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
Email: jcrisp@u-bourgogne.fr
URL: http://math.u-bourgogne.fr/IMB/crisp
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