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.
Similar content being viewed by others
References
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).
Gruenberg K. W., “Residual properties of infinite soluble groups,” Proc. Lond. Math. Soc., vol. 7, 29–62 (1957).
Sokolov E. V., “A characterization of root classes of groups,” Comm. Algebra, vol. 43, 856–860 (2015).
Azarov D. N. and Tieudjo D., “On root-class residuality of amalgamated free products,” Nauch. Tr. Ivanovsk. Gos. Univ., vol. 5, 6–10 (2002).
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).
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).
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).
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).
Tumanova E. A., “On the residual π-finiteness of generalized free products of groups,” Math. Notes, vol. 95, no. 4, 544–551 (2014).
Tieudjo D., “On root-class residuality of some free constructions,” JP J. Algebra, Number Theory Appl., vol. 18, no. 2, 125–143 (2010).
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).
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).
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).
Tumanova E. A., “Residual p-finiteness of some HNN-extensions of groups,” Model. Anal. Inform. Sist., vol. 21, no. 4, 148–180 (2014).
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).
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).
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).
Tumanova E. A., “The root class residuality of Baumslag-Solitar groups,” Sib. Math. J., vol. 58, no. 3, 546–552 (2017).
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).
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).
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).
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).
Appel K. and Schupp P., “Artin groups and infinite Coxeter groups,” Invent. Math., vol. 72, 201–220 (1983).
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).
Magnus W., “Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring,” Math. Ann., vol. 111, 259–280 (1935).
Magnus W., Karrass A., and Solitar D., Combinatorial Group Theory, Dover Publications, Mineola (2004).
Neumann H., “Generalized free products with amalgamated subgroups. II,” Amer. J. Math., vol. 31, 491–540 (1949).
Karras A. and Solitar D., “Subgroups of HNN groups and groups with one defining relation,” Can. J. Math., vol. 23, 627–643 (1971).
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
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).
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446619040153