A Dual Braid Monoid for the Free Group

We construct a quasi-Garside monoid structure for the free group. This monoid should be thought of as a dual braid monoid for the free group, generalising the constructions by Birman-Ko-Lee and by the author of new Garside monoids for Artin groups of spherical type. Conjecturally, an analog construction should be available for arbitrary Artin groups and for braid groups of well-generated complex reflection groups. This article continues the exploration of the theory of Artin groups and generalised braid groups from the new point of view introduced by Birman-Ko-Lee in [BKL] for the classical braid group on n strings. In [B1], we generalised their construction to Artin groups of spherical type. In the current article, we study the case of the free group, which is the Artin group associated with the universal Coxeter group. The formal analogs of the main statements in [B1] turn out to be elementary consequences of classical material (some of which was known to Hurwitz and Artin). In an attempt to interpolate some recent generalisations of the dual monoid construction (by Digne for the Artin group of type Ãn, [D]; by Corran and the author for the braid group of the complex reflection group G(e, e, n), [BC]), we propose two conjectures describing properties of a generalised dual braid monoid, in the contexts of (a) arbitrary Artin groups and (b) braid groups of well-generated finite complex reflection groups. This would provide the first uniform combinatorial approach to these objects. The initial motivation for the current work was to understand the situation (b) from a natural geometric viewpoint; the conjectures about complex reflection groups will be studied in the sequel [B2], answering some questions raised in [BMR].