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