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 × n, 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 S1 are cohomologically rigid, the products G × n 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
Milestones
Received: 29 August 2020
Revised: 26 April 2021
Accepted: 26 April 2021
Published: 31 August 2021
Authors
Adam Clay
Department of Mathematics
University of Manitoba
Winnipeg
Canada
Tyrone Ghaswala
Département de Mathématiques
Université du Québec à Montréal
Montréal
Canada