Spaces of orders and their Turing degree spectra

We investigate computability theoretic and topological properties of spaces of orders on computable orderable groups. 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. In particular, we study groups for which the spaces of left orders are homeomorphic to the Cantor set, and their Turing degree spectra contain certain upper cones of degrees. Our approach unifies and extends Sikora’s investigation of orders on groups in topology [28] and Solomon’s investigation of these orders in computable algebra [31]. Furthermore, we establish that a computable free group Fn of rank n > 1 has a bi-order in every Turing degree.