Affine reflection subgroups of Coxeter groups

Abstract

In this paper we study affine reflection subgroups in arbitrary infinite Coxeter groups of finite rank. In particular, we study the distribution of roots of Coxeter groups in the root subsystems associated with affine reflection subgroups. We give a characterization of limit roots arising from affine reflection subgroups. We also give a characterization of when a Coxeter group may possess affine reflection subgroups. We show that the intersection of the normalized isotropic cone (associated with the Tits representation of a Coxeter group) and the imaginary cone consists of limit roots closely related to affine reflection subgroups.