Asymptotical behaviour of roots of infinite Coxeter groups

TL;DR

This paper explores the asymptotic distribution of roots in infinite Coxeter groups, showing that their limit points lie within a specific geometric cone and are densely populated by certain subgroup limit points.

Contribution

It introduces the set of limit points of normalized roots, proves their containment in the isotropic cone, and demonstrates the density of a subset formed by dihedral subgroup limit points.

Findings

Limit points of roots lie in the isotropic cone.

A countable dense subset of limit points is generated by dihedral subgroups.

The geometric action of the Coxeter group on the limit set is characterized.

Abstract

Let W be an infinite Coxeter group. We initiate the study of the set E of limit points of "normalized" positive roots (representing the directions of the roots) of W. We show that E is contained in the isotropic cone of the bilinear form B associated to a geometric representation, and illustrate this property with numerous examples and pictures in rank 3 and 4. We also define a natural geometric action of W on E, and then we exhibit a countable subset of E, formed by limit points for the dihedral reflection subgroups of W. We explain that this subset is built from the intersection with Q of the lines passing through two positive roots, and finally we establish that it is dense in E.