I am interested in applications of the combination of Stallings theorem and Dunwoody's accessibility, which can be summarized as follow:
Theorem: Let $G$ be a finitely presented group. Then $G$ is the fundamental group of a graph of groups such that:
The underlying simplicial graph is finite.
The edge groups are finite.
The vertex groups are finitely generated and have at most one end.
I already know the two applications below:
Application 1: A group that is quasi-isometric to a free group is virtually free.
A simple proof can be found in Kapovich and Drutu's Lectures on Geometric Group Theory (theorem 18.38, p. 473). Another proof, based on boundaries of hyperbolic groups, can be found in Ghys and de la Harpe's book, Les groupes hyperboliques d'après Misha Gromov.
Application 2: The fundamental group of a compact Riemannian manifold whose universal covering has a bounded fill radius is virtually free.
A proof can be found in Ramachandran and Wolfson's article, Fill radius and the fundamental group. The main idea is to prove that such a group has no finitely-generated subgroup with exactly one end.
Do you know other applications?
[Nota bene: I already asked the question on math.stackexchange, but it does not seem to interest a lot of people there.]