Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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CONJUGACY SEPARABILITY OF CERTAIN GENERALIZED FREE PRODUCTS OF NILPOTENT GROUPS

  • Kim, Goansu (Department of Mathematics, Yeungnam University) ;
  • Tang, C.Y. (Department of Mathematics, University of Waterloo)
  • Received : 2011.07.04
  • Published : 2013.07.01
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Abstract

It is known that generalized free products of finitely generated nilpotent groups are conjugacy separable when the amalgamated subgroups are cyclic or central in both factor groups. However, those generalized free products may not be conjugacy separable when the amalgamated subgroup is a direct product of two infinite cycles. In this paper we show that generalized free products of finitely generated nilpotent groups are conjugacy separable when the amalgamated subgroup is ${\langle}h{\rangle}{\times}D$, where D is in the center of both factors.

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References

  1. G. Baumslag, A non-Hopfian group, Bull. Amer. Math. Soc. 68 (1962), 196-198. https://doi.org/10.1090/S0002-9904-1962-10743-5
  2. N. Blackburn, Conjugacy in nilpotent groups, Proc. Amer. Math. Soc. 16 (1965), 143-148. https://doi.org/10.1090/S0002-9939-1965-0172925-5
  3. J. L. Dyer, Separating conjugates in free-by-finite groups, J. London Math. Soc. (2) 20 (1979), no. 2, 215-221.
  4. J. L. Dyer, Separating conjugates in amalgamated free products and HNN extensions, J. Austral. Math. Soc. Ser. A 29 (1980), no. 1, 35-51.
  5. B. Fine and G. Rosenberger, Conjugacy separability of Fuchsian groups and related questions, Combinatorial group theory (College Park, MD, 1988), 11-18, Contemp. Math., 109, Amer. Math. Soc., Providence, RI, 1990.
  6. E. Formanek, Conjugate separability in polycyclic groups, J. Algebra 42 (1976), no. 1, 1-10. https://doi.org/10.1016/0021-8693(76)90021-1
  7. G. Kim and C. Y. Tang, Conjugacy separability of generalized free products of finite extensions of residually nilpotent groups, In Group Theory (Proc. of the '96 Beijing Int'l Symposium), pages 10-24. Springer-Verlag, 1998.
  8. G. Kim and C. Y. Tang, Separability properties of certain tree products of groups, J. Algebra 251 (2002), no. 1, 323-349. https://doi.org/10.1006/jabr.2001.9134
  9. John C. Lennox and John S. Wilson, On products of subgroups in polycyclic groups, Arch. Math. 33 (1979), no. 4, 305-309. https://doi.org/10.1007/BF01222760
  10. W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory, Pure and Applied Math. Vol. XIII, Wiley-Interscience, New York-London-Sydney, 1966.
  11. A. I. Mal'cev, Homomorphisms of finite groups, Ivanov Gos. Ped. Inst. Ucen. Zap. Uchen. Zap. Karel. Ped. Inst. Ser. Fiz.-Mat. Nauk 18 (1958), 49-60.
  12. A. W. Mostowski, On the decidability of some problems in special classes of groups, Fund. Math. 59 (1966), 123-135. https://doi.org/10.4064/fm-59-2-123-135
  13. G. A. Niblo, Separability properties of free groups and surface groups, J. Pure and Appl. Algebra 78 (1992), no. 1, 77-84. https://doi.org/10.1016/0022-4049(92)90019-C
  14. V. N. Remeslennikov, Conjugacy in polycyclic groups, Algebra Log. 8 (1969), 712-725.
  15. V. N. Remeslennikov, Groups that are residually finite with respect to conjugacy, Siberian Math. J. 12 (1971), 783-792.
  16. L. Ribes, D. Segal, and P. A. Zalesskii, Conjugacy separability and free products of groups with cyclic amalgamation, J. London Math. Soc. 57 (1998), no. 3, 609-628. https://doi.org/10.1112/S0024610798006267
  17. L. Ribes and P. A. Zalesskii, On the profinite topology on a free group, Bull. London Math. Soc. 25 (1993), no. 1, 37-43. https://doi.org/10.1112/blms/25.1.37
  18. P. Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. 17 (1978), no. 3, 555-565.
  19. P. F. Stebe, A residual property of certain groups, Proc. Amer. Math. Soc. 26 (1970), 37-42. https://doi.org/10.1090/S0002-9939-1970-0260874-5
  20. C. Y. Tang, Conjugacy separability of generalized free products of surface groups, J. Pure Appl. Algebra 120 (1997), no. 2, 187-194. https://doi.org/10.1016/S0022-4049(96)00060-6
  21. J. S. Wilson and P. A. Zalesskii, Conjugacy separability of certain Bianchi groups and HNN extensions, Math. Proc. Cambridge Philos. Soc. 123 (1998), no. 2, 227-242. https://doi.org/10.1017/S0305004197002193