## Geometry & Topology

### 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.

#### Article information

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

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

https://projecteuclid.org/euclid.gt/1513799640

Digital Object Identifier
doi:10.2140/gt.2005.9.1381

Mathematical Reviews number (MathSciNet)
MR2174269

Zentralblatt MATH identifier
1135.20027

#### Citation

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. https://projecteuclid.org/euclid.gt/1513799640

#### References

• E Artin, Braids and permutations, Ann. of Math. 48 (1947) 643–649
• P Bahls, Automorphisms of Coxeter groups, Trans. Amer. Math. Soc. (to appear)
• P Bahls, Rigidity of 2–dimensional Coxeter groups, preprint (2004)
• N Brady, J McCammond, B Mühlherr, W Neumann, Rigidity of Coxeter groups and Artin groups, Geom. Dedicata 94 (2002) 91–109
• N Brady, J Crisp, Two dimensional Artin groups with CAT(0) dimension three, Geometriae Dedicata 94 (2002) 185–214
• M R Bridson, On the semisimplicity of polyhedral isometries, Proc. Amer. Math. Soc. 127 (1999) 2143–2146
• M R Bridson, A Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren series 319, Springer-Verlag (1999)
• R Charney, J. Crisp, Automorphism groups of some affine and finite type Artin groups, Math. Res. Letters 12 (2005) 321–333
• R Charney, M W Davis, The $K(\pi,1)$–problem for hyperplane complements associated to infinite reflection groups, J. Amer. Math. Soc. 8 (1995) 597–627
• A M Cohen, L Paris, On a theorem of Artin, J. Group Theory 6 (2003) 421–441
• C Droms, Isomorphisms of graph groups, Proc. Amer. Math. Soc. 100 (1987) 407–408
• N D Gilbert, J Howie, V Metaftsis, E Raptis, Tree actions of automorphism groups, J. Group Theory 3 (2000) 213–223
• E Godelle, Parabolic subgroups of Artin groups of type FC, Pacific J. Math. 208 (2003) 243–254
• E Godelle, Artin-Tits groups with CAT(0) Deligne complex, preprint
• H van der Lek, The Homotopy Type of Complex Hyperplane Complements, PhD Thesis, University of Nijmegen (1983)
• G Moussong, Hyperbolic Coxeter groups, PhD Thesis, Ohio State University (1988)
• B Mühlherr, The isomorphism problem for Coxeter groups, from: “The Coxeter Legacy: Reflections and Projections”, Fields Institute Communications (to appear)
• B Mühlherr, R Weidmann, Rigidity of skew-angled Coxeter groups, Adv. Geom. 2 (2002) 391–451
• G Niblo, L Reeves, The geometry of cube complexes and the complexity of their fundamental groups, Topology 37 (1998) 621–633
• L Paris, Artin groups of spherical type up to isomorphism, J. Algebra 281 (2004) 666–678
• M Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. 71 (1995) 585–617