Coxeter Groups

Abstract

This chapter considers Coxeter groups and how to find a space on which a group acts by building a space using combinatorics from the group. It first describes groups generated by reflections, focusing on Euclidean spaces and showing that some natural, beautiful, and important subsets of Euclidean spaces have symmetric groups that are discrete and are generated by reflections. It then explores discrete groups generated by reflections, beginning with irreducible finite groups generated by reflections followed by infinite reflection groups. It also looks at relations in finite groups generated by reflections before concluding with an analysis of special subgroups of the Coxeter group and how to construct a geometric space for a Coxeter group. The discussion includes exercises and research projects.