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 biinvariant circular ordering and iterated direct products whose factor groups are amenable.

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