Mapping the Davis complex into the imaginary cone

TL;DR

This paper introduces a natural mapping from the Davis complex of a Coxeter group into the normalized imaginary cone, linking geometric and algebraic structures in Coxeter group theory.

Contribution

It defines a new natural map from the Davis complex to the imaginary cone, enhancing understanding of Coxeter groups' geometric and algebraic properties.

Findings

Established a map from Davis complex to imaginary cone

Connected Davis complex geometry with root system structures

Provided new insights into Coxeter group representations

Abstract

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.