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.


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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