Automorphisms of Coxeter Groups of Rank Three

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...

Petrie-Coxeter Maps Revisited

This paper presents a technique for constructing new chiral or regular polyhedra (or maps) from self-dual abstract chiral polytopes of rank 4. From improperly self-dual chiral polytopes we derive “Petrie-Coxeter-type” polyhedra (abstract chiral analogues of the classical Petrie-Coxeter polyhedra) and investigate their groups of automorphisms.

Almost central involutions in split extensions of Coxeter groups by graph automorphisms

In this paper, given a split extension of an arbitrary Coxeter group by automorphisms of the Coxeter graph, we determine the involutions in that extension whose centralizer has finite index. Our result has applications to many problems such as the isomorphism problem of general Coxeter groups. In the argument, some properties of certain special elements and of the fixed-point subgroups by graph...

Complete Growth Series of Coxeter Groups with more than Three Generators

Fixed Points of Involutive Automorphisms of the Bruhat Order

Applying a classical theorem of Smith, we show that the poset property of being Gorenstein∗ over Z2 is inherited by the subposet of fixed points under an involutive poset automorphism. As an application, we prove that every interval in the Bruhat order on (twisted) involutions in an arbitrary Coxeter group has this property, and we find the rank function. This implies results conjectured by F. ...