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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.]

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    $\begingroup$ It is key to the Muller-Schupp theorem that groups with a context-free word problem are virtually free. $\endgroup$ – Benjamin Steinberg May 16 '14 at 12:42
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The theorem of Papasoglu and Whyte, which says the following. Given a finitely presented group $G$ and any graph of groups presentation as in the theorem you quote, let $V(G)$ denote the set of quasi-isometry types of one-ended vertex groups. Then:

  • $V(G)$ is a quasi-isometry invariant of $G$ (well-defined independent of the graph of groups chosen).
  • Two finitely presented groups $G,H$ are quasi-isometric if and only if they have the same number of ends, and $V(G)=V(H)$.

This theorem is from their paper "Quasi-isometries between groups with infinitely many ends." Comment. Math. Helv. 77 (2002), no. 1, 133–144.

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The theorem is used to prove that finetely presented groups of asymptotic dimension 1 are virtually free. This was proved independently by Fujiwara-Whyte, Gentimis and Januszkiewicz-Swiatkowski.

To show this result, application 1, or Muller-Schupp one doesn't really need the accessiblity of finitely presented groups. A previous result of Dunwoody is enough. See http://arxiv.org/abs/0911.2177v2

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There's an application, of the slight improvement of these results to locally compact groups, which yields, for instance that many virtually free groups do not have proper cocompact isometric actions on the same proper metric space.

It requires 1) the generalization of Stallings to locally compact groups, which essentially consists in replacing a Cayley graph by a vertex-transtive (locally finite connected) graph, which was done by Abels in 1974, and 2) the extension of Dunwoody's result to simply connected vertex-transitive 2-complexes, which is a straightfoward extension of his argument, and was observed by Mosher-Sageev-Whyte (who also obtained this application).

The outcome is that if 2 virtually free f.g. groups have proper cocompact isometric actions on a single proper metric space, then they also have proper cocompact action on a single tree, and the latter can be discarded by easy Bass-Serre theory arguments, e.g. for $C_2*C_3$ and $C_2*C_5$ where $C_k$ is cyclic of order $k$.

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