Automorphisms of nearly finite Coxeter groups

Abstract

Suppose that W is an infinite Coxeter group of finite rank n, and suppose that W has a finite parabolic subgroup WJ of rank n � 1. Suppose also that the Coxeter diagram of W has no edges with infinite labels. Then any automorphism of W that preserves reflections lies in the subgroup of AutðW Þ generated by the inner automorphisms and the automorphisms induced by symmetries of the Coxeter graph. If, in addition, WJ is irreducible and every conjugacy class of reflections in W has nonempty intersection with WJ , then all automorphisms of W preserve reflections, and it follows that AutðW Þ is the semi-direct product of InnðW Þ by the group of graph automorphisms.