Garside families and Garside germs

Abstract. Garside families have recently emerged as a relevant context for extending results involving Garside monoids and groups, which themselves extend the classical theory of (generalized) braid groups. Here we establish various characterizations of Garside families, that is, equivalently, various cri-teria for establishing the existence of normal decompositions of a certain type. In 1969, F.A.Garside [21] solved the word and conjugacy problems of Artin’s braid groups by using convenient monoids. This approach was pursued [1, 19, 25, 20, 8] and extended in several steps, first to Artin-Tits groups of spherical type [7, 15], then to a larger family of groups now known as Garside groups [12, 9, 10]. More recently, it was realized that going to a categorical context allows for capturing further examples [18, 3, 11, 23], and a coherent theory now emerges around a central unifying notion called a Garside family. The aim of this paper is to present the main basic results of this approach. A more comprehensive text, including examples and many further developments, will be found in [14]. Algorithmic issues are addressed in [13]