Automorphisms of Coxeter groups of rank three (Q2716103)

Let \(W\) be a Coxeter group. Then \(W\) is a group with presentation of the form NEWLINE\[NEWLINEW=\langle S\mid s^2=e\;\forall s\in S,\;(st)^{m_{st}}=e\;\forall s\not=t\in S\rangle,NEWLINE\]NEWLINE where \(m_{st}\in \mathbb{Z}\cup\{\infty\}\) for all \(s,t\in S\), \(s\not=t\), and \(m_{st}=\infty\) means that there is no relation for this pair \(s,t\). The Coxeter diagram of \(W\) has vertices labelled by \(S\), and, for each \(s,t\in S\) there is an edge joining \(s\) and \(t\) labelled by \(m_{st}\). It is conventional to omit this edge if \(m_{st}=2\), and to omit the label if \(m_{st}=3\). The finite Coxeter groups are reflection groups, and the crystallographic ones (most of them) are the Weyl groups of the finite-dimensional simple Lie algebras over the complex numbers. The rank of a Coxeter group is defined to be the cardinality of \(S\).NEWLINENEWLINENEWLINEIf \(\alpha\) is an automorphism of the Coxeter diagram of \(W\), then it induces a bijection of \(S\) which can be extended uniquely to an automorphism of \(W\). Such an automorphism is called a graph automorphism of \(W\). The authors show that if \(W\) is an infinite, rank \(3\) Coxeter group, whose Coxeter diagram has no edges labelled \(\infty\), then the automorphism group of \(W\) is the semi-direct product of \(\text{Inn}(W)\) (the inner automorphism group of \(W\)) and the group of graph automorphisms. The proof involves studying the effect of a group automorphism of \(W\) on maximal finite parabolic subgroups of \(W\); the degenerate case (when \(W\) is an affine Weyl group) is dealt with separately.