Asymptotical behaviour of roots of infinite Coxeter groups I

Details Asymptotical behaviour of roots of infinite Coxeter groups I Asymptotical behaviour of roots of infinite Coxeter groups I

2011-12-22

arxiv.org/abs/1112.5415v1

Mathematics

Group Theory

19 pages, 11 figures

Scientific paper

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 can be built from the intersection with Q of the lines passing through two positive roots, and we establish that it is dense in E.

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