Affine Reflection Subgroups of Coxeter Groups

In this paper, we study affine reflection subgroups and their root subsystems in arbitrary infinite Coxeter groups of finite rank. We give a characterization of limit roots (the accumulation points of the projections of roots onto suitably chosen hyperplanes) associated with affine reflection subgroups. We describe the relationship between these limit roots and all other limit roots. In particular, we show that each limit root associated with an affine reflection subgroup may only produce positive values with finitely many positive roots under the bilinear form afforded by the Tits representation (thus satisfying a so-called Pos-finite condition); and we show that an arbitrary limit root producing positive bilinear form values with only finitely many positive roots must be a convex combination of limit roots associated with affine reflection subgroups. As an application of this Pos-finite condition, we give an independent characterization of when a finitely generated infinite Coxeter group may possess affine reflection subgroups. We also use this Pos-finite analysis to partially resolve the question of whether the projections of an infinite injective sequence of roots totally ordered by the dominance partial ordering possesses or not a unique point of accumulation.