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.]