Circularly ordering direct products and the obstruction to left-orderability
| Autor: | Adam Clay, Tyrone Ghaswala |
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| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Class (set theory)
Pure mathematics Fundamental group Group (mathematics) General Mathematics 010102 general mathematics Geometric Topology (math.GT) Group Theory (math.GR) Dynamical Systems (math.DS) Homology (mathematics) Characterization (mathematics) 01 natural sciences 20F60 (Primary) 37E10 57M27 (Secondary) Mathematics - Geometric Topology Iterated function 0103 physical sciences FOS: Mathematics 010307 mathematical physics 0101 mathematics Mathematics - Dynamical Systems Mathematics - Group Theory Direct product Mathematics |
| Popis: | 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 |
| Databáze: | OpenAIRE |
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