On the Number of Circular Orders on a Group

We give a classification and complete algebraic description of groups allowing only finitely many (left multiplication invariant) circular orders. In particular, they are all solvable groups with a specific semi-direct product decomposition. This allows us to also show that the space of circular orders of any group is either finite or uncountable. As a special case and first step, we show that the space of circular orders of an infinite Abelian group has no isolated points, hence is homeomorphic to a Cantor set. © 2018 Elsevier Inc.