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Automorphisms and abstract commensurators of 2–dimensional Artin groups

John Crisp

Geometry & Topology 9 (2005) 1381–1441

arXiv: math.GR/0410205


In this paper we consider the class of 2–dimensional Artin groups with connected, large type, triangle-free defining graphs (type CLTTF). We classify these groups up to isomorphism, and describe a generating set for the automorphism group of each such Artin group. In the case where the defining graph has no separating edge or vertex we show that the Artin group is not abstractly commensurable to any other CLTTF Artin group. If, moreover, the defining graph satisfies a further “vertex rigidity” condition, then the abstract commensurator group of the Artin group is isomorphic to its automorphism group and generated by inner automorphisms, graph automorphisms (induced from automorphisms of the defining graph), and the involution which maps each standard generator to its inverse.

We observe that the techniques used here to study automorphisms carry over easily to the Coxeter group situation. We thus obtain a classification of the CLTTF type Coxeter groups up to isomorphism and a description of their automorphism groups analogous to that given for the Artin groups.

2–dimensional Artin group, Coxeter group, commensurator group, graph automorphisms, triangle free
Mathematical Subject Classification 2000
Primary: 20F36, 20F55
Secondary: 20F65, 20F67
Forward citations
Received: 18 December 2004
Revised: 2 August 2005
Accepted: 4 July 2005
Published: 5 August 2005
Proposed: Joan Birman
Seconded: Walter Neumann, Martin Bridson
John Crisp
IMB (UMR 5584 du CNRS)
Université de Bourgogne
BP 47 870
21078 Dijon