The topology on the space of left orderings of a group
Abstract
Let G be a group and let OG denote the set of left orderings on G. Then OG can be topologized in a natural way, and we shall study this topology to answer three conjectures. In particular we shall show that OG can never be countably infinite. Furthermore in the case G is a countable nonabelian free group, we shall show that OG is homeomorphic to the Cantor set and that the positive cone of a left order on G is not finitely generated. Generalizations to locally indicable groups will also be considered.
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