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.
