2005/01/01 by Steven Boyer, Dale Rolfsen, Bert Wiest · 179 citations
Mathematics · #Geometric and Algebraic Topology #Advanced Combinatorial Mathematics #Homotopy and Cohomology in Algebraic Topology #Betti number #Mathematics #3-manifold #Invariant (physics) #Manifold (fluid mechanics) #Pure mathematics #Group (periodic table) #Relatively hyperbolic group #Hyperbolic manifold #Mathematical analysis #Physics #Mathematical physics #Hyperbolic function
paper · pdf · doi:10.5802/aif.2098
published in Annales de l’institut Fourier 55(1), 243-288 (Association of the Annals of the Fourier Institute)
openalex publication_date 2005/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the orderability properties of fundamental groups of 3-dimensional manifolds. Many 3-manifold groups support left-invariant orderings, including all compact <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>P</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -irreducible manifolds with positive first Betti number. For seven of the eight geometries (excluding hyperbolic) we are able to characterize which manifolds’ groups support a left-invariant or bi-invariant ordering. We also show that manifolds modelled on these geometries have virtually bi-orderable groups. The question of virtual orderability of 3-manifold groups in general, and even hyperbolic manifolds, remains open, and is closely related to conjectures of Waldhausen and others.