Orderable 3-manifold groups

2002/11/30 by Steven Boyer, Dale Rolfsen, Bert Wiest · 4 citations
Mathematics · #math.GT

paper · pdf

published as Ann. Inst. Fourier, Grenoble, 55, 1 (2005), 243-288 · 37 pages. Published version. Improvements in the organisation and presentation of the material

arxiv created 2005/06/14 · arxiv updated 2009/11/30

Abstract

We investigate the orderability properties of fundamental groups of 3-dimensional manifolds. Many 3-manifold groups support left-invariant orderings, including all compact P²-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.

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