The hull metric on Coxeter groups

Essays on Coxeter groups Coxeter elements in finite Coxeter groups

A finite Coxeter group possesses a distinguished conjugacy class of Coxeter elements. The literature about these is very large, but it seems to me that there is still room for a better motivated account than what exists. The standard references on thismaterial are [Bourbaki:1968] and [Humphreys:1990], butmy treatment follows [Steinberg:1959] and [Steinberg:1985], from which the clever parts of ...

Coxeter Groups, Salem Numbers and the Hilbert Metric

On Outer Automorphism Groups of Coxeter Groups

It is shown that the outer automorphism group of a Coxeter group W of nite rank is nite if the Coxeter graph contains no innnite bonds. A key step in the proof is to show that if the group is irreducible and 1 and 2 any two bases of the root system of W, then 2 = w 1 for some w 2 W. The proof of this latter fact employs some properties of the dominance order on the root system introduced by Bri...

Coxeter Covers of the Classical Coxeter Groups

Let C(T ) be a generalized Coxeter group, which has a natural map onto one of the classical Coxeter groups, either Bn or Dn. Let CY (T ) be a natural quotient of C(T ), and if C(T ) is simply-laced (which means all the relations between the generators has order 2 or 3), CY (T ) is a generalized Coxeter group, too . Let At,n be a group which contains t Abelian groups generated by n elements. The...

On the Direct Indecomposability of Infinite Irreducible Coxeter Groups and the Isomorphism Problem of Coxeter Groups

In this paper we prove that any irreducible Coxeter group of infinite order is directly indecomposable as an abstract group, without the finite rank assumption. The key ingredient of the proof is that we can determine, for an irreducible Coxeter group W , the centralizers in W of the normal subgroups of W that are generated by involutions. As a consequence, we show that the problem of deciding ...