A Coxeter group W is an abstract group generated by a set S consisting of elements of order 2, subject only to relations specifying the orders of the products of pairs of generators.
Cayley graphs and root systems are two major tools used to study Coxeter groups.
The Cayley graph Γ of a Coxeter group W with generating set S is an undirected graph whose vertex set is identified with W, and the edge set is identified with S, subject to the requirement that the vertices corresponding to a pair of elements w and wl of W are joined by the edge corresponding to s of S if and only if w = wls.
If W is a Coxeter group, then there exists a real vector space V such that W is isomorphic to a subgroup of GL (V) generated by reflections with respect to a collection of hyperplanes in V. The root system of this representation of W consists of representative normal vectors of these reflecting hyperplanes, called roots.
It is well-known that the root systems of Coxeter groups are discrete. However, if an infinite Coxeter group W has a finite generating set S, then the projection of the roots of W onto suitably chosen hyperplanes may exhibit asymptotic behaviors.
In this talk we present the recently discovered connection between the visual bound- aries of Cayley graphs and the limits of sequences of projected roots. This is a joint work with L. Reeves of Melbourne University.