Abstract
LetG be a finitely generated group acting on anR-treeT. First assume that the action is free, and minimal (there is no proper invariant subtree), or more generally that it satisfies a certain finiteness condition. Then it may be described as agraph of transitive actions: the action may be recovered from a finite graph, together with additional data; in particular, every vertexv carries an action (G v, Tv) whose orbits are dense. For the action (G, T), it follows for instance that the closure of any orbit is a discrete union of closed subtrees: it cannot meet a segment in a Cantor set.
Now let ℓ be the length function for an arbitrary action ofG. For ɛ>0 small enough, the subgroupG(ɛ)⊂G generated by elementsg withg is independent of ɛ, andG/G(ɛ) is free. Several interpretations are given for the rank ofG/G(ɛ).
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Levitt, G. Graphs of actions on R-trees. Commentarii Mathematici Helvetici 69, 28–38 (1994). https://doi.org/10.1007/BF02564472
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DOI: https://doi.org/10.1007/BF02564472