Tree actions of automorphism groups

Gilbert, N. D.; Howie, James; Metaftsis, V.; Raptis, E.

doi:10.1515/jgth.2000.017

Cited by 23 publications

(40 citation statements)

References 10 publications

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“…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%

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: Collapses and Algebraic Rigiditymentioning

confidence: 99%

On the automorphism group of generalized Baumslag–Solitar groups

Levitt

2007

Geom. Topol.

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“…It can also happen that there is only one reduced graph, or finitely many. In these latter cases, useful information about Out.G/ can be obtained, as in Gilbert-Howie-Metaftsis-Raptis [8], Pettet [15] and Levitt [13]. Other aspects of GBS groups have been studied by Kropholler, Whyte, Levitt and others.…”

Section: Introductionmentioning

confidence: 95%

On the isomorphism problem for generalized Baumslag–Solitar groups

Clay

Forester

2008

Algebr. Geom. Topol.

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Generalized Baumslag-Solitar groups (GBS groups) are groups that act on trees with infinite cyclic edge and vertex stabilizers. Such an action is described by a labeled graph (essentially, the quotient graph of groups). This paper addresses the problem of determining whether two given labeled graphs define isomorphic groups; this is the isomorphism problem for GBS groups. There are two main results and some applications. First, we find necessary and sufficient conditions for a GBS group to be represented by only finitely many reduced labeled graphs. These conditions can be checked effectively from any labeled graph. Then we show that the isomorphism problem is solvable for GBS groups whose labeled graphs have first Betti number at most one. 20E08; 20F10, 20F28

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“…The strongly-slide free condition gives the following fact (see [7]): Lemma 2.1 Assume that e 1 , e 2 ∈ E(T ) are two edges sharing a common vertex v and that f (e 1 ) ∩ f (e 2 ) is not reduced to one point. Then e 1 and e 2 are in the same G v -orbit and f (e 1 ) ∩ f (e 2 ) is strictly contained in f (e 1 ) (resp.…”

Section: Let's Now Define a G-equivariant Map F : T → T For Each Vementioning

confidence: 99%

A very short proof of Forester’s rigidity result

Guirardel

2003

Geom. Topol.

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The deformation space of a simplicial G-tree T is the set of G-trees which can be obtained from T by some collapse and expansion moves, or equivalently, which have the same elliptic subgroups as T . We give a short proof of a rigidity result by Forester which gives a sufficient condition for a deformation space to contain an Aut(G)-invariant G-tree. This gives a sufficient condition for a JSJ splitting to be invariant under automorphisms of G. More precisely, the theorem claims that a deformation space contains at most one strongly slidefree G-tree, where strongly slide-free means the following: whenever two edges e 1 , e 2 incident on a same vertex v are such that G e 1 ⊂ G e 2 , then e 1 and e 2 are in the same orbit under G v .

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Tree actions of automorphism groups

Cited by 23 publications

References 10 publications

On the automorphism group of generalized Baumslag–Solitar groups

On the automorphism group of generalized Baumslag–Solitar groups

On the isomorphism problem for generalized Baumslag–Solitar groups

A very short proof of Forester’s rigidity result

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