Geometry & Topology, Vol. 9 (2005) Paper no. 32, pages 1381--1441.

Automorphisms and abstract commensurators of 2-dimensional Artin groups

John Crisp

Abstract. 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.

Keywords. 2-dimensional Artin group, Coxeter group, commensurator group, graph automorphisms, triangle free

AMS subject classification. Primary: 20F36, 20F55. Secondary: 20F65, 20F67.

E-print: arXiv:math.GR/0410205

DOI: 10.2140/gt.2005.9.1381

Submitted to GT on 18 December 2004. (Revised 2 August 2005.) Paper accepted 4 July 2005. Paper published 5 August 2005.

Notes on file formats

John Crisp
IMB(UMR 5584 du CNRS), Universite de Bourgogne
BP 47 870, 21078 Dijon, France

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