#### Vol. 312, No. 2, 2021

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Circularly ordering direct products and the obstruction to left-orderability

### Adam Clay and Tyrone Ghaswala

Vol. 312 (2021), No. 2, 401–419
##### Abstract

Motivated by the recent result that left-orderability of a group $G$ is intimately connected to circular orderability of direct products $G×ℤ∕nℤ$, 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 $G×ℤ∕nℤ$ 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.

##### Keywords
orderable groups, 3-manifold groups
##### Mathematical Subject Classification
Primary: 20F60
Secondary: 37E10, 57K31