Dominance and regularity in Coxeter groups

The main purpose of this thesis is to generalize many of the results of Dyer, [18]. Dyer introduces a large family of finite state automata that are demonstrated to recognize many natural subsets of Coxeter groups. In the current work, we consider generalized length functions in Coxeter systems and show that these length functions lead to a larger family of finite state automata, which in turn recognize a larger class of natural subsets of Coxeter groups. By a well-known result, these subsets (called regular subsets) will then have a rational multivariate PoincarÌÄå© series.

Dominance order on the set of reflections (or positive roots) of a Coxeter system plays a major role in the study of regular subsets. We draw the connection between dominance in root systems and a notion of closure in root systems. We describe some conjectures, due to Dyer, [17], and introduce a conjectural normal form for Coxeter group elements related to closure and dominance.

In [14], Dyer introduced the notion of the imaginary cone to characterize dominance. We investigate the imaginary cone in the special case of hyperbolic Coxeter systems. We show that any infinite, irreducible, non-affne Coxeter system has universal reflection subgroups of arbitrarily large rank. This allows us to deduce some consequences about the growth type of Coxeter systems.