Automorphisms of nearly finite Coxeter groups

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 AutðWÞ 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. 2000 Mathematics Subject Classification. Primary 20F55 There is not much literature dealing with the automorphism groups of infinite Coxeter groups.1 It seems that complete results are known only for rank 3 Coxeter groups and the so-called right-angled Coxeter groups. A Coxeter group is right-angled if the labels on all edges in the Coxeter diagram are y. These were investigated by James, [12], who described the automorphism groups of Coxeter groups whose diagrams have the following form: