Algebra pp 63-74 | Cite as

Around Automorphisms of Relatively Free Groups

  • C. K. Gupta
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

This article is intended to be a survey on some topics within the framework of automorphisms of free groups and relatively free groups of certain soluble varieties. The bibliography at the end is neither claimed to be exhaustive, nor it is necessarily connected with a reference in the text. I include it as 1 see it revolves around the concepts emerging from the investigation of automorphisms of free groups. The interested reader may find it useful to browse over the list occasionally.

Keywords

Free Group Automorphism Group Nilpotent Group Primitive Element Invariant Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References

  1. Andreadakis, S. On the automorphisms of free groups and free nilpotent groups, Proc. London Math. Soc. (3) 15, 239–269, 1965.MathSciNetMATHCrossRefGoogle Scholar
  2. Andreadakis, S. Generators for Aut G., G free nilpotent, Arch. Math. 42, 296–300, 1984.MathSciNetMATHCrossRefGoogle Scholar
  3. Andreadakis, S. and Gupta, C.K. Automorphisms of free metabelian nilpotent groups, Algebra i Logika, 29, 746–751, 1990.MathSciNetCrossRefGoogle Scholar
  4. Bachmuth, S. Automorphisms of free metabelian groups, Trans. Amen Math. Soc. 118, 93–104, 1965.MathSciNetMATHCrossRefGoogle Scholar
  5. Bachmuth, S. Induced automorphisms of free groups and free metabelian groups, Trans. Amer. Math. Soc. 122, 1–17, 1966.MathSciNetMATHCrossRefGoogle Scholar
  6. Bachmuth, S., Baumslag, G., Dyer, J. and Mochizuki, H.Y. Automorphism groups of two-generator metabelian groups, J. London Math. Soc. 36, 393–406, 1987.MathSciNetMATHCrossRefGoogle Scholar
  7. Bachmuth, S. and Mochizuki, H.Y. IA-automorphisms of free metabelian group of rank 3 J.Alg. 55, 106–115, 1979.MathSciNetCrossRefGoogle Scholar
  8. Bachmuth, S. and Mochizuki, H.Y. Aut (F) → Aut(F/F″) is surjective for free group F of rank ≥ 4, Trans. Amer. Math. Soc. 292, 81–101, 1985.MathSciNetMATHGoogle Scholar
  9. Bachmuth, S., Mochizuki, H.Y. The tame range of automorphism groups and GL n , Proc. Singapore Group Theory Conf. 241–251, de Gruyter, New York, 1987/89.Google Scholar
  10. Bachmuth, S. Foremanek, E. and Mochizuki, H.Y. IA-automorphisms of certain two generator torsion free groups, J. Alg. 40, 19–30, 1967.CrossRefGoogle Scholar
  11. Baumslag, G. Automorphism groups of residually finite groups, J. London Math. Soc. 38, 117–118, 1963.MathSciNetMATHCrossRefGoogle Scholar
  12. Birman, Joan S. An inverse function theorem for free groups, Proc. Amer. Math. Soc. 41, 634–638, 1974.MathSciNetCrossRefGoogle Scholar
  13. Bryant, R.M. and Groves, J.R.J. Automorphisms of free metabelian groups of infinite rank, Comm. Alg. 20, 783–814, 1992.MathSciNetMATHCrossRefGoogle Scholar
  14. Bryant, R.M. and Gupta, C.K. Automorphism groups of free nilpotent groups, Arch. Math, 52, 313–320, 1989.MathSciNetMATHCrossRefGoogle Scholar
  15. Bryant, R.M. and Gupta, C.K. Automorphisms of free nilpotent-by-abelian groups, Math. Proc. Comb. phil. Soc. 114, 143–147, 1993.MathSciNetMATHCrossRefGoogle Scholar
  16. Bryant, R.M., Gupta, C.K., Levin, F. and Mochizuki, H.Y. Non-tame automorphisms of free nilpotent groups, Comm. Alg. 18, 3619–3631, 1990.MathSciNetMATHCrossRefGoogle Scholar
  17. Bryant, Roger M. and Olga Macedonska, Automorphisms of relatively free nilpotent groups of infinite rank, J. Alg. 121, 388–398, 1989.MathSciNetMATHCrossRefGoogle Scholar
  18. Caranti, A. and Scoppola, C.M. Endomorphisms of two-generator metabelian groups that induce the identity modulo the derived subgroup, Arch. Math. 56, 218–227, 1991.MathSciNetMATHCrossRefGoogle Scholar
  19. Caranti, A. and Scoppola, C.M. Two-generator metabelian groups that have many IA- automorphisms, (1989 Preprint). Google Scholar
  20. Chein Orin, IA Automorphisms of free and free metabelian groups, Comm. pure appl Math. 21, 605–629, 1968.MathSciNetCrossRefGoogle Scholar
  21. Drensky, V. and Gupta, C.K. Automorphisms of free nilpotent Lie algebras, Canad. J. Math. 42, 259–279, 1990.MathSciNetMATHCrossRefGoogle Scholar
