Abstract:

I will introduce Coxeter groups, a rich class of finitely generated groups with connections to geometry, combinatorics, number theory, Lie theory, and much more. One definition (if made suitable precise), is that they are groups generated by reflections which act discretely. Studying the geometry of how they act on various spaces is often the key to understanding them. I will focus on one question: how can we classify the sets of reflections which generate a Coxeter group (up to a suitable notion of equivalence)? The suitable notion of equivalence is a version of Nielsen equivalence adapted to the reflection setting, and it has an elegant interpretation in terms of hyperplane arrangements. I will prove that any reflect generating set of a Coxeter group is equivalent to a “geometrically simple” generating set, and survey how this relates to other results in surface and Fuchsian groups.