P. S. Novikov and S. I. Adyan [Adjan], “Infinite periodic groups. III,” Izv. Akad. Nauk SSSR Ser. Mat. 32(3), 709–731 (1968) [Math. USSR-Izv. 2, 665–685 (1968)].MathSciNetGoogle Scholar
  3. 3.
    S. I. Adyan, “Studies in the Burnside problem and related questions,” in Trudy Mat. Inst. Steklov, Vol. 168: Algebra, Mathematical Logic, Number Theory, Topology, Collection of survey papers, 1 (Nauka, Moscow, 1984), pp. 171–196 [in Russian].Google Scholar
  4. 4.
    S. I. Adyan, “Random walks on free periodic groups,” Izv. Akad. Nauk SSSR Ser. Mat. 46(6), 1139–1149 (1982) [Math. USSR-Izv. 21 (3), 425–434 (1983)].zbMATHMathSciNetGoogle Scholar
  5. 5.
    G. N. Arzhantseva, J. Burillo, M. Lustig, L. Reeves, H. Short, and E. Ventura, “Uniform non-amenability,” Adv. Math. 197(2), 499–522 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    J. von Neumann, “Zur allgemeinen Theorie des Masses,” Fundam. Math. 13, 73–116 (1929).zbMATHGoogle Scholar
  7. 7.
    S. V. Ivanov and A. Yu. Ol’shanskii, “Some applications of graded diagrams in combinatorial group theory,” in London Math. Soc. Lecture Note Ser., Vol. 160: Groups — St. Andrews, 1989, Vol. 2 (Cambridge Univ. Press, Cambridge, 1991), pp. 258–308.Google Scholar
  8. 8.
    D. V. Osin, “Uniform non-amenability of free Burnside groups,” Arch. Math. (Basel) 88(5), 403–412 (2007).zbMATHMathSciNetGoogle Scholar
  9. 9.
    V. S. Atabekyan, “Uniform nonamenability of subgroups of free Burnside groups of odd period,” Mat. Zametki 85(4), 516–523 (2009) [Math. Notes 85 (3–4), 496–502 (2009)].Google Scholar
  10. 10.
    H. Neumann, Varieties of Groups (Springer-Verlag New York, Inc., New York, 1967; Mir, Moscow, 1969).zbMATHGoogle Scholar
  11. 11.
    A. Yu. Ol’shanskii, “Self-normalization of free subgroups in the free Burnside groups,” in Groups, Rings, Lie and Hopf Algebras, Math. Appl., St. John’s, NF, 2001 (Kluwer Acad. Publ., Dordrecht, 2003), Vol. 555, pp. 179–187.Google Scholar
  12. 12.
    V. Atabekyan, “Normal subgroups in free Burnside groups of odd period,” Armen. J. Math. 1(2), 25–29 (2008).MathSciNetGoogle Scholar
  13. 13.
    M. F. Newman, “Problems,” in Lecture Notes in Math., Vol. 806: Burnside groups, Proc. Workshop, Univ. Bielefeld, Bielefeld, 1977 (Springer-Verlag, Berlin, 1980), pp. 249–254. Lecture Notes in Math.Google Scholar
  14. 14.
    S. I. Adyan, “Periodic Products of Groups,” in Trudy Mat. Inst. Steklov, Vol. 142: Number Theory, Mathematical Analysis and Their Applications, Collected papers dedicated to the 85th birthday of Academician Ivan Matveevich Vinogradov (Nauka, Moscow, 1976), pp. 3–21 [in Russian].Google Scholar
  15. 15.
    V. L. Shirvanyan, “Embedding of the group B(∞, n) in the group B(2, n),” Izv. Akad. Nauk SSSR Ser. Mat. 40(1), 190–208 (1976) [Math. USSR-Izv. 10 (1), 181–199 (1976)].zbMATHMathSciNetGoogle Scholar
  16. 16.
    V. S. Atabekyan, On Approximation and Subgroups of Free Periodic Groups, Available from VINITI, No. 5380-V86 (1981) [in Russian].Google Scholar
  17. 17.
    V. S. Atabekyan, “Simple and free periodic groups,” VestnikMoskov. Univ. Ser. I Mat. Mekh., No. 6, 76–78 (1987) [Moscow Univ. Math. Bull. 42 (6), 80–82 (1987)].Google Scholar
  18. 18.
    A. Yu. Ol’shanskii and M. V. Sapir, “Non-amenable finitely presented torsion-by-cyclic groups,” Publ. Math. Inst. Hautes Études Sci. 96, 43–169 (2002).zbMATHMathSciNetGoogle Scholar
  19. 19.
    D. Sonkin, “CEP-subgroups of free Burnside groups of large odd exponents,” Comm. Algebra 31(10), 4687–4695 (2003).zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    S. V. Ivanov, “On subgroups of free Burnside groups of large odd exponent,” Illinois J. Math. 47(1–2), 299–304 (2003).zbMATHMathSciNetGoogle Scholar
  21. 21.
    V. S. Atabekyan, “On free subgroups of periodic groups of odd period n ≥ 1003,” Izv. Ross. Akad. Nauk Ser. Mat. (in press).Google Scholar
  22. 22.
    S. I. Adyan and I. G. Lysenok, “Groups, all of whose proper subgroups are finite cyclic,” Izv. Akad. Nauk SSSR Ser. Mat. 55(5), 933–990 (1991) [Math. USSR-Izv. 39 (2), 905–957 (1992)].Google Scholar
  23. 23.
    P. M. Neumann and J. Wiegold, “Schreier varieties of groups,” Math. Z. 85, 392–400 (1964).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Yerevan State UniversityYerevanArmenia

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