Automorphisms of Hyperbolic Groups and Graphs of Groups

By Gilbert Levitt. 2002

114 citations recorded by the source corpus.

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 decomposition of an arbitrary group G such that edge groups Ge are rigid (i.e. Out(Ge) is finite). We describe the group of automorphisms of G preserving Γ, by comparing it to direct products of suitably defined mapping class groups of vertex groups.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4294a4d6c3ac6ba4ab9c9474a8c0ef5c5ed5ba8e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Vincent Guirardel
2. Franccois Dahmani
3. Gilbert Levitt
4. Radhika Gupta
5. Benjamin Brück
6. E. Troitsky
7. Alexander Fel'shtyn

In Discrete Mathematics and Combinatorics

1. Vincent Guirardel
2. Franccois Dahmani
3. Gilbert Levitt
4. Alessandro Sisto
5. Denis V. Osin
6. Jason A. Behrstock
7. Radhika Gupta
8. Benjamin Brück

In Geometry and Topology

1. Vincent Guirardel
2. Gilbert Levitt
3. Franccois Dahmani
4. Denis V. Osin
5. Alessandro Sisto
6. D. Groves
7. Henry Wilton
8. Nathanial P. Brown Narutaka Ozawa

Citations in the corpus

• On the automorphism group of generalized Baumslag–Solitar groups (2005, methodology): “We may now study Out(G) = Out(G) using the results of [20] recalled in Section 3.”
• Jsj Decompositions of Groups (2016, background): “twocornerreflectorscarrying D 4,ityieldsasplittingoverD 1butnotwist. Wereferto[GL15b](especiallyTheorem7.14)forageneraldiscussionoftherelations between splittings and automorphisms (see also [DG11] or [Lev05a], and the discussion in Subsection 9.5). Here we limit ourselves to classifying orbifold gro”
• Deformation spaces of trees (2006, methodology): “We study its stabilizer Out (G) ⊂ Out(G) using results and notations from [24].”
• The Isomorphism Problem for All Hyperbolic Groups (2010, background): “3. There exists R> 0 such that, modulo inner automorphisms, there are only finitely many endomorphisms of G injective on the ball of radius R. 4. For all hyperbolic group H, there exists R> 0 such that, modulo inner automorphisms of H, there are only finitely many morphisms G → H injective on the ba”