Asymptotical Behaviour of Roots of Infinite Coxeter Groups

Source abstract

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 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 .