Circularly ordering direct products and the obstruction to left-orderability

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.

Minor changes made to accommodate the referee's comments. To appear in the Pacific Journal of Mathematics

References in corpus (1)

• Rigidity of mapping class group actions on $S^1$

Cited by in corpus (1)

• Circular orderability of 3-manifold groups