Lobachevskii Journal of Mathematics

, Volume 39, Issue 2, pp 281–285 | Cite as

A Necessary Condition for The Residual Nilpotence of HNN-Extensions

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Abstract

Let G be amultiple HNN-extension of a group A, and let all its associated subgroups be properly contained in some locally nilpotent subgroup of A. We prove that if G is residually nilpotent, then all the associated subgroups are p′-isolated in A for some prime p. Moreover, if q is a prime such that G is residually a q′-torsion-free nilpotent group, then p = q.

Keywords and phrases

multiple HNN-extension residual nilpotence residual p-finiteness 

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References

  1. 1.
    R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory (Springer, Berlin, Heidelberg, New York, 1977).MATHGoogle Scholar
  2. 2.
    W. Magnus, “Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring,” Math. Ann. 111, 259–280 (1935).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    K.W. Gruenberg, “Residual properties of infinite soluble groups,” Proc. LondonMath. Soc., Ser. 3 7, 29–62 (1957).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    E. Raptis and D. Varsos, “Residual properties ofHNN-extensions with base group an Abelian group,” J. Pure Appl. Algebra 59, 285–290 (1989).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    E. Raptis and D. Varsos, “The residual nilpotence of HNN-extensions with base group a finite or a f. g. Abelian group,” J. Pure Appl. Algebra 76, 167–178 (1991).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    D. Varsos, “The residual nilpotence of the fundamental group of certain graphs of groups,” Houston J.Math. 22, 233–248 (1996).MathSciNetMATHGoogle Scholar
  7. 7.
    D. I. Moldavanskii, “The residual p-finiteness of HNN-extensions,” Vestnik Ivanovsk. Gos. Univ., Ser. Biol. Khim. Fiz. Mat., No. 3, 129–140 (2000).Google Scholar
  8. 8.
    D. I. Moldavanskii, “On the residuality a finite p-group of HNN-extensions,” arXiv: math/0701498.Google Scholar
  9. 9.
    M. Aschenbrenner and S. Friedl, “A criterion forHNNextensions of finite p-groups to be residually p,” J. Pure Appl. Algebra 215, 2280–2289 (2011).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    D. I. Moldavanskii, “The residual p-finiteness of some HNN-extensions of groups,” Vestnik Ivanovsk. Gos. Univ., Ser. Biol. Khim. Fiz.Mat., No. 3, 102–116 (2003).Google Scholar
  11. 11.
    D. I. Moldavanskii, “On the residual p-finiteness of HNN-extensions of nilpotent groups,” Vestnik Ivanovsk. Gos. Univ., Ser. Biol. Khim. Fiz.Mat., (3), 128–132 (2006).Google Scholar
  12. 12.
    G. Baumslag, “On the residual finiteness of generalized free products of nilpotent groups,” Trans. Am.Math. Soc. 106, 193–209 (1963).CrossRefMATHGoogle Scholar
  13. 13.
    M. Aschenbrenner and S. Friedl, 3-Manifold Groups are Virtually Residually p, Vol. 225 of Memoirs of Am. Math. Soc. (Am.Math. Soc., Providence, RI, 2013).MATHGoogle Scholar
  14. 14.
    G. Baumslag and D. Solitar, “Some two-generator one-relator non-Hopfian groups,” Bull. Am. Math. Soc. 68, 199–201 (1962).MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    A.M. Brunner, “On a class of one-relator groups,” Can. J. Math. 32, 414–420 (1980).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    M. Aschenbrenner and S. Friedl, “Residual properties of graph manifold groups,” Topol. Appl. 158, 1179–1191 (2011).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    D. N. Azarov and E. A. Ivanova, “On the problem of the residual nilpotence of a free product with an amalgamation of locally nilpotent groups,” Tr. Ivanovsk. Gos. Univ.Mat. 2, 5–7 (1999).Google Scholar
  18. 18.
    G. Baumslag, “On the residual nilpotence of certain one-relator groups,” Commun. Pure Appl. Math. 21, 491–506 (1968).MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Ivanovo State UniversityIvanovoRussia

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