Monoids of O-type, Subword Reversing, and Ordered Groups

We describe a simple scheme for constructing finitely generated monoids in which left-divisibility is a linear ordering and for practically investigating these monoids. The approach is based on subword reversing, a general method of combinatorial group theory, and connected with Garside theory, here in a non-Noetherian context. As an application we describe several families of ordered groups whose space of left-invariant orderings has an isolated point, including torus knot groups and some of their amalgamated products. A group G is left-orderable if there exists a linear ordering on G that is left-invariant, i.e., g < g implies hg < hg for every h in G. Viewing an ordering on G as a subset of G ×G, one equips the family LO(G) of all left-invariant orderings of G with a topology induced by the product topology of P(G×G). Then LO(G) is a compact space and, in many cases, in particular when G is a countable non-abelian free group, LO(G) has no isolated points and it is a Cantor set [21, 4]. By contrast, apart from the cases when LO(G) is finite and therefore discrete, as is the case for the Klein bottle group and, more generally, for the Tararin groups [22, 15], not so many examples are known when LO(G) contains isolated points. By the results of [12], this happens when G is an Artin braid group (see also [16]), and, by those of [17, 13], when G is a torus knot group, i.e., a group of presentation 〈x, y | x = y〉 with m,n > 2. These results, as well as the further results of [14], use non-elementary techniques. The aim of this paper is to observe that a number of ordered groups with similar properties, including the above ones, can be constructed easily using a monoid approach. A necessary and sufficient condition for a submonoid M of a group G to be, when 1 is removed, the positive cone of a left-invariant ordering on G is that M is what will be called of O-type, namely it is cancellative, has no nontrivial invertible element, and its left-and right-divisibility relations (see Definition 1.1) are linear orderings. Moreover, the involved ordering is isolated in the corresponding space LO(G) whenever M is finitely generated. We are thus naturally led to the question of recognizing which (finite) presentations define monoids of O-type. Here we focus on presentations of a certain syntactical type called triangular. Although no complete decidability result can probably be expected, the situation is that, in practice, many cases can be successfully addressed, actually all cases in the samples we tried. The main tool we use here is subword reversing [5, 6, 7, 8], a general method of combinatorial group theory that is especially suitable for investigating divisibility in a presented monoid and provides efficient algorithms that make experiments easy. Both in the positive case (when the defined monoid is of O-type) and in the negative one (when it is not), the approach leads to sufficient Σ1-conditions, i.e., provides effective procedures returning a result when the conditions are met but possibly running forever otherwise. At a technical level, the Work partially supported by the ANR grant ANR-08-BLAN-0269-02 1991 Mathematics Subject Classification. 06F15, 20M05, 20F60.