Inventiones mathematicae

, Volume 117, Issue 1, pp 373–389 | Cite as

Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture

  • Warren Dicks
Article

Summary

We show that Walter Neumann's strengthened form of Hanna Neumann's conjecture on the best possible upper bound for the rank of the intersection of two subgroups of a free group is equivalent to a conjecture on the best possible upper bound for the number of edges in a bipartite graph with a certain weak symmetry condition. We illustrate the usefulness of this equivalence by deriving relatively easily certain previously known results.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Burns, R.G.: On the intersection of finitely generated subgroups of a free group. Math. Z.119, 121–130 (1971)Google Scholar
  2. 2.
    Dicks, W., Dunwoody, M.J.: Groups acting on graphs (Camb. Stud. Adv. Math., vol. 17) Cambridge: Cambridge University Press 1989Google Scholar
  3. 3.
    Gersten, S.M.: Intersections of finitely generated subgroups of free groups and resolutions of graphs. Invent. Math.71 567–591 (1983)Google Scholar
  4. 4.
    Howson, A.G.: On the intersection of finitely generated free groups. J. Lond. Math. Soc.29, 428–434 (1954)Google Scholar
  5. 5.
    Imrich, W.: On finitely generated subgroups of free groups. Arch. Math.28, 21–24 (1977)Google Scholar
  6. 6.
    Imrich, W.: Subgroup theorems and graphs. In: Little, C.H.C. (ed.) Combinatorial Mathematics V, Melbourne 1976. (Lect. Notes Math., vol. 622, pp. 1–27), Berlin Heidelberg New York: Springer 1977Google Scholar
  7. 7.
    Neumann, H.: On intersections of finitely generated subgroups of free groups. Publ. Math., Debrecen4, 186–189 (1956)Google Scholar
  8. 8.
    Neumann, H.: On intersections of finitely generated subgroups of free groups. Addendum. Publ. Math., Debrecen5, 128 (1958)Google Scholar
  9. 9.
    Neumann, W.D.: On intersections of finitely generated subgroups of free groups. In: Kovács, L.G. (ed.) Groups—Canberra 1989. (Lect. Notes Math., vol. 1456, pp. 161–170) Berlin Heidelberg New York: Springer 1990Google Scholar
  10. 10.
    Nickolas, P.: Intersections of finitely generated free groups. Bull. Aust. Math. Soc.31, 339–348 (1985)Google Scholar
  11. 11.
    Reidemeister, K.: Einführing in die kombinatorische Topologie. Braunschweig 1932; reprinted: New York: Chelsea 1950Google Scholar
  12. 12.
    Serre, J.P.: Arbres, amalgames, SL2. (Astérisque, no. 46) Paris: Soc. Math. Fr. 1977; English translation: Trees. Berlin Heidelberg New York: Springer 1980Google Scholar
  13. 13.
    Servatius, B.: A short proof of a theorem of Burns. Math. Z.184, 133–137 (1983)Google Scholar
  14. 14.
    Stallings, J.R.: On torsion-free groups with infinitely many ends. Ann. Math.88, 312–334 (1968)Google Scholar
  15. 15.
    Stallings, J.R.: Topology of finite graphs. Invent. Math.71, 551–565 (1983)Google Scholar
  16. 16.
    Tardos, G.: On the intersection of subgroups of a free group. Invent. Math.108, 29–36 (1992)Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Warren Dicks
    • 1
  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBarcelonaSpain

Personalised recommendations