Automorphisms of Coxeter groups

Ruth Charney

Abstract: Let (W,S) be a Coxeter system and let \phi : W \to W be an automorphism of W. We show that under appropriate hypotheses (eg., if W acts as a cocompact reflection group on a contractible manifold), then \phi (S) is conjugate to S. It follows that the outer automorphism group of W, Aut(W)/Inn(W), can be naturally identified with the group of symmetries of the Coxeter diagram for (W,S). It also follows that if (W',S') is another Coxeter system with W and W' isomorphic as abstract groups, then (W,S) and (W',S') are isomorphic as Coxeter systems, that is, there is an isomorphism \phi : W \to W' taking S to S'.

created Thu May 18 14:51:05 PDT 2000