Axioms for Translation Length Functions
This paper arose from the paper  of Culler and Morgan concerning group actions on ℝ-trees. An ℝ-tree is a nonempty metric space T such that every two points in T are joined by a unique arc (image of an injective continuous function from a closed interval in ℝ to T) and that arc is the image of an isometry from a closed interval in ℝ to T. This generalizes the usual notion of simplicial tree. Roughly speaking, the difference is that in a simplicial tree, the branching occurs at a discrete set of points, namely, the vertices, whereas in an ℝ-tree, there is no restriction on where branching may occur. The notion of ℝ-tree arose indirectly in the work of Lyndon and Chiswell concerning Lyndon length functions. The first definition was given by Tits in .
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