Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank \(1\) Lie groups. II.

*(English)*Zbl 0884.20025In a series of papers, the author has developed group-theoretic analogs of several results in the topology of two and three-manifolds, including the notion of a characteristic manifold and of the existence of a canonical Jaco-Shalen-Johannson decomposition of a three-manifold [for part I cf. E. Rips and the author, Geom. Funct. Anal. 4, No. 3, 337-371 (1994; Zbl 0818.20042)]. The author has obtained in this way a canonical splitting of a (Gromov) hyperbolic group. For instance, this decomposition is a key tool in the author’s solution of the isomorphism problem for torsion-free hyperbolic groups. The author showed then that this decomposition is valid in a category which is much larger than that of hyperbolic groups, and in joint work with E. Rips [Cyclic splittings of finitely presented groups and a canonical JSJ decomposition, to appear in Ann. Math., II. Ser.], this canonical decomposition is generalized to all finitely presented groups.

In this paper, the author derives corollaries from the existence of such a structure, both for hyperbolic groups and for finitely generated discrete groups of rank one Lie groups.

For hyperbolic groups, the author proves several results on the dynamics of the automorphisms and the structure of the group of automorphisms of such a group, some of which are related to Thurston’s classification of surface mapping classes.

For finitely generated discrete groups in rank one Lie groups, the canonical decomposition suggests a generalization of the Teichmüller modular group of a surface, and a generalization of the Riemann moduli space as a quotient space. The author proves here a generalization of the Mumford lemma to this generalized moduli space as well as a general version of Thurston’s bounded image theorem.

The author considers then this decomposition in the particular case of geometrically finite Kleinian groups in arbitrary dimension and obtains some results on their automorphism groups. He proves then that torsion-free hyperbolic groups are co-Hopf if and only if they are freely indecomposable.

In his introduction, the author suggests several other applications in different directions, for instance carrying on the results of Bestvina-Handel on the automorphisms of free groups to the general context of automorphisms of (torsion-free) hyperbolic groups.

In this paper, the author derives corollaries from the existence of such a structure, both for hyperbolic groups and for finitely generated discrete groups of rank one Lie groups.

For hyperbolic groups, the author proves several results on the dynamics of the automorphisms and the structure of the group of automorphisms of such a group, some of which are related to Thurston’s classification of surface mapping classes.

For finitely generated discrete groups in rank one Lie groups, the canonical decomposition suggests a generalization of the Teichmüller modular group of a surface, and a generalization of the Riemann moduli space as a quotient space. The author proves here a generalization of the Mumford lemma to this generalized moduli space as well as a general version of Thurston’s bounded image theorem.

The author considers then this decomposition in the particular case of geometrically finite Kleinian groups in arbitrary dimension and obtains some results on their automorphism groups. He proves then that torsion-free hyperbolic groups are co-Hopf if and only if they are freely indecomposable.

In his introduction, the author suggests several other applications in different directions, for instance carrying on the results of Bestvina-Handel on the automorphisms of free groups to the general context of automorphisms of (torsion-free) hyperbolic groups.

Reviewer: A.Papadopoulos (Strasbourg)

##### MSC:

20F65 | Geometric group theory |

20F28 | Automorphism groups of groups |

22E40 | Discrete subgroups of Lie groups |

57M07 | Topological methods in group theory |

20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |

20E07 | Subgroup theorems; subgroup growth |

57M60 | Group actions on manifolds and cell complexes in low dimensions |

57M05 | Fundamental group, presentations, free differential calculus |