Geometry & Topology

Automorphisms and abstract commensurators of 2–dimensional Artin groups

John Crisp

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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.

Article information

Geom. Topol., Volume 9, Number 3 (2005), 1381-1441.

Received: 18 December 2004
Revised: 2 August 2005
Accepted: 4 July 2005
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F36: Braid groups; Artin groups 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20F67: Hyperbolic groups and nonpositively curved groups

2–dimensional Artin group Coxeter group commensurator group graph automorphisms triangle free


Crisp, John. Automorphisms and abstract commensurators of 2–dimensional Artin groups. Geom. Topol. 9 (2005), no. 3, 1381--1441. doi:10.2140/gt.2005.9.1381.

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