  22. Drensky, V. and Gupta, C.K. New automorphisms of generic matrix algebras and polynomial algebras, J. Alg. 194, 408–414, 1997.MathSciNetMATHCrossRefGoogle Scholar
  23. Evans, M.J. Presentations of the free metabelian group of rank 2, Canad. Math. Bull. 37, 468–472, 1994.MathSciNetMATHCrossRefGoogle Scholar
  24. Formanek, E. and Procesi, C The automorphism group of free group is not linear, J. Algebra 149, 494–499, 1992.MathSciNetMATHCrossRefGoogle Scholar
  25. Fox, R.H. Free differential caculus 1. Derivations in the free group ring, Ann. of Math. 57, 547–560, 1953.MathSciNetCrossRefGoogle Scholar
  26. Gawron, Piotr Wlodzimierz and Olga Macedonska, All automorphisms of the 3-nilpotent free group of countably infinite rank can be lifted, J. Alg. 118,120–128, 1988.MathSciNetMATHCrossRefGoogle Scholar
  27. Goryaga, A.V. Generators of the automorphism group of a free nilpotent group, Alg. and Logic, 15, 289–292 [English Translation], 1976.CrossRefGoogle Scholar
  28. Grigorchuk, R.I. and Kurchanko, P.F. Certain questions of group theory related to Geometry, In: Encyclopaedia of Mathematical Sciences, Algebra VII, Combinatorial Group Theory, Applications to Geometry, Springer, 173–231, 1993.Google Scholar
  29. Grossman, E.K. On the residual finiteness of certain mapping class groups, J. London Math.Soc. 9, 160–164, 1974.MathSciNetMATHCrossRefGoogle Scholar
  30. Gupta, C.K. IA-automorphisms of two generator metabelian groups, Arch. Math. 37, 106–112, 1981.MATHCrossRefGoogle Scholar
  31. Gupta, C.K. and Gupta, N.D. Lifting primitivity of free nilpotent groups, Proc. Amer. Math. Soc. 114, 617–621, 1992.MathSciNetMATHCrossRefGoogle Scholar
  32. Gupta, C.K., Gupta, N.D. and Noskov, G.A. Some applications of Artamonov-Quillen-Suslin theorems to metabelian inner-rank and primitivity, Canad. J. Math. 46, 298–307, 1994.MathSciNetMATHCrossRefGoogle Scholar
  33. Gupta, C.K., Gupta, N.D. and Roman’kov, V.A. Primitivity in free groups and free metabelian groups, Canad. J. Math. 44, 516–523, 1992.MathSciNetMATHCrossRefGoogle Scholar
  34. Gupta, C.K. and Frank Levin, Automorphisms of free class-2 by abelian groups, Bull. Austral. Math. Soc. 40, 207–214, 1989.MathSciNetMATHCrossRefGoogle Scholar
  35. Gupta, C.K. and Frank Levin, Tame range of automorphism groups of free polynilpotent groups, Comm. Alg. 19, 2497–2500, 1991.MathSciNetMATHCrossRefGoogle Scholar
  36. Gupta, C.K. and Noskov, G.A. Splitting epimorphisms of free metabelian groups, Internat. J. Algebra and Computation 7, 697–711, 1997.MathSciNetMATHCrossRefGoogle Scholar
  37. Gupta, C.K. and Roman’kov, V.A. Finite separability of tameness and primitivity in certain relatively free groups, Comm. Alg. 23, 4101–4108, 1995.MathSciNetMATHCrossRefGoogle Scholar
  38. Gupta, C.K. and Shpilrain, V. Lifting automorphisms — a survey, London Math. Soc. Lecture Notes Ser. 211, 249–263, 1995 [Groups 93, Galway/St. Andrews].MathSciNetGoogle Scholar
  39. Gupta, C.K. and Timoshenko, E.I. Primitivity in the free groups of the variety A m A n , Comm. Alg. 24, 2859–2876, 1996.MathSciNetMATHCrossRefGoogle Scholar
  40. Gupta, C.K. and Timoshenko, E.I. Automorphic and endomorphic reducibility and primitive endomorphisms of free metabeilan groups, Comm. Alg. 25, 3057–3070, 1997.MathSciNetMATHCrossRefGoogle Scholar
  41. Gupta, C.K. and Timoshenko, E.I. Primitive system in the variety A m A n: the criteria and lifting (to appear). Google Scholar
  42. Gupta, Narain. Free Group Rings, Contemporary Math. 66, Amer. Math. Soc. 1987.MATHCrossRefGoogle Scholar
  43. Gupta, N.D. and Shpilrain, V. Nielsen’s commutator test for two-generator groups, Math. Proc. Camb. phil. Soc. 114, 295–301, 1993.MathSciNetMATHCrossRefGoogle Scholar
  44. Imrich, W. and Turner, E.C. Endomorphisms of free groups and their fixed points, Math. Proc. Camb. phil. Soc. 105, 421–422, 1989.MathSciNetMATHCrossRefGoogle Scholar
  45. Ivanov, S.V. On endomorphisms of free groups that preserve primitivity, (Preprint). Google Scholar
