Free lattice-ordered groups and the space of left orderings (Q714965)

Let \(G\) be a left-orderable group, \(\text{LO}(G)\) be its space of left orderings and \(\text{F}(G)\) be the free lattice-ordered group over \(G\). In the current paper a connection between the topology of \(\text{LO}(G)\) and the group \(\text{F}(G)\) is given. More precisely, its main result establishes a correspondence between the kernels of certain maps in \(\text{F}(G)\) and the closures of orbits in \(\text{LO}(G)\) under the natural \(G\)-action. It is also proved that \(\text{LO}(F_n)\) is homeomorphic with the Cantor set, where \(F_n\) is the free group on \(n>1\) generators.