Mapping the Davis complex into the imaginary cone

The study of the set of limit roots associated to an infinite Coxeter group was initiated by Hohlweg, Labb\'{e} and Ripoll and further developed by Dyer, Hohlweg, P\'eaux 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.