Automorphisms of Hyperbolic Groups and Graphs of Groups

  title={Automorphisms of Hyperbolic Groups and Graphs of Groups},
  author={G. Levitt},
  journal={Geometriae Dedicata},
  • G. Levitt
  • Published 2002
  • Mathematics
  • Geometriae Dedicata
Using the canonical JSJ splitting, we describe the outer automorphism group Out(G) of a one-ended word hyperbolic group G. In particular, we discuss to what extent Out(G) is virtually a direct product of mapping class groups and a free abelian group, and we determine for which groups Out(G) is infinite. We also show that there are only finitely many conjugacy classes of torsion elements in Out(G), for G any torsion-free hyperbolic group. More generally, let Γ be a finite graph of groups… Expand
Splittings and automorphisms of relatively hyperbolic groups
We study automorphisms of a relatively hyperbolic group G. When G is one-ended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular,Expand
The automorphism group of accessible groups
It is shown that Out(G) is essentially obtained by taking extensions of relative automorphism groups of vertex groups, groups of Dehn twists and groups of automorphisms of free products, and a criterion for Out( G) to be finitely presented is obtained. Expand
Automorphisms of free groups with boundaries
The automorphisms of free groups with boundaries form a fam- ily of groups An,k closely related to mapping class groups, with the stan- dard automorphisms of free groups as An,0 and (essentially) theExpand
On the automorphisms of a graph product of abelian groups
We study the automorphisms of a graph product of finitely generated abelian groups W. More precisely, we study a natural subgroup Aut* W of Aut W, with Aut* W = Aut W whenever vertex groups areExpand
McCool groups of toral relatively hyperbolic groups
The outer automorphism group Out(G) of a group G acts on the set of conjugacy classes of elements of G. McCool proved that the stabilizer $Mc(c_1,...,c_n)$ of a finite set of conjugacy classes isExpand
A fixed point theorem for deformation spaces of G-trees
For a finitely generated free group Fn, of rank at least 2, any finite subgroup of Out(Fn) can be realized as a group of automorphisms of a graph with fundamental group Fn. This result, known asExpand
The Outer Automorphism Groups of Three Classes of Groups
The theory of outer automorphism groups allows us to better understand groups through their symmetries, and in this thesis we approach outer automorphism groups from two directions. In the firstExpand
On finitely generated normal subgroups of K\"ahler groups
We prove that if a surface group embeds as a normal subgroup in a Kähler group and the conjugation action of the Kähler group on the surface group preserves the conjugacy class of a non-trivialExpand
Free-by-cyclic groups, automorphisms and actions on nearly canonical trees
We study the automorphism groups of free-by-cyclic groups and show these are finitely generated in the following cases: (i) when defining automorphism has linear growth and (ii) when the rank of theExpand
A tale of two groups: arithmetic groups and mapping class groups
In this chapter, we discuss similarities, differences and interaction between two natural and important classes of groups: arithmetic subgroups Γ of Lie groups G and mapping class groups Modg,n ofExpand


Virtually free groups with finitely many outer automorphisms
Let G be a finitely generated virtually free group. From a presentation of G as the fundamental group of a finite graph of finite-by-cyclic groups, necessary and sufficient conditions are derived forExpand
Automorphisms of free groups with boundaries
The automorphisms of free groups with boundaries form a fam- ily of groups An,k closely related to mapping class groups, with the stan- dard automorphisms of free groups as An,0 and (essentially) theExpand
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 paperExpand
Structure and Rigidity in (Gromov) Hyperbolic Groups and Discrete Groups in Rank 1 Lie Groups II
Abstract. We borrow the Jaco-Shalen-Johannson notion of characteristic sub-manifold from 3-dimensional topology to study cyclic splittings of torsion-free (Gromov) hyperbolic groups and finitelyExpand
Automorphism groups of tree actions and of graphs of groups
Let Γ be a group. The minimal non-abelian Γ-actions on real trees can be parametrized by the projective space of the associated length functions. The outer automorphism group of Γ, Out(Γ) =Expand
Cut points and canonical splittings of hyperbolic groups
In this paper, we give a construction of the JSJ splitting of a one-ended hyperbolic group (in the sense of Gromov [Gr]), using the local cut point structure of the boundary. In particular, thisExpand
Automorphisms of free groups have asymptotically periodic dynamics
Abstract We show that every automorphism α of a free group Fk of finite rank k has asymptotically periodic dynamics on Fk and its boundary ∂Fk : there exists a positive power α q such that everyExpand
On the outer automorphism group of a hyperbolic group
LetG be a one-ended, word-hyperbolic group. Let Γ be an irreducible lattice in a connected semi-simple Lie group of rank at least 2. Ifh: Γ→Out(G) is a homomorphism, then Im(h) is finite.
On injective homomorphisms between Teichmüller modular groups I
Abstract. In this paper and its sequel, we prove that injective homomorphisms between Teichmüller modular groups of compact orientable surfaces are necessarily isomorphisms, if an appropriatelyExpand
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 studyExpand