Neighbourhoods in root systems of infinite Coxeter groups

Abstract

Let $W$ be a finitely generated infinite Coxeter group, with $\Phi$ and $\Pi$ being the corresponding root system and set of simple roots respectively. It has been observed by Hohlweg et la that the projections of elements of $\Phi$ onto suitably chosen hyperplanes, called \emph{normalized roots}, are contained in the convex hull of $\Pi$ (which is a compact set), and hence the set of all normalized roots may exhibit interesting asymptotical behaviours. In this paper we investigate the topology of the limit set of the normalized roots and demonstrate a natural system of neighbourhoods around each limit point arising from a non-affine infinite dihedral reflection subgroup of $W$.