Free Lattice Ordered Groups and the Topology on the Space of Left Orderings of a Group

For any left orderable group G, we recall from work of McCleary that isolated points in the space LO(G) correspond to basic elements in the free lattice ordered group F (G). We then establish a new connection between the kernels of certain maps in the free lattice ordered group F (G), and the topology on the space of left orderings LO(G). This connection yields a simple proof that no left orderable group has countably many left orderings. When we take G to be the free group Fn of rank n, this connection sheds new light on the space of left orderings LO(Fn): by applying a result of Kopytov, we show that there exists a left ordering of the free group whose orbit is dense in the space of left orderings. From this, we obtain a new proof that LO(Fn) contains no isolated points, and equivalently, a new proof that F (Fn) contains no basic elements. 1. The space of left orderings A group G is said to be left orderable if there exists a strict total ordering < of the elements of G, such that g < h implies fg < fh for all f, g, h in G. Associated to each left ordering of a group G is its positive cone defined by P = {g ∈ G|g > 1}, elements of the positive cone are said to be positive in the ordering <. The positive cone P of a left ordering < of G satisfies P · P ⊂ P , and P ⊔ P−1 ⊔ {1} = G; conversely, any P ⊂ G satisfying these two properties defines a left invariant total ordering of the elements of G, according to the prescription g < h if and only if g−1h ∈ P . We will denote the left ordering of a group G arising from a positive cone P by <P . If the positive cone P of a left ordering additionally satisfies gPg−1 = P for all g ∈ G, then the associated left ordering of G satisfies g <P h implies gf <P hf and fg <P fh for all f, g, h in G. In this case, <P is a bi-ordering of G. Finally, a left ordering <P of G is called Conradian if for every pair of positive elements g, h ∈ P , there exists an integer n such that g <P hg . In fact, we can equivalently require that g <P hg 2 for all pairs of positive elements g, h in G [18]. Observe that all bi-orderings are also Conradian left orderings, but not vice versa. A group G is Conradian left-orderable if and Date: September 1, 2009.