Deformation and rigidity of simplicial group actions on trees

Forester, Max

doi:10.2140/gt.2002.6.219

Cited by 67 publications

(129 citation statements)

References 15 publications

(33 reference statements)

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“…As any two GBS trees have the same elliptic subgroups, Forester's deformation theorem [11] yields the following.…”

Section: Corollary 22mentioning

confidence: 99%

“…Labels near o.e/ get multiplied by x e when we collapse an edge e with e D 1 (see Figure 5). The graph , or the tree T , is reduced (in the sense of Forester [11]) if there is no collapsible edge. In terms of trees, T is reduced if and only if any edge e D vw satisfying G e D G v has its endpoints in the same G -orbit.…”

Section: Collapses and Algebraic Rigiditymentioning

confidence: 99%

“…Building on work from Forester [11], Gilbert et al [14], Guirardel [15] and Pettet [30], it is shown in Levitt [23] that, if G is not solvable, T is rigid if and only if satisfies the following condition (see Figure 6): if e; f are distinct oriented edges of with the same origin v , and the label of f near v divides that of e , then either e D x f is a .p;˙p/-loop with p 2, or v has valence 3 and bounds a .1;˙1/-loop.…”

Section: Collapses and Algebraic Rigiditymentioning

confidence: 99%

“…Lemma 2.1 (Forester [11]) Let T be a GBS tree, with G non-elementary. Any two non-trivial elliptic elements g; g 0 are commensurable.…”

Section: Elliptic Elementsmentioning

confidence: 99%

“…In the language of [11], we first show that all possible trees S belong to the same deformation space, and then using [23] that there is only one reduced tree in that space. In particular Out.G/ D Out S .G/, and we shall conclude the proof of Theorem 7.1 by showing that Out S .G/ is virtually nilpotent.…”

Section: Labels Near a 2 ; 2/-segmentmentioning

confidence: 99%

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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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“…As any two GBS trees have the same elliptic subgroups, Forester's deformation theorem [11] yields the following.…”

Section: Corollary 22mentioning

confidence: 99%

Section: Collapses and Algebraic Rigiditymentioning

confidence: 99%

Section: Collapses and Algebraic Rigiditymentioning

confidence: 99%

“…Lemma 2.1 (Forester [11]) Let T be a GBS tree, with G non-elementary. Any two non-trivial elliptic elements g; g 0 are commensurable.…”

Section: Elliptic Elementsmentioning

confidence: 99%

Section: Labels Near a 2 ; 2/-segmentmentioning

confidence: 99%

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On the automorphism group of generalized Baumslag–Solitar groups

Levitt

2007

Geom. Topol.

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A General Construction of JSJ Decompositions

Guirardel¹,

Levitt²

Trends in Mathematics

153

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We define the compatibility JSJ tree of a group G over a class of subgroups. It exists whenever G is finitely presented and leads to a canonical tree (not just a deformation space) which is invariant under automorphisms. Under acylindricity hypotheses, we prove that the (usual) JSJ deformation space and the compatibility JSJ tree both exist when G is finitely generated, and we describe their flexible subgroups. We apply these results to CSA groups, Γ-limit groups (allowing torsion), and relatively hyperbolic groups.This paper and its companion arXiv:0911.3173 have been replaced by arXiv:1602.05139.

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$$\mathcal {F}_\pi $$-residuality of generalized Baumslag–Solitar groups

Dudkin

2019

Arch. Math.

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Deformation and rigidity of simplicial group actions on trees

Cited by 67 publications

References 15 publications

On the automorphism group of generalized Baumslag–Solitar groups

On the automorphism group of generalized Baumslag–Solitar groups

A General Construction of JSJ Decompositions

$$\mathcal {F}_\pi $$-residuality of generalized Baumslag–Solitar groups

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