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.

Abstract. Let (W, S) be an infinite Coxeter system. To each geometric representation of W is associated a root system. While a root system lives in the positive side of the isotropic cone of its associated bilinear form, an imaginary cone lives in the negative side of the isotropic cone. Precisely on the isotropic cone, between root systems and imaginary cones, lives the set E of limit points of the directions of roots. In this article we study the close relations of the imaginary cone with the set E, which leads to new fundamental results about the structure of geometric representations of infinite Coxeter groups. In particular, we show that the W -action on E is minimal and faithful, and that E and the imaginary cone can be approximated arbitrarily well by sets of limit roots and imaginary cones of universal root subsystems of W , i.e., root systems for Coxeter groups without braid relations (the free object for Coxeter groups). Finally, we discuss open questions as well as the possible relevance of our framework in other areas such as geometric group theory.

No abstract

Abstract. Given an irreducible well-generated complex reflection group W with Coxeter number h, we call a Coxeter element any regular element (in the sense of Springer) of order h in W ; this is a slight extension of the most common notion of Coxeter element. We show that the class of these Coxeter elements forms a single orbit in W under the action of reflection automorphisms. For Coxeter and Shephard groups, this implies that an element c is a Coxeter element if and only if there exists a simple system S of reflections such that c is the product of the generators in S. We moreover deduce multiple further implications of this property. In particular, we obtain that all noncrossing partition lattices of W associated to different Coxeter elements are isomorphic. We also prove that there is a simply transitive action of the Galois group of the field of definition of W on the set of conjugacy classes of Coxeter elements. Finally, we extend several of these properties to Springer's regular elements of arbitrary order.

No abstract