  46. Jaco, W. Heegard splittings and splitting homomorphisms, Trans. Amer. Math. Soc. 144, 365–379, 1969.MathSciNetMATHCrossRefGoogle Scholar
  47. Krasnikov, A.F. Generators of the group F/[N, N], Alg. Logika, 17, 167–173, 1978.MathSciNetGoogle Scholar
  48. Lyndon, Roger C. and Schupp, Paul E. Combinatorial group theory, Ergebnisse Math. Grenzgeb. 89, Springer-Verlag, 1977.MATHGoogle Scholar
  49. Magnus, W. Über n-dimensionale Gitter transformationen. Acta Math. 64, 353–367, 1934.MathSciNetCrossRefGoogle Scholar
  50. Magnus, W. Karrass, A. and Solitar, D. Combinatorial Group Theory, Interscience Publ., New York, 1966.MATHGoogle Scholar
  51. Magnus, W. and Tretkoff, C. Representations of automorphism groups of free groups, Word Problems II, Studies in Logic and Foundations of Mathematics, 95, 255–260, North Holland, Oxford, 1980.MathSciNetCrossRefGoogle Scholar
  52. McCool, J. A presentation for the automorphism group of finite rank, J. London Math. Soc. 8(2) 259–266, 1974.MathSciNetMATHCrossRefGoogle Scholar
  53. Meskin, S. Periodic automorphisms of the two-generator free group, Proc. Second int. Group Theory Conf. Canberra 1973, Springer LN. 372, 494–498, 1974.MathSciNetCrossRefGoogle Scholar
  54. Mikhalev, A. A. and Zolotykh, A. A. Automorphisms and primitive elements in free LieGoogle Scholar
  55. algebras, Comm. Alg. 22, 5889–5901, 1994.Google Scholar
  56. Neumann, B.H. Die Automorphismerigruppe der freien Gruppen, Math. Ann. 107, 367–376, 1932.CrossRefGoogle Scholar
  57. Nielsen, J. Die Isomorphismender allegmeinen unendlichen Gruppen mit zwei Erzugenden, Math. Ann. 78, 385–397, 1918.CrossRefGoogle Scholar
  58. Nielsen, J. Die isomorphismengruppe der freien Gruppen, Math. Ann. 91, 169–209, 1924.MathSciNetMATHCrossRefGoogle Scholar
  59. Papistas, A.I. Automorphisms of metabelian groups, Canad. Math. Bull. 41, 98–104, 1988.MathSciNetCrossRefGoogle Scholar
  60. Papistas, A.I. Free nilpotent groups of rank 2, Comm. Alg. 25, 2141–2145, 1997.MathSciNetMATHCrossRefGoogle Scholar
  61. Rapaport, E.S. On free groups and their automorphisms, Acta Math. 99, 139–163, 1958.MathSciNetMATHCrossRefGoogle Scholar
  62. Roman’kov, V.A. Primitive elements of free groups of rank 3, Math. Sb. 182, 1074–1085, 1991.MATHGoogle Scholar
  63. Roman’kov, V.A. Automorphisms of groups, Acta appl. Math. 29, 241–280, 1992.MathSciNetMATHCrossRefGoogle Scholar
  64. Roman’kov, V.A. The automorphism groups of free metabelian groups, [Questions on pure and applied algebra] Proc. Computer Centre, USSR Academy of Sciences, Novosibirsk, 35–81 [Russian], 1985.Google Scholar
  65. Shpilrain, V.E. Automorphisms of F/R′ groups, Int. J. Alg. Comput. 1, 177–184, 1991.MathSciNetMATHCrossRefGoogle Scholar
  66. Shpilrain, V.E. On monomorphisms of free groups, Arch. Math. 64,465–470, 1995.MathSciNetMATHCrossRefGoogle Scholar
  67. Elena Stöhr, On automorphisms of free centre-by-metabelian groups, Arch. Math. 48, 376–380, 1987.MATHCrossRefGoogle Scholar
  68. Timoshenko, E.I. Algorithmic problems for metabelian groups, Alg. Logic 12, 132–137, [Russian Edition: Alg. Logika 12, 232–240], 1973.CrossRefGoogle Scholar
  69. Timoshenko, E.I. On embedding of given elements into a basis of free metabelian groups, Sibirsk. Math. Zh, Novosibirsk, 1988.Google Scholar
  70. Turner, E.C. Test words for automorphisms of free groups, Bull. London Math. Soc. 28, 255–263, 1996.MathSciNetMATHCrossRefGoogle Scholar
  71. Umirbaev, U.U. On primitive elements of free groups, Russian Math. Surv. 49, 184–185, 1994.MathSciNetCrossRefGoogle Scholar
  72. Whitehead, J.H.C. On certain set of elements in a free group, Proc. London Math. Soc. 41, 48–56, 1936.CrossRefGoogle Scholar
  73. Whitehead, J.H.C. On equivalent sets of elements in a free group, Ann. Math. 37, 782–800, 1936.Google Scholar

Copyright information

© Hindustan Book Agency (India) and Indian National Science Academy 1999

Authors and Affiliations

  • C. K. Gupta
    • 1
  1. 1.Department of MathematicsUniversity of ManitobaWinnipegCanada

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