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The Root Class Residuality of the Tree Product of Groups with Amalgamated Retracts

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Abstract

Given a root class \(\mathscr{K}\) of groups, we prove that the tree product of residually \(\mathscr{K}\)-groups with amalgamated retracts is a residually \(\mathscr{K}\)-group. This yields a criterion for the \(\mathscr{K}\)-residuality of Artin and Coxeter groups with tree structure. We also prove that the HNN-extension X of a residually \(\mathscr{K}\)-group B is a residually \(\mathscr{K}\)-group provided that the associated subgroups of X are retracts in B and \(\mathscr{K}\) contains at least one nonperiodic group.

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References

  1. Karras A. and Solitar D., “The subgroups of a free products of two groups with an amalgamated subgroup,” Trans. Amer. Math. Soc., vol. 150, 227–254 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  2. Gruenberg K. W., “Residual properties of infinite soluble groups,” Proc. Lond. Math. Soc., vol. 7, 29–62 (1957).

    Article  MathSciNet  MATH  Google Scholar 

  3. Sokolov E. V., “A characterization of root classes of groups,” Comm. Algebra, vol. 43, 856–860 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  4. 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 

  5. Raptis E. and Varsos D., “The residual finiteness of HNN-extensions and generalized free products of nilpotent groups: A characterization,” J. Austral. Math. Soc. Ser. A, vol. 53, 408–420 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  6. Azarov D. N., “On the residual nilpotence of free products of free groups with cyclic amalgamation,” Math. Notes, vol. 64, no. 1, 3–7 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  7. Aschenbrenner M. and Friedl S., “A criterion for HNN extensions of finite p-groups to be residually p,” J. Pure Appl. Algebra vol. 215, 2280–2289 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  8. Azarov D. N., “On the residual finiteness of the HNN-extensions and generalized free products of finite rank groups,” Sib. Math. J., vol. 54, no. 6, 959–967 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  9. Tumanova E. A., “On the residual π-finiteness of generalized free products of groups,” Math. Notes, vol. 95, no. 4, 544–551 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  10. Tieudjo D., “On root-class residuality of some free constructions,” JP J. Algebra, Number Theory Appl., vol. 18, no. 2, 125–143 (2010).

    MathSciNet  MATH  Google Scholar 

  11. Tumanova E. A., “On residuality of generalized free products by root classes of groups,” Model. Anal. Inform. Sist., vol. 20, no. 1, 133–137 (2013).

    Article  Google Scholar 

  12. Goltsov D. V., “On the virtual root-class residuality of generalized free products and HNN-extensions of groups,” Chebyshevskii Sb., vol. 14, no. 3, 34–41 (2013).

    MathSciNet  Google Scholar 

  13. Tumanova E. A., “Certain conditions of the root-class residuality of generalized free products with a normal amalgamated subgroups,” Chebyshevskii Sb., vol. 14, no. 3, 134–141 (2013).

    Google Scholar 

  14. Tumanova E. A., “Residual p-finiteness of some HNN-extensions of groups,” Model. Anal. Inform. Sist., vol. 21, no. 4, 148–180 (2014).

    Article  Google Scholar 

  15. Goltsov D. V., “Approximability of HNN-extensions with central associated subgroups by a root class of groups,” Math. Notes, vol. 97, no. 5, 679–683 (2015).

    Article  MathSciNet  Google Scholar 

  16. 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).

    Article  MathSciNet  MATH  Google Scholar 

  17. Sokolov E. V. and Tumanova E. A., “Sufficient conditions for the root-class residuality of certain generalized free products,” Sib. Math. J., vol. 57, no. 1, 135–144 (2016).

    Article  MATH  Google Scholar 

  18. Tumanova E. A., “The root class residuality of Baumslag-Solitar groups,” Sib. Math. J., vol. 58, no. 3, 546–552 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  19. Sokolov E. V. and Tumanova E. A., “Root class residuality of HNN-extensions with central cyclic associated subgroups,” Math. Notes, vol. 102, no. 4, 556–568 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  20. Boler J. and Evans B., “The free product of residually finite groups amalgamated along retracts is residually finite,” Proc. Amer. Math. Soc., vol. 37, no. 1, 50–52 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  21. Bobrovskii P. A. and Sokolov E. V., “The cyclic subgroup separability of certain generalized free products of two groups,” Algebra Colloq., vol. 17, no. 4, 577–582 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  22. Azarov D. N. and Tumanova E. A., “On the residuality of generalized free products by root classes of groups,” Nauch. Tr. Ivanovsk. Gos. Univ., no. 6, 29–42 (2008).

  23. Appel K. and Schupp P., “Artin groups and infinite Coxeter groups,” Invent. Math., vol. 72, 201–220 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  24. Bezverkhnii V. N. and Inchenko O. V., “On the word problem and the conjugacy problem in Coxeter groups with a tree structure,” Chebyshevskii Sb., vol. 6, no. 2, 81–90 (2005).

    MathSciNet  MATH  Google Scholar 

  25. Magnus W., “Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring,” Math. Ann., vol. 111, 259–280 (1935).

    Article  MathSciNet  MATH  Google Scholar 

  26. Magnus W., Karrass A., and Solitar D., Combinatorial Group Theory, Dover Publications, Mineola (2004).

    MATH  Google Scholar 

  27. Neumann H., “Generalized free products with amalgamated subgroups. II,” Amer. J. Math., vol. 31, 491–540 (1949).

    Article  MathSciNet  MATH  Google Scholar 

  28. Karras A. and Solitar D., “Subgroups of HNN groups and groups with one defining relation,” Can. J. Math., vol. 23, 627–643 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  29. Bezverkhnii V. N., “A solution of the conjugacy problem in Artin and Coxeter groups of large type,” in: Algorithmic Problems of the Theory of Groups and Subgroups [Russian], Tula State Lev Tolstoy Pedagogical University, Tula, 1986, 26–61.

    Google Scholar 

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Correspondence to E. A. Tumanova.

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Russian Text © The Author(s), 2019, published in Sibirskii Matematicheskii Zhurnal, 2019, Vol. 60, No. 4, pp. 891–906.

The author was supported by the Russian Foundation for Basic Research (Grant 18-31-00187).

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Tumanova, E.A. The Root Class Residuality of the Tree Product of Groups with Amalgamated Retracts. Sib Math J 60, 699–708 (2019). https://doi.org/10.1134/S0037446619040153

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  • DOI: https://doi.org/10.1134/S0037446619040153

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