Geometric Presentations for Thompson’s Groups

Starting from the observation that Thompson’s groups F and V are the geometry groups respectively of associativity, and of associativity together with commutativity, we deduce new presentations of these groups. These presentations naturally lead to introducing a new subgroup S• of V and a torsion free extension B• of S•. We prove that S• and B• are the geometry groups of associativity together with the law x(yz) = y(xz), and of associativity together with a twisted version of this law involving self-distributivity, respectively. Previous work showed that associating to an algebraic law a so-called geometry group that captures some specific geometrical features gives useful information about that law: the approach proved instrumental for studying exotic laws like self-distributivity x(yz) = (xy)(xz) [7] or x(yz) = (xy)(yz) [8]. In the case of associativity [6], the geometry group turns out to be Thompson’s group F , not a surprise as the connection of the latter with associativity has been known for long time [20]. In this paper, we develop a rather general method for constructing geometry groups and, chiefly, finding presentations for these groups, and we apply this method in the case of associativity—thus finding presentations of F—and of associativity plus commutativity, thus finding new presentations of Thompson’s group V , as the latter happens to be the involved geometry group. In the case of F , the new presentation, which is centered around MacLane’s pentagon relation, is more symmetric than the usual ones and it leads to an interesting lattice structure connected with Stasheff’s associahedra; this structure will be investigated in [10]. In the case of V , on which we concentrate here, we describe several new presentations corresponding to various choices of the generators. In each case, once some preliminary combinatorial results are established, proving that a candidate list of relations actually makes a presentation is a straightforward application of our general method and a very simple argument. Perhaps the main merit of the above presentations of V is to naturally lead to introducing two new groups which seem interesting in themselves. Indeed, one of these presentations explicitly includes the Coxeter presentation of the symmetric group S∞ (direct limit of the Sn’s), thus emphasizing the existence of a copy of S∞ inside V . When we extract those generators and relations that correspond to F and to that copy of S∞, we obtain a subgroup S• of V , and, when we remove the torsion relations si = 1 in the involved Coxeter presentation, we obtain an extension B• of S•: the connection between B• and S• is the same as the one between Artin’s braid group B∞ and S∞. The algebraic and geometric properties of the groups S• and, specially, B• are very rich. In the current paper, we address these groups only from the viewpoint of geometry groups, and we prove two results: on the one hand, the group S• is itself a geometry group, namely that of associativity together with the left semi-commutativity law x(yz) = y(xz); on the other hand, in some convenient sense, B• is the geometry group for associativity together with a twisted version of semi-commutativity in which x(yz) = y(xz) is weakened into x(yz) = x[y](xz), where x, y 7→ x[y] is a second binary operation obeying a self-distributivity condition. The groups S• and B• to which our approach leads turn out to be (isomorphic to) the groups V̂ and B̂V recently introduced and investigated by M.Brin in [1, 2, 3]. The current 1991 Mathematics Subject Classification. 20F05, 20F36, 20B07.