Abstract
LetΓ be infinite connected graph with more than one end. It is shown that there is a subsetd ⊂V Γ which has the following properties. (i) Bothd andd*=VΓ\d are infinite. (ii) there are only finitely many edges joiningd andd*. (iii) For eachgε AutΓ at least one ofd⊂dg, d*⊂dg, d⊂d* g, d*⊂d* g holds. Any group acting on Γ has a decomposition as a free product with amalgamation or as an HNN-group.
Similar content being viewed by others
References
M. G. Brin, Splitting manifold covering spaces,Preprint, State University of New York, Binghampton.
W. Dicks,Groups, trees and projective modules, Lecture Notes in Mathematics790, Springer, Berlin-Heidelberg-New York 1980.
M. J. Dunwoody, Accessibility and groups of cohomological dimension one,Proc. London Math. Soc. 38 (1979) 193–215.
H. D. Macpherson, Infinite distance transitive graphs of finite valency,Combinatorica 2 (1) (1982) 63–69.
J. R. Stallings,Group theory and three-dimensional manifolds, Yale Mathematical Monographs4 (Yale University Press, 1971).
L. Babai andM. E. Watkins, Connectivity of infinite graphs having a transitive torsion group action,Arch. Math. 34 (1980), 90–96.
H. A. Jung, A note on fragments of infinite graphs,Combinatorica 1 (1981), 285–288.