Free lattice-ordered groups and the space of left orderings

For a left-orderable group G, let LO(G) denote its space of left orderings, and F(G) the free lattice-ordered group over G. This paper establishes a connection between the topology of LO(G) and the group F(G). The main result is a correspondence between the kernels of certain maps in F(G), and the closures of orbits in LO(G) under the natural G-action. The proof of this correspondence is motivated by earlier work of McCleary, which essentially shows that isolated points in LO(G) correspond to basic elements in F(G). As an application, we will study this new correspondence between kernels and the closures of orbits to show that LO(G) is either finite or uncountable. We will also show that LO(Fn) is homeomorphic to the Cantor set, where Fn is the free group on n > 1 generators.