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Graphs of actions on $$\mathbf R$$-trees. (English) Zbl 0802.05044
The main results are the following theorems:
Let $$G$$ be a finitely generated group acting minimally on an R-tree $$T$$. If the action is free, then the associated metric space $$\widehat{T/G}$$ is a finite graph.
Let $$G$$ be a finitely generated group acting on an R-tree $$T$$, with length function $$l$$.
(1) There exists $$\varepsilon_ 0>0$$ such that $$G(\varepsilon)$$ is independent of $$\varepsilon$$ for $$\varepsilon<\varepsilon_ 0$$. Denote it by $$G_ 0$$.
(2) The quotient $$G/G_ 0$$ is free of rank $$m(l)$$.
(3) The space $$T/G$$ has the homotopy type of a wedge of $$m(l)$$ circles. It has a universal covering: the R-tree $$\widehat{T/G_ 0}$$.
Let $$G$$ be a finitely generated group acting freely and minimally on an R-tree $$T$$. The graph $$\widehat{T/G}$$ is homeomorphic to a segment iff, given $$g\in G$$ and $$\varepsilon>0$$, there exist $$g_ 1$$ and $$g_ 2$$ such that $$g= g_ 1 g_ 2$$ and $$\max(l(g_ 1),l(g_ 2))<{1\over 2} l(g)+\varepsilon$$.

##### MSC:
 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 05C05 Trees
##### Keywords:
R-tree; metric space; homotopy; universal covering
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