Outer space for untwisted automorphisms of right-angled Artin groups

Charney, Ruth; Stambaugh, Nathaniel; Vogtmann, Karen

doi:10.2140/gt.2017.21.1131

Cited by 26 publications

(105 citation statements)

References 16 publications

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“…The untwisted subgroup U (A Γ ) of Out(A Γ ) is defined to be the subgroup generated by So U (A Γ ) is generated by the Laurence generating set with the adjacent transvections taken out. Charney-Stambaugh-Vogtmann [11] have shown that U (A Γ ) has a proper action on a space, called an outer space for A Γ . Let U 0 (A Γ ) be the intersection of the untwisted subgroup with Out 0 (A Γ ).…”

Section: Previously Studied Relative Automorphism Groupsmentioning

confidence: 99%

Relative automorphism groups of right‐angled Artin groups

Day

Wade

2019

Journal of Topology

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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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Section: Previously Studied Relative Automorphism Groupsmentioning

confidence: 99%

Relative automorphism groups of right‐angled Artin groups

Day

Wade

2019

Journal of Topology

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“…Some of the terminology has been altered slightly, and we will point this out when it occurs. We refer to [5] for more details and all proofs. 3.1.…”

Section: 2mentioning

confidence: 99%

“…For the subgroup U (A Γ ) associated to a general RAAG, an analogous outer space O Γ and spine K Γ were defined in [5]. The dimension of K Γ gives an obvious upper bound on the vcd of U (A Γ ).…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2 we review basic facts and notation about right-angled Artin groups and their automorphisms, and define the subgroup U (A Γ ). In Section 3 we review the definitions and results from [5] that we will need in this paper. In Section 4 we construct free abelian subgroups of U (A Γ ) using Γ-Whitehead automorphisms, and show that these subgroups have maximal possible rank among all such subgroups.…”

Section: Introductionmentioning

confidence: 99%

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Cube complexes and abelian subgroups of automorphism groups of RAAGs

Millard¹,

Vogtmann²

2019

Math. Proc. Camb. Phil. Soc.

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We construct free abelian subgroups of the group U (AΓ) of untwisted outer automorphisms of a right-angled Artin group, thus giving lower bounds on the virtual cohomological dimension. The group U (AΓ) was studied in [5] by constructing a contractible cube complex on which it acts properly and cocompactly, giving an upper bound for the virtual cohomological dimension. The ranks of our free abelian subgroups are equal to the dimensions of principal cubes in this complex. These are often of maximal dimension, so that the upper and lower bounds agree. In many cases when the principal cubes are not of maximal dimension we show there is an invariant contractible subcomplex of strictly lower dimension.

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“…The group