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Siberian Mathematical Journal

, Volume 58, Issue 3, pp 546–552 | Cite as

The root class residuality of Baumslag–Solitar groups

  • E. A. TumanovaEmail author
Article

Abstract

Given a homomorphically closed root class K of groups, we find a criterion for a Baumslag–Solitar group to be a residually K-group. In particular, we establish that all Baumslag–Solitar groups are residually soluble and a Baumslag–Solitar group is residually finite soluble if and only if it is residually finite.

Keywords

root class residuality residual solubility residual π-finiteness Baumslag–Solitar groups HNN-extension 

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References

  1. 1.
    Mal’tsev A. I., “On isomorphic matrix representations of infinite groups,” Mat. Sb., vol. 8, no. 3, 405–422 (1940).MathSciNetzbMATHGoogle Scholar
  2. 2.
    Andreadakis S., Raptis E., and Varsos D., “Residual finiteness and Hopficity of certain HNN extensions,” Arch. Math., vol. 47, 1–5 (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baumslag G. and Solitar D., “Some two-generator one-relator non-Hopfian groups,” Bull. Amer. Math. Soc., vol. 68, 199–201 (1962).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Moldavanskiĭ D. I., “On the isomorphism of the Baumslag–Solitar groups,” Ukr. Mat. Zh., vol. 43, no. 12, 1684–1686 (1991).MathSciNetGoogle Scholar
  5. 5.
    Karrass A. and Solitar D., “Subgroups of HNN groups and groups with one defining relation,” Canad. J. Math., vol. 23, no. 4, 627–643 (1971).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Meskin S., “Non-residually finite one-relator groups,” Trans. Amer. Math. Soc., vol. 164, 105–114 (1972).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Moldavanskiĭ D. I., “Approximability of HNN-extensions by finite p-groups,” Vestnik Ivanovsk. Gos. Univ. Ser. Biol., Khim., Fiz., Mat., no. 3, 129–140 (2000).Google Scholar
  8. 8.
    Varlamova I. A. and Moldavanskiĭ D. I., “Approximability of Baumslag–Solitar groups by finite groups,” Vestnik Ivanovsk. Gos. Univ. Ser. Estestv., Obshchestv. Nauki, no. 2, 107–114 (2012).Google Scholar
  9. 9.
    Ivanova O. A. and Moldavanskiĭ D. I., “Approximability of some groups with one defining relation by finite π-groups,” Nauch. Tr. Ivanovsk. Gos. Univ. Mat., vol. 6, 51–58 (2008).Google Scholar
  10. 10.
    Moldavanskiĭ D. I., “The intersection of the subgroups of finite index in Baumslag–Solitar groups,” Math. Notes, vol. 87, no. 1, 88–95 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Moldavanskiĭ D. I., “The intersection of the subgroups of finite p-index in Baumslag–Solitar groups,” Vestnik Ivanovsk. Gos. Univ. Ser. Estestv., Obshchestv. Nauki, no. 2, 106–111 (2010).zbMATHGoogle Scholar
  12. 12.
    Moldavanskii D., “On some residual properties of Baumslag–Solitar groups,” arXiv:1310.3585 [math.GR], 2013.zbMATHGoogle Scholar
  13. 13.
    Gruenberg K. W., “Residual properties of infinite soluble groups,” Proc. London Math. Soc., vol. 7, 29–62 (1957).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Sokolov E. V., “A characterization of root classes of groups,” Comm. Algebra, vol. 43, 856–860 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gol’tsov D. V. and Yatskin N. I., “Classes of groups and subgroup topologies,” Vestnik Ivanovsk. Gos. Univ. Ser. Estestv., Obshchestv. Nauki, no. 2, 115–128 (2011).Google Scholar
  16. 16.
    Baumslag G., “Positive one-relator groups,” Trans. Amer. Math. Soc., vol. 156, 165–183 (1971).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Azarov D. N. and Tieudjo D., “On root-class residuality of amalgamated free products,” Nauch. Tr. Ivanovsk. Gos. Univ., vol. 5, 6–10 (2002).Google Scholar
  18. 18.
    Tumanova E. A., “On the root-class residuality of generalized free products with a normal amalgamation,” Russian Math. (Iz. VUZ), vol. 59, no. 10, 23–37 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mal’tsev A. I., “On homomorphisms onto finite groups,” Ivanovo Gos. Ped. Inst. Uchen. Zap., vol. 18, no. 5, 49–60 (1958).Google Scholar
  20. 20.
    Tieudjo D., “Root-class residuality of some free constructions,” J. Algebra, Number Theory, Appl., vol. 18, no. 2, 125–143 (2010).MathSciNetzbMATHGoogle Scholar
  21. 21.
    Magnus W., Karrass A., and Solitar D., Combinatorial Group Theory, Dover Publications, Mineola (2004).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Ivanovo State UniversityIvanovoRussia

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