UC Santa Barbara

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 group generated by such transformations that acts properly on the sphere of unit vectors. 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 a reflection group.

A Coxeter group is a generalization of a finite reflection group, whose 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. Each such group has an associated symmetric bilinear form that it preserves, and the signature of this bilinear form contains valuable information about W. When 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, Labbe 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 the vector space V, but the projective version of the reflection vectors has limit points. Understanding these limit points is the primary goal of this text. They lie within the light cone of the Lorentz space, and have intimate connections with the boundary of the group.