An improved proof of the Almost Stability Theorem

In 1989, Dicks and Dunwoody proved the Almost Stability Theorem, which has among its corollaries the Stallings-Swan theorem that groups of cohomological dimension one are free. In this article, we use a nestedness result of Bergman, Bowditch, and Dunwoody to simplify somewhat the proof of the finitely generable case of the Almost Stability Theorem. We also simplify the proof of the non finitely generable case. The proof we give here of the Almost Stability Theorem is essentially self contained, except that in the non finitely generable case we refer the reader to the original argument for the proofs of two technical lemmas about groups acting on trees.