## Geometry & Topology

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

#### Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513799981

Digital Object Identifier
doi:10.2140/gt.2007.11.2227

Mathematical Reviews number (MathSciNet)
MR2372847

Zentralblatt MATH identifier
1152.20032

#### Citation

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. https://projecteuclid.org/euclid.gt/1513799981

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