Mapping the Davis complex into the imaginary cone

Abstract

The study of the set of limit roots associated to an infinite Coxeter group was initiated by Hohlweg, Labbé and Ripoll and further developed by Dyer, Hohlweg, Péaux and Ripoll. The Davis complex associated to a finitely generated Coxeter group $W$ is a piecewise Euclidean CAT(0) space on which $W$ acts properly, cocompactly by isometries. The one skeleton of the Davis complex can be identified with the Cayley graph of $W$. In this paper we define a natural map from the Davis complex into the normalised imaginary cone of a based root system.