Geometry & Topology

Automorphisms of $2$–dimensional right-angled Artin groups

Ruth Charney, John Crisp, and Karen Vogtmann

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We study the outer automorphism group of a right-angled Artin group AΓ in the case where the defining graph Γ is connected and triangle-free. We give an algebraic description of Out(AΓ) in terms of maximal join subgraphs in Γ and prove that the Tits’ alternative holds for Out(AΓ). We construct an analogue of outer space for Out(AΓ) and prove that it is finite dimensional, contractible, and has a proper action of Out(AΓ). We show that Out(AΓ) has finite virtual cohomological dimension, give upper and lower bounds on this dimension and construct a spine for outer space realizing the most general upper bound.

Article information

Geom. Topol., Volume 11, Number 4 (2007), 2227-2264.

Received: 4 August 2007
Accepted: 7 September 2007
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F36: Braid groups; Artin groups
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20F28: Automorphism groups of groups [See also 20E36]

right-angled Artin groups outer automorphisms outer space


Charney, Ruth; Crisp, John; Vogtmann, Karen. Automorphisms of $2$–dimensional right-angled Artin groups. Geom. Topol. 11 (2007), no. 4, 2227--2264. doi:10.2140/gt.2007.11.2227.

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