Cutting up graphs

Abstract

LetΓ be infinite connected graph with more than one end. It is shown that there is a subsetdV Γ which has the following properties. (i) Bothd andd*=VΓ\d are infinite. (ii) there are only finitely many edges joiningd andd*. (iii) For each AutΓ at least one ofddg, d*⊂dg, dd* g, d*d* g holds. Any group acting on Γ has a decomposition as a free product with amalgamation or as an HNN-group.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    M. G. Brin, Splitting manifold covering spaces,Preprint, State University of New York, Binghampton.

  2. [2]

    W. Dicks,Groups, trees and projective modules, Lecture Notes in Mathematics790, Springer, Berlin-Heidelberg-New York 1980.

    MATH  Google Scholar 

  3. [3]

    M. J. Dunwoody, Accessibility and groups of cohomological dimension one,Proc. London Math. Soc. 38 (1979) 193–215.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    H. D. Macpherson, Infinite distance transitive graphs of finite valency,Combinatorica 2 (1) (1982) 63–69.

    MATH  MathSciNet  Google Scholar 

  5. [5]

    J. R. Stallings,Group theory and three-dimensional manifolds, Yale Mathematical Monographs4 (Yale University Press, 1971).

  6. [6]

    L. Babai andM. E. Watkins, Connectivity of infinite graphs having a transitive torsion group action,Arch. Math. 34 (1980), 90–96.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    H. A. Jung, A note on fragments of infinite graphs,Combinatorica 1 (1981), 285–288.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Dunwoody, M.J. Cutting up graphs. Combinatorica 2, 15–23 (1982). https://doi.org/10.1007/BF02579278

Download citation

AMS subject classification (1980)

  • 05 C 25
  • 20 F 05