Most automorphisms of a hyperbolic group have very simple dynamics

Levitt, Gilbert; Lustig, Martin

doi:10.1016/s0012-9593(00)00120-8

Cited by 79 publications

(105 citation statements)

References 15 publications

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“…The subgroup Z D Z.ker.// is characteristic, soį nduces an automorphismˇon the virtually free group G=Z . As G=Z is a nonelementary (word) hyperbolic group, R.ˇ/ is infinite (Levitt-Lustig [24], Fel'shtyn [9]). This implies that R.˛/ is infinite.…”

Section: This Generalizes Results Of [10] About Baumslag-solitar Groupsmentioning

confidence: 99%

On the automorphism group of generalized Baumslag–Solitar groups

Levitt

2007

Geom. Topol.

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A generalized Baumslag-Solitar group (GBS group) is a finitely generated group G which acts on a tree with all edge and vertex stabilizers infinite cyclic. We show that Out(G) either contains non-abelian free groups or is virtually nilpotent of class ≤ 2. It has torsion only at finitely many primes.One may decide algorithmically whether Out(G) is virtually nilpotent or not. If it is, one may decide whether it is virtually abelian, or finitely generated. The isomorphism problem is solvable among GBS groups with Out(G) virtually nilpotent.If G is unimodular (virtually F n × Z), then Out(G) is commensurable with a semi-direct product Z k ⋊ Out(H) with H virtually free. Contents1. Introduction and statement of results 2. Basic facts about GBS groups 3. The automorphism group of a GBS tree 4. Unimodular groups 5. The deformation space 6. Free subgroups in Out(G) 7. Groups with Out(G) ⊃ F 2 8. Further results References Given Γ, let T be the associated Bass-Serre tree, which we call a GBS tree. Let Out T (G) ⊂ Out(G) be the subgroup leaving T invariant. Most elements of Out T (G) may be viewed as "twists" (see Section 3). Algebraic rigidity implies Out T (G) = Out(G), but in general Out T (G) is smaller.Theorem 1.1. Let G be a GBS group, represented by a labelled graph Γ, and let T be the Bass-Serre tree. Define k as the first Betti number b of Γ if G has a nontrivial center, as b − 1 if the center is trivial.(1) The torsion-free rank of the abelianization of(3) Up to commensurability within Out(G), the subgroup Out T (G) does not depend on Γ.Conversely, any subgroup of Out(G) commensurable with a subgroup of Out T (G) is contained in Out T ′ (G) for some GBS tree T ′ [4].For G = BS(m, n), one has k = 0 if m = n, and k = 1 if m = n. For G = BS(2, 4) with the presentation (1 p ), the group Out T (G) has order 2 p+1 .The converse is also true (see Theorem 8.5). Unimodular groups.A GBS group G is unimodular if xy p x −1 = y q with y = 1 implies |p| = |q|, or equivalently if G is virtually F n × Z (with F n a free group of rank n). The group G then has a normal infinite cyclic subgroup with virtually free quotient, and we show:where k is as above, H is virtually free, and Out 0 has finite index in Out. Since Out(H) is VF [19], we get:Corollary 1.4. Out(G) is virtually torsion-free and VF (it has a finite index subgroup admitting a finite classifying space).Groups with no nontrivial integral modulus. Now consider groups G which do not contain a solvable Baumslag-Solitar group BS(1, n) with n ≥ 2 (there is an equivalent characterization in terms of the modular homomorphism ∆ : G → Q * , see Section 2).Given any GBS group G, the group Out(G) acts on the space P D of all GBS trees (see Section 5), with stabilizers virtually Z k by Theorem 1.1. Clay [3] proved that the space P D is contractible (see also [16]), and Forester [12] proved that the quotient is a finite complex if G does not contain BS(1, n) with n ≥ 2. This gives: 3

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Section: This Generalizes Results Of [10] About Baumslag-solitar Groupsmentioning

confidence: 99%

On the automorphism group of generalized Baumslag–Solitar groups

Levitt

2007

Geom. Topol.

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“…Following [Taback and Wong 2007], we say a group G has the property R ∞ if R(ϕ) is infinite for every automorphism ϕ of G. Many classes of groups have the property R ∞ : nonelementary Gromov hyperbolic groups [Levitt and Lustig 2000;Fel'shtyn 2001], nonabelian generalized Baumslag-Solitar groups [Levitt 2007], saturated weakly branch groups (including the Grigorchuk group and the Gupta-Sidki group) [Fel'shtyn et al 2008a], and the Thompson's group F [Bleak et al 2008].…”

Section: Introductionmentioning

confidence: 99%

Sigma theory and twisted conjugacy classes

Gonçalves¹,

Kochloukova²

2010

Pacific J. Math.

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“…There are only finitely many isogredience classes of principal automorphisms. In fact by Levitt and Lustig [14], for all but finitely many isogredience classes, the only fixed points of ŷ are a source and a sink.…”

Section: Lines and Laminationsmentioning

confidence: 98%

Abelian subgroups of Out(F_n)

Feighn

Handel

2009

Geom. Topol.

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We classify abelian subgroups of Out(F n ) up to finite index in an algorithmic and computationally friendly way. A process called disintegration is used to canonically decompose a single rotationless element φ into a composition of finitely many elements and then use these elements to generate an abelian subgroup A(φ) that contains φ. The main theorem is that up to finite index every abelian subgroup is realized by this construction. As an application we classify, up to finite index, abelian subgroups of Out(F n ) and of IA n with maximal rank.

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Most automorphisms of a hyperbolic group have very simple dynamics

Cited by 79 publications

References 15 publications

On the automorphism group of generalized Baumslag–Solitar groups

On the automorphism group of generalized Baumslag–Solitar groups

Sigma theory and twisted conjugacy classes

Abelian subgroups of Out(F_n)

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Most automorphisms of a hyperbolic group have very simple dynamics

Cited by 79 publications

References 15 publications

On the automorphism group of generalized Baumslag–Solitar groups

On the automorphism group of generalized Baumslag–Solitar groups

Sigma theory and twisted conjugacy classes

Abelian subgroups of Out(Fn)

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Abelian subgroups of Out(F_n)