Automorphisms of 2–dimensional right-angled Artin groups

Charney, Ruth¹ ; Crisp, John² ; Vogtmann, Karen³

¹ Department of Mathematics, Brandeis University, Waltham MA 02454-9110, USA
² Institut de Mathematiques de Bourgogne, Université de Bourgogne, B P 47 870, 21078 Dijon Cedex, France
³ Department of Mathematics, Cornell University, Ithaca NY 14853-4201, USA

Geometry & topology, Tome 11 (2007) no. 4, pp. 2227-2264.

Résumé

We study the outer automorphism group of a right-angled Artin group $A_{\Gamma }$ in the case where the defining graph $\Gamma$ is connected and triangle-free. We give an algebraic description of $Out\left(A_{\Gamma }\right)$ in terms of maximal join subgraphs in $\Gamma$ and prove that the Tits’ alternative holds for $Out\left(A_{\Gamma }\right)$. We construct an analogue of outer space for $Out\left(A_{\Gamma }\right)$ and prove that it is finite dimensional, contractible, and has a proper action of $Out\left(A_{\Gamma }\right)$. We show that $Out\left(A_{\Gamma }\right)$ 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.

Détail

DOI : 10.2140/gt.2007.11.2227

Keywords: right-angled Artin groups, outer automorphisms, outer space

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