Calculating the virtual cohomological dimension of the automorphism group of a RAAG

Day, Matthew B.; Sale, Andrew W.; Wade, Richard D.

doi:10.1112/blms.12418

Cited by 8 publications

(6 citation statements)

References 25 publications

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“…If all the consecutive quotients are virtual duality groups, then so is Out(A Γ ) (when working with more general subnormal series one has to be a little bit careful about the passage to finite index subgroups, however here we can use congruence subgroups). This lends us to ask the following question, which also appeared briefly in [15]. Question 3.2.…”

Section: Further Discussionmentioning

confidence: 99%

A note on virtual duality and automorphism groups of right-angled Artin groups

Wade¹,

Brück²

2021

Preprint

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A theorem of Brady and Meier states that a right-angled Artin group is a duality group if and only if the flag complex of the defining graph is Cohen-Macaulay. We use this to give an example of a RAAG with the property that its outer automorphism group is not a virtual duality group. This gives a partial answer to a question of Vogtmann. In an appendix, Brück describes how he used a computer-assisted search to find further examples.

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Section: Further Discussionmentioning

confidence: 99%

A note on virtual duality and automorphism groups of right-angled Artin groups

Wade¹,

Brück²

2021

Preprint

Self Cite

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“…If all the consecutive quotients are virtual duality groups, then so is Out(A ) (when working with more general subnormal series one has to be a little bit careful about the passage to finite index subgroups, however here we can use congruence subgroups). This leads us to ask the following question, which also appeared briefly in [15].…”

Section: Obstructions To Duality: Fouxe-rabinovitch Groupsmentioning

confidence: 99%

A note on virtual duality and automorphism groups of right-angled Artin groups

Wade

Brück

2023

Glasgow Math. J.

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A theorem of Brady and Meier states that a right-angled Artin group is a duality group if and only if the flag complex of the defining graph is Cohen–Macaulay. We use this to give an example of a RAAG with the property that its outer automorphism group is not a virtual duality group. This gives a partial answer to a question of Vogtmann. In an appendix, Brück describes how he used a computer-assisted search to find further examples.

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“…This has proved a successful approach in some cases, remarkably with the definition of analogues of Outer Space (see Bregman, Charney and Vogtmann [20] and Charney, Stambaugh and Vogtmann [25]) and its consequences for the study of homological properties. However, there are limits to such analogies: in practice, techniques that are tailored to general RAAGs and based on induction on the complexity of the graph seem to provide the most effective approach to many problems; see for instance Charney and Vogtmann [27;28], Day and Wade [43], Day, Sale and Wade [42] and Guirardel and Sale [61].…”

Section: Introductionmentioning

confidence: 99%

Coarse-median preserving automorphisms

Fioravanti

2024

Geom. Topol.

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This paper has three main goals.First, we study fixed subgroups of automorphisms of right-angled Artin and Coxeter groups. If ' is an untwisted automorphism of a RAAG, or an arbitrary automorphism of a RACG, we prove that Fix ' is finitely generated and undistorted. Up to replacing ' with a power, we show that Fix ' is quasiconvex with respect to the standard word metric. This implies that Fix ' is a virtual retract and a special group in the sense of Haglund and Wise. By contrast, there exist "twisted" automorphisms of RAAGs for which Fix ' is undistorted but not of type F (hence not special), of type F but distorted, or even infinitely generated.Secondly, we introduce the notion of "coarse-median preserving" automorphism of a coarse median group, which plays a key role in the above results. We show that automorphisms of RAAGs are coarse-median preserving if and only if they are untwisted. On the other hand, all automorphisms of Gromov-hyperbolic groups and right-angled Coxeter groups are coarse-median preserving. These facts also yield new or more elementary proofs of Nielsen realisation for RAAGs and RACGs.Finally, we show that, for every special group G (in the sense of Haglund and Wise), every infinite-order, coarse-median preserving outer automorphism of G can be realised as a homothety of a finite-rank median space X equipped with a "moderate" isometric G-action. This generalises the classical result, due to Paulin, that every infinite-order outer automorphism of a hyperbolic group H projectively stabilises a small H -tree.

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Calculating the virtual cohomological dimension of the automorphism group of a RAAG

Cited by 8 publications

References 25 publications

A note on virtual duality and automorphism groups of right-angled Artin groups

A note on virtual duality and automorphism groups of right-angled Artin groups

A note on virtual duality and automorphism groups of right-angled Artin groups

Coarse-median preserving automorphisms

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