Stable actions of groups on real trees
type: Item
instance of: scholarly article
title: Stable actions of groups on real trees
zbMATH Open document ID: 0837.20047
DOI: 10.1007/BF01884300
author: Mladen Bestvina
author: Mark Feighn
published in: Inventiones Mathematicae
publication date: 1996-04-21T00:00:00Z
full work available at URL: https://eudml.org/doc/144300
review text: 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.
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20E08
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 20F28
zbMATH DE Number: 807305
zbMATH Keywords: groups acting on simplicial trees
zbMATH Keywords: groups acting on real trees
zbMATH Keywords: free product of surface groups
zbMATH Keywords: finitely presented group
zbMATH Keywords: word hyperbolic group
zbMATH Keywords: virtually cyclic \(G\)-tree
zbMATH Keywords: virtually cyclic subgroup
zbMATH Keywords: outer automorphism group
zbMATH Keywords: virtually cyclic group
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
cites work: Degenerations of the hyperbolic space
cites work: Group Actions On R-Trees
cites work: Moduli of graphs and automorphisms of free groups
cites work: Measured lamination spaces for surfaces, from the topological viewpoint
cites work: Constructing free actions on \(\mathbf R\)-trees
cites work: EQUATIONS IN A FREE GROUP
cites work: Group actions on trees and the compactification of the space of classes of SO(n,1)-representations
cites work: Λ-trees and their applications
cites work: Degenerations of hyperbolic structures. II: Measured laminations in 3- manifolds
cites work: Topologie de Gromov équivariante, structures hyperboliques et arbres réels. (Equivariant Gromov topology, hyperbolic structures, and \({\mathbb{R}}\)-trees)
cites work: Q4028727
cites work: ON SYSTEMS OF EQUATIONS IN A FREE GROUP
OpenAlex ID: W2065236013
Wikidata QID: Q56453202
description: scientific article; zbMATH DE number 807305