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CONJUGACY SEPARABILITY OF GENERALIZED FREE PRODUCTS OF FINITELY GENERATED NILPOTENT GROUPS

  • Zhou, Wei (SCHOOL OF MATHEMATICS AND STATISTICS SOUTHWEST UNIVERSITY) ;
  • Kim, Goan-Su (DEPARTMENT OF MATHEMATICS YEUNGNAM UNIVERSITY) ;
  • Shi, Wujie (DEPARTMENT OF MATHEMATICS AND STATISTICS CHONGQING UNIVERSITY OF ARTS AND SCIENCES) ;
  • Tang, C.Y. (DEPARTMENT OF PURE MATHEMATICS UNIVERSITY OF WATERLOO)
  • Received : 2009.04.09
  • Published : 2010.11.30

Abstract

In this paper, we prove a criterion of conjugacy separability of generalized free products of polycyclic-by-finite groups with a non cyclic amalgamated subgroup. Applying this criterion, we prove that certain generalized free products of polycyclic-by-finite groups are conjugacy separable.

Keywords

generalized free products;residually finite;conjugacy separable;nilpotent groups

References

  1. R. B. J. T. Allenby, G. Kim, and C. Y. Tang, Conjugacy separability of certain Seifert 3-manifold groups, J. Algebra 285 (2005), no. 2, 481-507. https://doi.org/10.1016/j.jalgebra.2004.10.022
  2. G. Baumslag, A non-Hopfian group, Bull. Amer. Math. Soc. 68 (1962), 196-198. https://doi.org/10.1090/S0002-9904-1962-10743-5
  3. G. Baumslag, On the residual finiteness of generalised free products of nilpotent groups, Trans. Amer. Math. Soc. 106 (1963), 193-209. https://doi.org/10.1090/S0002-9947-1963-0144949-8
  4. N. Blackburn, Conjugacy in nilpotent groups, Proc. Amer. Math. Soc. 16 (1965), 143-148. https://doi.org/10.1090/S0002-9939-1965-0172925-5
  5. J. L. Dyer, Separating conjugates in amalgamated free products and HNN extensions, J. Austral. Math. Soc. Ser. A 29 (1980), no. 1, 35-51. https://doi.org/10.1017/S1446788700020917
  6. 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. https://doi.org/10.1090/conm/109/1076372
  7. 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
  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. W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory, Wiley-Interscience, New York-London-Sydney, 1966.
  10. A. I. Mal'cev, On homomorphisms onto finite groups, Amer. Math. Soc. Transl. 119 (1983), no. 2, 67-79.
  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 Appl. Algebra 78 (1992), no. 1, 77-84. https://doi.org/10.1016/0022-4049(92)90019-C
  14. L. Ribes, D. Segal, and P. A. Zalesskii, Conjugacy separability and free products of groups with cyclic amalgamation, J. London Math. Soc. (2) 57 (1998), no. 3, 609-628. https://doi.org/10.1112/S0024610798006267
  15. P. Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17 (1978), no. 3, 555-565. https://doi.org/10.1112/jlms/s2-17.3.555