Rigidity of Graph Products of Abelian Groups

Gutiérrez, Mauricio; Piggott, Adam

doi:10.1017/s0004972708000105

Cited by 5 publications

(12 citation statements)

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“…Our purpose is to find a set of generators for Aut G using the properties of (Γ, o). The new feature in the present paper is, as in [9], the mixture of infinite and finite cyclic groups.…”

Section: Introductionmentioning

confidence: 91%

“…As usual the support of a CR element u is S. Assume v ∈ S. If S lies in some complete subgraph, the result is found in [9,Lemma 2.2]. Assume then that S does not lie in a complete subgraph.…”

Section: Appendix a A Proof Of The Centralizer Theoremmentioning

confidence: 98%

“…In [9] it is shown that if G W (Λ, {H x }), then (1) there exists a labeled graph (Γ, o) with G W (Γ, o).…”

Section: Introductionmentioning

confidence: 99%

“…This result has been proved for RAAGs. The version we generalize in [9] is presented in [12] and Droms in [4] has an earlier and slightly weaker proof. For graph products of finite groups, in particular for RACGs, see [17].…”

Section: Introductionmentioning

confidence: 99%

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A Generating Set for the Automorphism Group of a Graph Product of Abelian Groups

Corredor

Gutiérrez

2012

Int. J. Algebra Comput.

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We find a set of generators for the automorphism group Aut G of a graph product G of finitely generated abelian groups entirely from a certain labeled graph. In addition, we find generators for the important subgroup Aut ⋆ G defined in [Automorphisms of graph products of abelian groups, to appear in Groups, Geometry and Dynamics]. We follow closely the plan of M. Laurence's paper [A generating set for the automorphism group of a graph group, J. London Math. Soc. (2)52(2) (1995) 318–334].

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“…Our purpose is to find a set of generators for Aut G using the properties of (Γ, o). The new feature in the present paper is, as in [9], the mixture of infinite and finite cyclic groups.…”

Section: Introductionmentioning

confidence: 91%

Section: Appendix a A Proof Of The Centralizer Theoremmentioning

confidence: 98%

“…In [9] it is shown that if G W (Λ, {H x }), then (1) there exists a labeled graph (Γ, o) with G W (Γ, o).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Generating Set for the Automorphism Group of a Graph Product of Abelian Groups

Corredor

Gutiérrez

2012

Int. J. Algebra Comput.

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“…The class of graph products of directly-indecomposable cyclic groups is identical to the class of graph products of finitely-generated abelian groups for the following reason: if G is group and G is isomorphic to a graph product of finitely-generated abelian groups, then there exists a unique isomorphism class of labeled-graphs (Γ, m) such that G ∼ = W (Γ, m) [12]. Empowered by this fact, we usually omit mention of Γ and m from the notation, writing W := W (Γ, m).…”

Section: Introductionmentioning

confidence: 99%

On the automorphisms of a graph product of abelian groups

Gutiérrez¹,

Piggott²,

Ruane³

2012

Groups Geom. Dyn.

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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 are finite and in a number of other cases. We prove a number of structure results, including a semi-direct product decomposition Aut * W = (Inn W ⋊ Out 0 W ) ⋊ Aut 1 W . We also give a number of applications, some of which are geometric in nature.complete subgroups are a set of representatives for the conjugacy classes of maximal finite subgroups of W [9, Lemma 4.5] and each automorphism of W maps each maximal complete subgroup to a conjugate of some maximal complete subgroup. This is not true in an arbitrary graph product of directlyindecomposable cyclic groups, but we may pretend that it is by restricting our attention to a natural subgroup of Aut W .Definition 1.1. Write Aut * W for the subgroup of Aut W consisting of those automorphisms which map each maximal complete subgroup to a conjugate of a maximal complete subgroup.The following lemma is immediate from the discussion above and the main result of [18]. For each 1 ≤ i ≤ N, we write L i (resp, S i ) for the link (resp. star ) of v i . Lemma 1.2. If W is a graph product of directly-indecomposable cyclic groups, then Aut * W = Aut W in each of the following cases:1. W is a graph product of primary cyclic groups; 2. W is a right-angled Artin group and L i ⊆ L j for each pair of distinct non-adjacent vertices v i , v j ∈ V ;3. W is a right-angled Artin group and Γ contains no vertices of valence less than two and no circuits of length less than 5.Remark 1.3. Case (2) can be substantially generalized to groups that are not right-angled Artin groups.We now report the main results of the present article. They concern the structure of Aut * W and shall make reference to the subgroups and quotients of Aut W defined in Figure 1. In writing Aut 0 W for the subgroup of 'conjugating automorphisms', we follow Tits [26]. Mühlherr [24] writes Spe(W ) for the same subgroup. Charney, Crisp and Vogtmann [5] use the notation Aut 0 W and Out 0 W for different subgroups of the automorphism group of a right-angled Artin group than described here.Tits [26] proved that if W is a right-angled Coxeter group, then Aut W = Aut 0 W ⋊ Aut 1 W . Our first main result is a generalization of Tits' splitting.Theorem 1.4 (cf. [26]). If W is a graph product of directly-indecomposable cyclic groups, then Aut * W = Aut 0 W ⋊ Aut 1 W .

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Actions of right-angled Coxeter groups on the Croke–Kleiner spaces

Qing

2016

Geom Dedicata

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It is an open question whether right-angled Coxeter groups have unique group-equivariant visual boundaries. In [4], Croke and Kleiner present a right-angled Artin group with more than one visual boundary. In this paper we present a right-angled Coxeter group with non-unique equivariant visual boundary. The main theorem is that if right-angled Coxeter groups act geometrically on a Croke-Kleiner spaces constructed in [4], then the local angles in those spaces all have to be π/2. We present a specific right-angled Coxeter group with non-unique equivariant visual boundary. However, we conjecture that the right angled Coxeter groups that can act geometrically on a given CAT(0) space are far from unique.This implies that the "right-angled" in the terminology "right-angled Coxeter group" turns out to be literal and is consistent with the "geometric" property of the group. Furthermore, given the result from [3] that if we fix the gluing angle of the Croke-Kleiner space at π/2 and change the side lengths of the tori, the resulting boundaries are not equivariantly homeomorphic to each other, we conclude the following: Corollary 1.2. There exists a right-angled Coxeter group that does not have unique equivariant visual boundary.

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Rigidity of Graph Products of Abelian Groups

Cited by 5 publications

References 5 publications

A Generating Set for the Automorphism Group of a Graph Product of Abelian Groups

A Generating Set for the Automorphism Group of a Graph Product of Abelian Groups

On the automorphisms of a graph product of abelian groups

Actions of right-angled Coxeter groups on the Croke–Kleiner spaces

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