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Stable actions of groups on real trees. (English) Zbl 0837.20047
The structure of groups acting on simplicial trees is well-understood, thanks to the theory of Bass-Serre. For groups acting on real trees, the main structure theorem (which had been conjectured by J. Morgan and P. Shalen) has been proved by E. Rips and states that a finitely generated group acting freely on a real tree is a free product of surface groups and free abelian groups. In this paper, the authors give a proof of Rips’ theorem. They obtain also several related new results, some of them generalizing work of Morgan-Shalen, Morgan, Paulin and the first author.
The results include the following: Let \(G\) be a finitely presented group with a nontrivial, stable and minimal action on a tree \(T\). Then either (i) \(G\) splits over an extension \(E\)-by-cyclic where \(E\) fixes an arc on \(T\), or (ii) \(T\) is a line, and in this case \(G\) splits over an extension of the kernel of the action by a finitely generated free abelian group. Let \(G\) be a word hyperbolic group. Suppose \(T\) is a virtually cyclic \(G\)-tree. Then, \(G\) splits over a virtually cyclic subgroup. – Let \(G\) be finitely presented and not virtually abelian. If \(G\) does not split over a virtually abelian subgroup then the space of conjugacy classes of discrete and faithful representations of \(G\) into \(\text{SO} (n, 1)\) is compact for all \(n\). – If the outer automorphism group of the word hyperbolic group \(G\) is infinite then \(G\) splits over a virtually cyclic group. The paper includes also structure theorems for measured foliations on 2-complexes.

20F65 Geometric group theory
20E08 Groups acting on trees
20F05 Generators, relations, and presentations of groups
57M07 Topological methods in group theory
57M60 Group actions on manifolds and cell complexes in low dimensions
20F28 Automorphism groups of groups
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