Geometriae Dedicata

, Volume 105, Issue 1, pp 61-76

First online:

A Geometric Proof of Stallings' Theorem on Groups with More than One End

  • Graham A. NibloAffiliated withFaculty of Mathematical Studies, University of Southampton

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Stallings showed that a finitely generated group which has more than one end splits as an amalgamated free product or an HNN extension over a finite subgroup. Dunwoody gave a new geometric proof of the theorem for the class of almost finitely presented groups, and separately, using somewhat different methods, generalised it to a larger class of splittings. Here we adapt the geometric method to the class of finitely generated groups using Sageev's generalisation of Bass Serre theory concerning group pairs with more than one end, and show that this new proof simultaneously establishes Dunwoody's generalisation.

amalgamated free product Bass–Serre theory CAT(0) cube complex ends HNN extension singularity obstruction Stallings' theorem