Abstract
This paper arose from the paper [3] 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 [5]and Chiswell [2]concerning Lyndon length functions. The first definition was given by Tits in [7].KeywordsAbelian GroupBranch PointLength FunctionCyclic PermutationOriented GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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