1. CJM Online first
|Asymptotical behaviour of roots of infinite Coxeter groups|
Let $W$ be an infinite Coxeter group. We initiate the study of the set $E$ of limit points of ``normalized'' roots (representing the directions of the roots) of W. We show that $E$ is contained in the isotropic cone $Q$ 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 how 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 $E$.
Keywords:Coxeter group, root system, roots, limit point, accumulation set