Mathematische Zeitschrift

, Volume 208, Issue 1, pp 225–237 | Cite as

The genus problem for one-relator products of locally indicable groups

  • Andrew J. Duncan
  • James Howie


Word Problem Boundary Component Free Product Isoperimetric Inequality Indicable Group 
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  1. 1.
    Baumslag, B.: Free products of locally indicable groups with a single relator. Bull. Aust. Math. Soc.29, 401–404 (1984)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Brodskiî, S.D.: Anomalous products of locally indicable groups (in Russian). In: Moldavanskiî, D.I. (ed.) Algebraicheskie Sistemy, pp. 51–77. Ivanovo: Ivanovo State University Publication 1981Google Scholar
  3. 3.
    Brodskiî, S.D.: Equations over groups and groups with a single defining relation. Sib. Math. J.25, 235–251 (1984)CrossRefGoogle Scholar
  4. 4.
    Brown, R., Huebschmann, J.: Identities among relations. In: Brown, R., Thickstun, T.L. (eds.) Low Dimensional Topology. (Lond. Math. Soc. Lect. Note Ser., vol. 48, pp. 153–202) Cambridge: Cambridge University Press 1982Google Scholar
  5. 5.
    Burns, R.G., Hale, V.W.D.: A note on group rings of certain torsion free groups. Can. math. Bull.15, 441–445 (1972)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Comerford, L.P., Edmunds, C.C.: Quadratic equations over free groups and free products. J. Algebra68, 276–297 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Comerford L.P., Edmunds, C.C.: Quadratic parametric equations over free groups. Contemp. Math.33, 159–222 (1984)MathSciNetGoogle Scholar
  8. 8.
    Culler, M.: Using surfaces to solve equations in free groups. Topology20, 133–145 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fenn, R.A.: Techniques of Geometric Topology. (Lond. Math. Soc. Lect. Note Ser. vol. 57) Cambridge: Cambridge University Press 1983zbMATHGoogle Scholar
  10. 10.
    Goldstein, R.Z., Turner, E.C.: Applications of topological graph theory to group theory. Math. Z.165, 1–10 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gonzalez-Acuña, F., Short, H.: Knot surgery and primeness. Math. Proc. Cambridge Philos. Soc.99, 89–102 (1986)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Gu, Z.-B.: Cancellation Diagrams on Surfaces and Quadratic Equations in Groups. Ph.D. Thesis, University of Michigan 1987Google Scholar
  13. 13.
    Howie, J.: On pairs of 2-complexes and systems of equations over groups. J. Reine Angew. Math.324, 165–174 (1981)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Howie, J.: On locally indicable groups. Math. Z.180, 445–461 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Howie, J.: How to generalize one-relator group theory. In: Gersten, S.M., Stallings, J.R. (eds.) Combinatorial Group Theory and Topology. (Ann. Math. Stud., vol. III, pp. 53–78) Princeton: Princeton University Press 1987Google Scholar
  16. 16.
    Howie, J.: The quotient of a free product of groups by a single high-powered relator. I. Pictures. Fifth and higher powers. Proc. Lond. Math. Soc.59, 507–540 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Howie, J., Pride, S.J.: A spelling theorem for staggered generalized 2-complexes, with applications. Invent. Math.76, 55–74 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lyndon, R.C., Schupp, P.E.: Combinatorial Group Theory. Berlin Heidelberg New York: Springer 1977zbMATHGoogle Scholar
  19. 19.
    Magnus, W.: Das Identitätsproblem für Gruppen mit einer definierenden Relation. Math. Ann.106, 295–307 (1932)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Moldavinskiî, D.I.: Certain subgroups of groups with one defining relation. Sib. Mat. Zh.8, 1370–1384 (1967)Google Scholar
  21. 21.
    Newman, B.B.: Some results on one-relator groups. Bull. Am. Math. Soc.74, 568–571 (1968)zbMATHCrossRefGoogle Scholar
  22. 22.
    Pride, S.J.: Small cancellation conditions satisfied by one-relator groups. Math. Z.184, 283–286 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Pride, S.J.: Star-complexes and the dependence problems for hyperbolic complexes. Glasg. Math. J.30, 155–170 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Rourke, C.P.: Presentations and the trivial group. In: Fenn, R.A. (ed.) Topology of Low Dimensional Manifolds. (Lect. Notes Math., vol. 722, pp. 134–143), Berlin Heidelberg New York: Springer 1979CrossRefGoogle Scholar
  25. 25.
    Schupp, P.E.: Quadratic equations in groups, cancellation diagrams on compact surfaces and automorphisms of surface groups. In: Adian, S.I., Boone, W.W., Higman, G. (eds.) Word Problems II. (The Oxford Book, pp. 347–371) Amsterdam: North-Holland 1980Google Scholar
  26. 26.
    Short, H.: Topological methods in group theory: the adjunction problem: Ph.D. Thesis, University of Warwick 1984Google Scholar
  27. 27.
    Wicks, M.J.: Commutators in free products. J. Lond. Math. Soc.37, 433–444 (1962)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Andrew J. Duncan
    • 1
  • James Howie
    • 1
  1. 1.Department of MathematicsHeriot-Watt UniversityEdinburghUK

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