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.
AMS subject classification (1980)05 C 25 20 F 05
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