Irreducible Coxeter groups

TL;DR

This paper establishes the indecomposability properties of non-spherical irreducible Coxeter groups and proves the uniqueness of their decompositions into indecomposable factors, including virtual decompositions into strongly indecomposable groups.

Contribution

It provides new results on the indecomposability and unique decomposition structures of Coxeter groups, including non-spherical, non-affine, and virtually decomposable cases.

Findings

Non-spherical irreducible Coxeter groups are directly indecomposable.

Non-spherical, non-affine Coxeter groups are strongly indecomposable.

Decompositions into indecomposable groups are unique up to automorphisms and permutations.

Abstract

We prove that a non-spherical irreducible Coxeter group is (directly) indecomposable and that a non-spherical and non-affine Coxeter group is strongly indecomposable in the sense that all its finite index subgroups are (directly) indecomposable. We prove that a Coxeter group has a decomposition as a direct product of indecomposable groups, and that such a decomposition is unique up to a central automorphism and a permutation of the factors. We prove that a Coxeter group has a virtual decomposition as a direct product of strongly indecomposable groups, and that such a decomposition is unique up to commensurability and a permutation of the factors.