Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank \(1\) Lie groups. II
type: Item
instance of: scholarly article
title: Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank \(1\) Lie groups. II
zbMATH Open document ID: 0884.20025
published in: Geometric and Functional Analysis. GAFA
publication date: 1998-01-28T00:00:00Z
review text: In 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. \textit{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 \textit{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 torsi…
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 1054110
zbMATH Keywords: hyperbolic groups
zbMATH Keywords: Gromov hyperbolicity
zbMATH Keywords: combinatorial group theory
zbMATH Keywords: characteristic submanifolds
zbMATH Keywords: discrete groups
zbMATH Keywords: rank one Lie groups
zbMATH Keywords: Kleinian groups
zbMATH Keywords: Jaco-Shalen-Johannson submanifolds
zbMATH Keywords: isomorphism problem for torsion-free hyperbolic groups
zbMATH Keywords: canonical splittings
zbMATH Keywords: canonical decompositions of finitely presented groups
zbMATH Keywords: automorphism groups
zbMATH Keywords: mapping class groups
zbMATH Keywords: freely indecomposable groups
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
full work available at URL: s000390050019
OpenAlex ID: W2005237754
Wikidata QID: Q115389257
DOI: 10.1007/S000390050019
author: Zlil Sela
description: scientific article; zbMATH DE number 1054110