On spaces of Conradian group orderings

Abstract

We classify C-orderable groups admitting only finitely many C-orderings. We show that if a C-orderable group has infinitely many C-orderings, then it actually has uncountably many C-orderings, and none of these is isolated in the space of C-orderings. As a relevant example, we carefully study the case of Baumslag-Solitar’s group B(1, 2). We show that B(1, 2) has four C-orderings, each of which is bi-invariant, but its space of left-orderings is homeomorphic to the Cantor set.