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Non-Hopfian Relatively Free Groups

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Abstract

To solve problems of Gilbert Baumslag and Hanna Neumann, posed in the 1960’s, we construct a nontrivial variety of groups all of whose noncyclic free groups are non-Hopfian.

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References

  1. S. I. Adian (1979) The Burnside Problem and Identities in Groups Springer-Verlag New York

    Google Scholar 

  2. G. Baumslag (1963) ArticleTitleWreath products and extensions Math Z 81 286–299

    Google Scholar 

  3. Gromov, M.: Hyperbolic groups, In: Essays in Group Theory, M.S.R.I. Publ. 8, Springer, 1987, pp. 75–263.

  4. Ph. Hall G. Higman (1956) ArticleTitleOn the p-length of p-soluble groups and reduction theorems for Burnside’s problem Proc. London. Math. Soc 6 1–42

    Google Scholar 

  5. S. V. Ivanov (1994) ArticleTitleThe free Burnside groups of sufficiently large exponents Internat. J. Algebra. Comput 4 1–308 Occurrence Handle10.1142/S0218196794000026

    Article  Google Scholar 

  6. S. V. Ivanov (1998) ArticleTitleOn the Burnside problem for groups of even exponent Documenta Math ICM-98 IssueIDII 67–76

    Google Scholar 

  7. S. V. Ivanov A. Yu. Ol’shanskii (1991) Some Applications of Graded Diagrams in Combinatorial Group Theory Cambridge Univ. Press Cambridge and New York 258–308

    Google Scholar 

  8. A. I. Kostrikin (1959) ArticleTitleOn a problem of Burnside Math. USSR Izvest 23 3–34

    Google Scholar 

  9. A. I. Kostrikin (1990) Around Burnside Springer-Verlag Berlin

    Google Scholar 

  10. Kourovka Notebook: Unsolved Problems in Group Theory, 12th edn, Novosibirsk, 1992.

  11. I. G. Lysenok (1996) ArticleTitleInfinite Burnside groups of even period Math Ross Izvest 60 3–224

    Google Scholar 

  12. H. Neumann (1967) Varieties of Groups Springer-Verlag New York

    Google Scholar 

  13. Novikov, P. S. and Adian, S. I.: On infinite periodic groups, I, II, III, Math. USSR Izvest. 32 (1968), 212–244, 251–524, 709–731.

  14. A. Yu. Ol’shanskii (1982) ArticleTitleOn the Novikov-Adian theorem Mat. Sb 118 203–235

    Google Scholar 

  15. A. Yu. Ol’shanskii (1985) ArticleTitleVarieties in which all finite groups are abelian Mat. Sb 126 59–82

    Google Scholar 

  16. A.Yu. Ol’shanskii (1991) Geometry of Defining Relations in Groups Kluwer Acad Publ Dordrecht

    Google Scholar 

  17. A. M. Storozhev (1994) ArticleTitleOn abelian subgroups of relatively free groups Comm Algebra 22 2677–2701

    Google Scholar 

  18. Z. Sela (1999) ArticleTitleEndomorphisms of hyperbolic groups. I. The Hopf property Topology 38 301–321 Occurrence Handle10.1016/S0040-9383(98)00015-9

    Article  Google Scholar 

  19. E. I. Zelmanov (1991) ArticleTitleSolution of the restricted Burnside problem for groups of odd exponent Math USSR Izvest 36 41–60

    Google Scholar 

  20. E. I. Zelmanov (1992) ArticleTitleA solution of the restricted Burnside problem for 2-groups Mat. Sb 72 543–565

    Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, Urbana, IL, 61801, U.S.A.

    S. V. Ivanov

  2. Australian Mathematics Trust, University of Canberra, Belconnen, ACT, 2601, Australia

    A. M. Storozhev

Authors
  1. S. V. Ivanov
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  2. A. M. Storozhev
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Correspondence to S. V. Ivanov.

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Ivanov, S.V., Storozhev, A.M. Non-Hopfian Relatively Free Groups. Geom Dedicata 114, 209–228 (2005). https://doi.org/10.1007/s10711-005-1726-x

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  • DOI: https://doi.org/10.1007/s10711-005-1726-x

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