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" 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.

19 pages, 11 figures. Version 2: 29 pages, 11 figures. Reorganisation of the paper, addition of many details (section 5 in particular). Version 3 : revised edition accepted in Journal of the CMS. The number "I" was removed from the title since number "II" paper was named differently, see http://arxiv.org/abs/1303.6710

References in corpus (4)

• On the weak order of Coxeter groups
• The dominance hierarchy in root systems of Coxeter groups
• Coxeter groups, imaginary cones and dominance
• On rigidity of abstract root systems of Coxeter systems

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• Limit Directions for Lorentzian Coxeter Systems
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• Even More Infinite Ball Packings from Lorentzian Root Systems
• Cannon-Thurston maps for Coxeter groups with signature $(n-1,1)$
• On paired root systems of Coxeter groups
• Cannon-Thurston maps for Coxeter groups including affine special subgroups
• Shi arrangements and low elements in Coxeter groups
• Infinite Reduced Words, Lattice Property And Braid Graph of Affine Weyl Groups
• Affine reflection subgroups of Coxeter groups
• Domains of Convergence for Polyhedral Packings