Garside monoids, and Garside structures on braid and reflection groups

Abstract

A Garside structure on a monoid (or group) relies on the existence of a so-called Garside element in the monoid, where understanding divisibility within this element suffices to understand divisibility in the monoid. Finding a Garside structure for a monoid (or its group of fractions, into which it always embeds) gives efficient solutions to the word and conjugacy problems, as well as implying other properties such as bi-automaticity, being torsion-free, and having finite homological dimension. I will recall the definition, describe some general properties and give various examples of Garside monoids. In particular, I will discuss some Garside structures for braid groups of (complex) reflection groups, obtained in work with Bessis, with Picantin, and with Lee & Lee, which give new insights into properties of these groups.