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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.

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)

### References:

[1] | [Bes88] M. Bestvina: Degenerations of the hyperbolic space. Duke Math. J.56, 143–161 (1988) · Zbl 0652.57009 |

[2] | [CM87] M. Culler, J. Morgan: Group actions on \(\mathbb{R}\)-trees. Proc. London. Math. Soc.55, 571–604 (1987) · Zbl 0658.20021 |

[3] | [CV86] M. Culler, K. Vogtmann: Moduli of graphs and automorphisms of free groups. Invent. Math.84, 91–119 (1986) · Zbl 0589.20022 |

[4] | [Hat88] A. Hatcher: Measured lamination spaces for surfaces, from the topological view-point. Top. and its Appl.30, 63–88 (1988) · Zbl 0662.57005 |

[5] | [Lev93] G. Levitt: Constructing free actions on \(\mathbb{R}\)-trees. Duke Math. J.69, 615–633 (1993) · Zbl 0794.57001 |

[6] | [Mak83] G.S. Makanin: Equations in a free groups. Math. USSR Izvestiya21, 145–163 (1983) · Zbl 0527.20018 |

[7] | [Mor86] J. Morgan: Group actions on trees and the compactification of the space of classes ofSO(n, 1)-representations. Topology25, 1–33 (1986) · Zbl 0595.57030 |

[8] | [Mor92] J. Morgan: {\(\Lambda\)}-trees and their applications. Bull. Amer. Math. Soc. (N.S.) (1)26, 87–112 (1992) · Zbl 0767.05054 |

[9] | [MS88] J. Morgan, P. Shalen: Valuations, trees, and degenerations of hyperbolic structures, II. Ann. Math. (2)127, 403–465 (1988) · Zbl 0656.57003 |

[10] | [Pau88] F. Paulin: Topologie de Gromov équivariant, structures hyperboliques et arbres réels. Invent. Math.94, 53–80 (1988) · Zbl 0673.57034 |

[11] | [Pau91] F. Paulin: Outer automorphisms of hyperbolic groups and small actions on \(\mathbb{R}\)-trees. Arboreal Group Theory (R.C. Alperin, ed.), MSRI Publ., vol. 19, pp. 331–343 Springer-Verlag, 1991 · Zbl 0804.57002 |

[12] | [Raz85] A.A. Razborov: On systems of equations in a free group. Math. USSR Izvestiya25, 115–162 (1985) · Zbl 0579.20019 |

[13] | [Ser80] J.P. Serre: Trees. Springer-Verlag, 1980 |

[14] | [Sha87] P. Shalen, Dendrology of groups: an introduction. Essays in Group Theory (S.M. Gersten, ed.), MSRI Publ., vol. 8, Springer-Verlag, 1987, pp. 265–319 |

[15] | [Sha91] P. Shalen, Dendrology and its applications. Group theory from a geometrical view-point (E. Ghys, A. Haefliger, A. Verjovsky, eds.). World Scientific, 1991 |

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