Exotic Left Orderings of the Free Groups from the Dehornoy Ordering

We show that the restriction of the Dehornoy ordering to an appropriate free subgroup of B3 defines a left ordering of the free group on k generators, k > 1, that has no convex subgroups. A group G is said to be left-orderable if there exists a strict total ordering of its elements such that g < h implies fg < fh for all f, g, h in G. To each left ordering < of a group G, we can associate the set P = {g ∈ G|g > 1}, which is called the positive cone associated to the left ordering <. The positive cone P satisfies P ·P ⊂ P , and P ⊔P−1⊔{1} = G. Conversely, any subset P satisfying these two properties defines a strict total ordering of the elements of G, via g < h if and only if g−1h ∈ P . Any ordering defined in this way is easily seen to be invariant under left multiplication. We may strengthen our conditions on a left ordering < of G by requiring that for all g, h > 1 in G, there must exist a positive integer n such that g < hg. In this case, the ordering is called Conradian (after the work of Conrad in [2]). It has since been observed that, equivalently, we may ask that this condition hold for n = 2 [9]. Finally, the strongest condition we may require of an ordering < of G is that the ordering be invariant under multiplication from both sides, that is, g < h implies fg < fh and gf < hf for all f, g, h in G. Equivalently, we may require that the positive cone associated to the ordering < of G be preserved by conjugation. If either of these equivalent conditions is satisfied by the ordering < of G, then the ordering is said to be a bi-ordering. An important structure associated to a given left ordering < of G is the set of convex subgroups of G. A subgroup H ⊂ G is said to be convex in G (with respect to the ordering <) if whenever f, h are in H and g is in G, the implication f < g < h ⇒ g ∈ H holds. Owing to work of Conrad and Hölder, the convex subgroups of bi-orderings and Conradian orderings are very well understood [2]. This leaves us with understanding the set of convex subgroups for the case of left orderings that are neither bi-orderings, nor Conradian orderings. This problem seems to be quite difficult, as constructing Conradian orderings and bi-orderings of a group G is in general somewhat easier than constructing left orderings of a group that are neither bi-orderings nor Conrad orderings. Date: June 26, 2009.