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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Automorphisms of $2$–dimensional right-angled Artin groups

Ruth Charney, John Crisp and Karen Vogtmann

Geometry & Topology 11 (2007) 2227–2264

arXiv: math/0610980v2

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.

Keywords
right-angled Artin groups, outer automorphisms, outer space
Mathematical Subject Classification 2000
Primary: 20F36
Secondary: 20F65, 20F28
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Publication
Received: 4 August 2007
Accepted: 7 September 2007
Published: 14 November 2007
Proposed: John Morgan
Seconded: Martin Bridson, Joan Birman
Authors
Ruth Charney
Department of Mathematics
Brandeis University
Waltham MA 02454-9110
USA
http://people.brandeis.edu/~charney/
John Crisp
Institut de Mathematiques de Bourgogne
Université de Bourgogne
B P 47 870
21078 Dijon Cedex
France
Karen Vogtmann
Department of Mathematics
Cornell University
Ithaca NY 14853-4201
USA
http://www.math.cornell.edu/~vogtmann/