Limit Roots of Lorentzian Coxeter Systems

A reflection in a real vector space equipped with a positive definite symmetric bilinear form is any automorphism that sends some nonzero vector v to its negative and pointwise fixes its orthogonal complement, and a finite reflection group is a discrete group generated by such transformations. We note two important classes of groups which occur as finite reflection groups: for a 2-dimensional vector space, we recover precisely the finite dihedral groups as reflection groups, and permuting basis vectors in an n-dimensional vector space gives a way of viewing a symmetric group as reflection group. A Coxeter group is a generalization of a finite reflection group, whose have rich geometric and algebraic properties interact in surprising ways. Any finite rank Coxeter group W acts faithfully on a finite dimensional real vector space V . To each group is an associated symmetric bilinear form which it preserves, and the signature of the bilinear form contains valuable information about W ; if it has type (n,1), we call such a group Lorentzian, and there is a natural action of such a group on a hyperbolic space. Inspired by a conjecture of Dyer in 2011, Hohlweg, Labbé and Ripoll have studied the set of reflection vectors in Lorentzian Coxeter groups. We summarize their results here. The reflection vectors form an infinite discrete subset of V , but if we projectivize, PV contains limit points, which have the appearance of a fractal.