David Sheard

Reflection and Nielsen equivalence in Coxeter groups

Nielsen equivalence – the natural notion of equivalence between generating sets of finitely generated groups – has been studied for the last century. Early techniques were often combinatorial, however more modern approaches use algebra and geometric/topological methods. In the last decade significant progress has been made studying it in surface and Fuchsian groups. In this talk I will introduce Nielsen equivalence in Coxeter groups, a class of groups with very rich geometry, and a related notion called reflection equivalence which is specialised for reflections. I will prove that any reflect generating set of a Coxeter group is equivalent to a “geometrically simple” generating set and provide a complete classification in some classes of Coxeter groups. Time permitting, I will also mention one approach to Nielsen equivalence in the right-angled case based on recent work generalising Stallings’ folds to RACGs.