Isolated points in the space of left orderings of a group

Abstract

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.