Stable actions of groups on real trees

  title={Stable actions of groups on real trees},
  author={Mladen Bestvina and Mark Feighn},
  journal={Inventiones mathematicae},
This paper further develops Rips's work on real trees. We study a class of actions called ‘stable’ which includes actions with trivial arc stabilizers and small actions of hyperbolic groups. 

Pseudogroups of isometries of ℝ and Rips’ theorem on free actions on ℝ-trees

We give a proof of Rips’ theorem that a finitely generated group acting freely on an ℝ-tree is a free product of free abelian groups and surface groups, using methods of dynamical systems and

Structure and rigidity in hyperbolic groups I

We introduce certain classes of hyperbolic groups according to their possible actions on real trees. Using these classes and results from the theory of (small) group actions on real trees, we study

Cyclic Splittings of Finitely Presented Groups and the Canonical JSJ-Decomposition

The classification of stable actions of finitely presented groups on ℝ-trees has found a number of applications. Perhaps one of the most striking of these applications is the theory of canonical

Connectedness properties of limit sets

We study convergence group actions on continua, and give a criterion which ensures that every global cut point is a parabolic fixed point. We apply this result to the case of boundaries of relatively

Botany of irreducible automorphisms of free groups

We give a classification of iwip outer automorphisms of the free group, by discussing the properties of their attracting and repelling trees.

On the failure of the co-hopf property for subgroups of word-hyperbolic groups

We provide an example of a finitely generated subgroupH of a torsion-free word-hyperbolic groupG such thatH is one-ended, andH does not split over a cyclic group, andH is isomorphic to one of its

Quasi-isometry rigidity of groups

This paper overviews recent developments in the classification up to quasi-isometry of finitely generated groups, and more specifically of relatively hyperbolic groups.

New examples of groups acting on real trees

We construct the first example of a finitely generated group that has Serre's property (FA) (that is, whenever it acts on a simplicial tree, it fixes a vertex), but admits a fixed point‐free action


We provide an example of a finitely generated subgroupH of a torsion-free word-hyperbolic group G such that H is one-ended, and H does not split over a cyclic group, and H is isomorphic to one of its

Nielsen Methods and Groups Acting on Hyperbolic Spaces

We show that for any n & in ℕ there exists a constant C(n) such that any n-generated group G which acts by isometries on a δ-hyperbolic space (with δ>0) is either free or has a nontrivial element



Outer Automorphisms of Hyperbolic Groups and Small Actions on ℝ-Trees

If Γ is a group, denote by Out(Γ) the group of outer automorphisms of Γ. The definitions of the notions used in this introduction are given in the first section. The main theorem of this paper


A description of the general solution of given bounded periodicity exponent is obtained for an arbitrary system of equations in a free group. On the basis of this result an algorithm is constructed


An algorithm for recognizing the solvability of arbitrary equations in a free group is constructed. Bibliography: 11 titles.

Moduli of graphs and automorphisms of free groups

This paper represents the beginning of an a t tempt to transfer, to the study of outer au tomorphisms of free groups, the powerful geometric techniques that were invented by Thurs ton to study

Dendrology of Groups: An Introduction

The study of group actions on “generalized trees” or “ℝ-trees” has recently been attracting the attention of mathematicians in several different fields and has been developed further by the above-mentioned people and also by H. Culler, H. Gillet, M. Rimlinger and J. Stallings.

Degenerations of hyperbolic structures, II: Measured laminations in 3-manifolds

That paper concerned the general theory of groups acting on R-trees and the relationship of these actions to representations into SL2(C). The purpose of the present paper is to develop the

$\Lambda $\<-Trees and Their Applications

To most mathematicians and computer scientists the word ``tree'' conjures up, in addition to the usual image, the image of a connected graph with no circuits. In the last few years various types of

Topologie de Gromov équivariante, structures hyperboliques et arbres réels

RésuméLes objets que nous étudions sont les espaces métriques munis d'une action par isométrie d'un groupe fixé Γ. Nous définissons une «topologie» naturelle sur «l'ensemble» de ces espaces. Nous