Automorphisms of nearly finite Coxeter groups

Abstract

ABSTRACT. Suppose that W is an infinite Coxeter group of finite rank n, and suppose that W has a finite parabolic subgroup W J 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 Aut(W ) generated by the inner automorphisms and the automorphisms induced by symmetries of the Coxeter graph. If, in addition, W J is irreducible and every conjugacy class of reflections in W has nonempty intersection with W J , then all automorphisms of W preserve reflections, and it follows that Aut(W ) is the semi-direct product of Inn(W ) by the group of graph automorphisms.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 ∞. These were investigated by James, [12], who described the automorphism groups of Coxeter groups whose diagrams have the following form:James's result was extended by Tits, [16], to include all irreducible right-angled Coxeter groups whose diagrams do not contain triangles. Finally, in [14], Mühlherr gave a presentation for the automorphism group of any right-angled Coxeter group. The automorphism groups of infinite rank 3 Coxeter groups whose diagrams have no edges with infinite labels are described in [9]; in this case the automorphism group is the semi-direct product of Inn(W ) and the group of graph automorphisms. The automorphism groups of rank 3 Coxeter groups with both finite and infinite edge labels are described in [7].For the purposes of this paper, we say that an infinite Coxeter group is nearly finite if it has finite rank n and has a finite parabolic subgroup of rank n − 1. It is shown that if W is nearly finite and does not have an edge labelled ∞ then the group of all automorphisms of W that preserve reflections is the semi-direct product of Inn(W ) and the group of graph automorphisms. In certain special cases we are able to show that all automorphisms of W preserve reflections. In fact, if we restrict attention to infinite irreducible Coxeter groups whose diagrams have no infinite edge labels, then we know of no example having an automorphism that does not preserve reflections. PRELIMINARIESRecall that a Coxeter group is a group with a presentation of the form 2000 Mathematics Subject Classification. Primary 20F55. 1 The closely related question of whether a Coxeter group may contain more than one class of Coxeter generating sets is investigated in [5].

Matching theorems for systems of a finitely generated Coxeter group

“…The remaining cases follow from Franzsen and Howlett [8,Theorem 31], since the automorphisms˛ ;˛ ;˛ in Theorem 31 induce the identity on OEW; W (since OEW; W is the kernel of ; and ), and the remaining automorphisms described in the generating sets are reflection preserving. The next lemma is known to experts; for a proof, see Paris [12].…”

Triality and étale algebras

“…We are grateful to Parimala for her unshakable interest in triality, in particular for many discussions at earlier stages of this work and we specially thank J-P. Serre for communicating to us his results on Witt and cohomological invariants of the group W (D 4 ). We also thank Emmanuel Kowalski who introduced us to Magma [2] with much patience, Jean Barge for his help with Galois cohomology and J. E. Humphreys and B. Mühlherr for the reference to the paper [8]. The paper [10] on octic fields was a very useful source of inspiration.…”

On the rank of a Coxeter group

“…Now y restricts to the identity on ½C n ; C n , and so by composing f with y in the latter case, we may assume that fðB n Þ ¼ B n . Every automorphism of B n is inner by [3,Theorem 31]. Hence f restricts to conjugation on ½B n ; B n by an element of B n .…”