Garside families and Garside germs

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 criteria 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]. The philosophy of Garside’s theory as developed in the past decades is that, in some cases, a group can be realized as a group of fractions for a monoid and that the divisibility relations of the latter provide a lot of information about the group. The key technical ingredient in the approach is a certain distinguished decomposition for the elements of the monoid and the group in terms of some fixed (finite) family, usually called the greedy normal form. Our current approach consists in analyzing the abstract mechanism underlying the greedy normal form and developing it in the general context of what we call Garside families. The leading principle is that, with Garside families, one should retrieve all results about Garside monoids and groups at no extra cost. In the current paper, we concentrate on one fundamental question, namely characterizing Garside families. As the latter are defined to be those families that guarantee the existence of the normal form, this exactly amounts to establishing various (necessary and sufficient) criteria for this existence. Two types of characterizations will be established here: extrinsic characterizations consist in recognizing whether a subfamily of a given category is a Garside family, whereas intrinsic ones consist in recognizing whether an abstract family (more precisely, a precategory) generates a category in which it embeds as a Garside family. Beyond the results themselves, one of our goals is to show that the new framework, which properly extends those previously considered in literature, works efficiently and provides arguments that are both simple and natural. In particular, 1991 Mathematics Subject Classification. 20M05, 18B40, 20F10, 20F36.