Circularly ordering direct products and the obstruction to left-orderability

TL;DR

This paper establishes cohomological criteria for when direct products of groups are circularly orderable, linking this to left-orderability and providing new characterizations for fundamental groups of rational homology 3-spheres.

Contribution

It offers necessary and sufficient cohomological conditions for circular orderability of direct products, connecting group orderability properties with topological and algebraic structures.

Findings

Cohomological conditions characterize circular orderability of direct products.

New criteria for left-orderability of fundamental groups of rational homology 3-spheres.

Products of certain groups with rigid actions on S^1 are rarely circularly orderable.

Abstract

Motivated by the recent result that left-orderability of a group $G$ is intimately connected to circular orderability of direct products $G \times \mathbb{Z}/n\mathbb{Z}$, we provide necessary and sufficient cohomological conditions that such a direct product be circularly orderable. As a consequence of the main theorem, we arrive at a new characterization for the fundamental group of a rational homology 3-sphere to be left-orderable. Our results imply that for mapping class groups of once-punctured surfaces, and other groups whose actions on $S^1$ are cohomologically rigid, the products $G \times \mathbb{Z}/n\mathbb{Z}$ are seldom circularly orderable. We also address circular orderability of direct products in general, dealing with the cases of factor groups admitting a bi-invariant circular ordering, and iterated direct products whose factor groups are amenable.