On spaces of Conradian group orderings

TL;DR

This paper classifies groups with finitely many or uncountably many Conradian orderings, revealing that infinite cases have a rich, Cantor-set structure and no isolated orderings, with detailed analysis of Baumslag-Solitar's group B(1,2).

Contribution

It provides a complete classification of C-orderable groups based on the number of C-orderings they admit, including the structure of their ordering spaces.

Findings

Groups with finitely many C-orderings are classified.

Infinite C-orderings form a Cantor set with no isolated points.

Baumslag-Solitar's group B(1,2) has four bi-invariant C-orderings and a Cantor set of left-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.