## Pseudogroups of isometries of $$\mathbb{R}$$ and Rips’ theorem on free actions on $$\mathbb{R}$$-trees.(English)Zbl 0824.57001

An $$\mathbb{R}$$-tree is an arcwise connected metric space in which every arc is isometric to an interval of $$\mathbb{R}$$. Using methods of dynamical systems and measured foliations, the authors present a proof of Rips’ theorem that a finitely generated group acting on an $$\mathbb{R}$$-tree is a free product of free abelian groups and surface groups. More generally they prove the following theorem: Let $$G$$ be a finitely generated group acting on an $$\mathbb{R}$$-tree $$T$$. Let $$G_ e$$ be the (normal) subgroup of $$G$$ generated by all elements acting with a fixed point (elliptic elements). Then $$G/G_ e$$ is a free product of free abelian groups and surface groups.

### MSC:

 57M07 Topological methods in group theory 57S30 Discontinuous groups of transformations 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20F05 Generators, relations, and presentations of groups
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