Asymptotical Behaviour of Roots of Infinite Coxeter Groups

Abstract 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 $\text{W}$ . We show that $E$ is contained in the isotropic cone $Q$ of the bilinear form $B$ associated with a geometric representation, and we 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 fromthe intersection with $Q$ of the lines passing through two positive roots, and finally we establish that it is dense in $E$ .