Subspace arrangements, BNS invariants, and pure symmetric outer automorphisms of right-angled Artin groups

We study the outer automorphism group of a right-angled Artin group AΓ with finite defining graph Γ. We construct a subnormal series for Out(AΓ) such that each consecutive quotient is either finite, free-abelian, GL(n, Z) or a Fouxe-Rabinovitch group. The last two types act, respectively, on a symmetric space or a deformation space of trees, so that there is a geometric way of studying each piece. As a consequence we prove that the group Out(AΓ) is type VF (it has a finite index subgroup with a finite classifying space).The main technical work is a study of relative outer automorphism groups of RAAGs and their restriction homomorphisms, refining work of Charney, Crisp and Vogtmann. We show that the images and kernels of restriction homomorphisms are always simpler examples of relative outer automorphism groups of RAAGs. We also give generators for relative automorphism groups of RAAGs, in the style of Laurence's theorem.

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“…Then

Aut (A_{normalΓ}; {scriptG}^{t})

is the pure symmetric automorphism group and

Out (A_{normalΓ}; {scriptG}^{t})

is the pure symmetric outer automorphism group . These groups have been studied by Day–Wade , Koban–Piggott and Toinet .…”

Section: Computation and Examplesmentioning

confidence: 99%

Relative automorphism groups of right‐angled Artin groups

Day

Wade

2019

Journal of Topology

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“…. , r 20 } in the free group F = F (a 1 , a 2 ), we denote by Γ(S) the group with generators a 1 , a 2 , a 3 , a 4 Note that this presentation is of the type described in Theorem 1.1. The group Γ(S) is an HNN extension of F × F with a stable letter t that commutes with the fibre product P < F × F associated to the presentation P = a, b | S .…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…In more detail, there is no algorithm that, given 22 words u i in the free group F (a 1 , a 2 , a 3 , a 4 ) can determine whether or not the group with presentation a 1 , a 2 , a 3 , a 4 is a RAAG. Nor is there an algorithm that can determine whether or not such a group is commensurable with a RAAG or quasi-isometric to a RAAG.…”

Section: Introductionmentioning

confidence: 99%

On the Recognition of Right-Angled Artin Groups

Bridson¹

2019

Glasgow Math. J.

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“…It says that when you have a SIL

(x_{1}, x_{2} | x_{3})

, there is a subset of

normalΓ

that appears as a connected component of each of the three sets

Γ ∖ st (x_{1})

Γ ∖ st (x_{2})

, and

Γ ∖ (lk false(x_{1} false) \cap lk false(x_{2} false))

. It follows from [, Lemma 4.5], and we also refer the reader to [, Section 2.3], however we include a short proof for completeness.…”

Section: Separating Intersections Of Linksmentioning

confidence: 99%

Vastness properties of automorphism groups of RAAGs

Guirardel

Sale

2018

Journal of Topology

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37 pages, 4 figuresInternational audienceOuter automorphism groups of RAAGs, denoted $Out(A_\Gamma)$, interpolate between $Out(F_n)$ and $GL_n(\mathbb{Z})$. We consider several vastness properties for which $Out(F_n)$ behaves very differently from $GL_n(\mathbb{Z})$: virtually mapping onto all finite groups, SQ-universality, virtually having an infinite dimensional space of homogeneous quasimorphisms, and not being boundedly generated. We give a neccessary and sufficient condition in terms of the defining graph $\Gamma$ for each of these properties to hold. Notably, the condition for all four properties is the same, meaning $Out(A_\Gamma)$ will either satisfy all four, or none. In proving this result, we describe conditions on $\Gamma$ that imply $Out(A_\Gamma)$ is large. Techniques used in this work are then applied to the case of McCool groups, defined as subgroups of $Out(F_n)$ that preserve a given family of conjugacy classes. In particular we show that any McCool group that is not virtually abelian virtually maps onto all finite groups, is SQ-universal, is not boundedly generated, and has a finite index subgroup whose space of homogeneous quasimorphisms is infinite dimensional

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Subspace arrangements, BNS invariants, and pure symmetric outer automorphisms of right-angled Artin groups

Cited by 9 publications

References 11 publications

Relative automorphism groups of right‐angled Artin groups

Relative automorphism groups of right‐angled Artin groups

On the Recognition of Right-Angled Artin Groups

Vastness properties of automorphism groups of RAAGs

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