Isolated Points in the Space of Left Orderings of a Group

Let G be a left orderable group and LO(G) the space of all left orderings. We investigate the circumstances under which a left ordering < of G can correspond to an isolated point in LO(G), in particular we extend the main result of [9] to the case of uncountable groups. With minor technical restrictions on the group G, we find that no dense left ordering is isolated in LO(G), and that the closure of the set of all dense left orderings of G yields a dense Gδ set within a Cantor set of left orderings in LO(G). Lastly, we show that certain conditions on a discrete left ordering of G can guarantee that it is not isolated in LO(G), and we illustrate these ideas using the Dehornoy ordering of the braid groups. 1. The space of left orderings of a group A group G is said to be left-orderable if there exists a strict total ordering < of its elements such that g < h ⇒ fg < fh for all f, g, h ∈ G. Given a left-orderable group G with ordering <, we can identify the left ordering < of G with its positive cone P = {g ∈ G|g > 1}, the set of all positive elements. The positive cone P of a left ordering of a group G satisfies the following two properties: (1) If g, h ∈ P then gh ∈ P . (2) For all g ∈ G, exactly one of g ∈ P, g−1 ∈ P , or g = 1 holds. Conversely, given a semigroup P ⊂ G satisfying the above two properties, we can order the elements of G by specifying that g < h if and only if g−1h ∈ P . A left ordering < of G is said to be a Conradian ordering if whenever g, h > 1, then there exists n ∈ N such that g < hg. Lastly, a left ordering of a group G is said to be a bi-ordering if the ordering is also invariant under multiplication from the right, namely g < h ⇒ gf < hf for all f, g, h ∈ G. It should be noted that the positive cone P ⊂ G of a bi-ordering also satsifies the additional property: (3) For all g ∈ G, we have gPg−1 = P . Analogous to the case of left orderings, a semigroup P ⊂ G satisfying properties (1)–(3) defines a bi-ordering of G. Date: December 12, 2008. 1