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Asymptotical behaviour of roots of infinite Coxeter groups

  Published:2013-08-10
 Printed: Apr 2014
  • Christophe Hohlweg,
    Université du Québec à Montréal, LaCIM et Département de Mathématiques, CP 8888 Succ. Centre-Ville, Montréal, Québec, H3C 3P8
  • Jean-Philippe Labbé,
    Freie Universität Berlin, Institut für Mathematik, Arnimallee 2, 14195, Berlin, Deutschland
  • Vivien Ripoll,
    Université du Québec à Montréal, LaCIM et Département de Mathématiques, CP 8888 Succ. Centre-Ville, Montréal, Québec, H3C 3P8
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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 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 Coxeter group, root system, roots, limit point, accumulation set
MSC Classifications: 17B22, 20F55 show english descriptions Root systems
Reflection and Coxeter groups [See also 22E40, 51F15]
17B22 - Root systems
20F55 - Reflection and Coxeter groups [See also 22E40, 51F15]
 

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