Actions of finitely generated groups on R-trees

By Vincent Guirardel. 2006

76 citations recorded by the source corpus.

We study actions of finitely generated groups on $\bbR$-trees under some stability hypotheses. We prove that either the group splits over some controlled subgroup (fixing an arc in particular), or the action can be obtained by gluing together actions of simple types: actions on simplicial trees, actions on lines, and actions coming from measured foliations on 2-orbifolds. This extends results by Sela and Rips-Sela. However, their results are misstated, and we give a counterexample to their statements. The proof relies on an extended version of Scott's Lemma of independent interest. This statement claims that if a group $G$ is a direct limit of groups having suitably compatible splittings, then $G$ splits.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d0e049b3efd95f46d04c46cb25046f145b91bad4

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The co‐surface graph and the geometry of hyperbolic free group extensions (2016, background): “Following Guirardel [19], we say that a tree T is indecomposable if for every pair of nondegenerate arcs τ, τ ′ ⊂ T there exist a1, .”
Limit groups for relatively hyperbolic groups. I. The basic tools (2004, background): “Guirardel [26] provided a counterexample to this, but pointed out that Sela’s proof works under the assumption of superstability.”