Orderable groups and their spaces of order

Abstract: A left order on a group G is a linear order of the domain of G, which is left-invariant under the group operation. Right orders and bi-orders are defined similarly. We investigate computability theoretic and topological properties of spaces of left orders on computable orderable groups. Topological properties of spaces of orders on groups were first studied by A. Sikora who showed that for free abelian groups of finite rank n >1 the space of orders is homeomorphic to the Cantor set. We study groups for which the spaces of left orders contain the Cantor set and we establish that a countable free group of rank n ≥2 and fundamental groups of oriented surfaces have a bi-order in every Turing degree.