Stable actions of groups on real trees.

*(English)*Zbl 0837.20047The 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.

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.

Reviewer: A.Papadopoulos (Strasbourg)

##### MSC:

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 |

##### Keywords:

groups acting on simplicial trees; groups acting on real trees; free product of surface groups; finitely presented group; word hyperbolic group; virtually cyclic \(G\)-tree; virtually cyclic subgroup; outer automorphism group; virtually cyclic group
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\textit{M. Bestvina} and \textit{M. Feighn}, Invent. Math. 121, No. 2, 287--321 (1995; Zbl 0837.20047)

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