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Cube complexes and abelian subgroups of automorphism groups of RAAGs

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  20 November 2019

BENJAMIN MILLARD  and
KAREN VOGTMANN
BENJAMIN MILLARD
Affiliation:
Department of Mathematics, University College London, 25 Gordon Street, London WC1H 0AY, U.K. e-mail: [email protected]
KAREN VOGTMANN
Affiliation:
Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, U.K. e-mail: [email protected]
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Abstract

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.

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 20F28: Automorphism groups of groups 20F36: Braid groups; Artin groups
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 3 , May 2021 , pp. 523 - 547
Copyright
© Cambridge Philosophical Society 2019

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