Imaginary cones and limit roots of infinite Coxeter groups

TL;DR

This paper explores the relationship between imaginary cones and limit roots in infinite Coxeter groups, revealing new structural insights and approximation properties of geometric representations.

Contribution

It establishes the minimal and faithful action of the group on the set of limit roots and shows how these sets can be approximated by universal root subsystems.

Findings

W-action on E is minimal and faithful

E and imaginary cones can be approximated by universal root subsystems

Provides new structural results about geometric representations of infinite Coxeter groups

Abstract

Let (W,S) be an infinite Coxeter system. To each geometric representation of W is associated a root system. While a root system lives in the positive side of the isotropy cone of its associated bilinear form, an imaginary cone lives in the negative side of the isotropic cone. Precisely on the isotropic cone, between root systems and imaginary cones, lives the set E of limit points of the directions of roots (see arXiv:1112.5415). In this article we study the close relations of the imaginary cone (see arXiv:1210.5206) with the set E, which leads to new fundamental results about the structure of geometric representations of infinite Coxeter groups. In particular, we show that the W-action on E is minimal and faithful, and that E and the imaginary cone can be approximated arbitrarily well by sets of limit roots and imaginary cones of universal root subsystems of W, i.e., root systems for Coxeter groups without braid relations (the free object for Coxeter groups). Finally, we discuss open questions as well as the possible relevance of our framework in other areas such as geometric group theory.