Orderable 3-manifold groups

Boyer, Steven; Rolfsen, Dale; Wiest, Bert

doi:10.5802/aif.2098

Boyer, Steven ¹; Rolfsen, Dale; Wiest, Bert

¹ UQAM, Département de mathématiques, P.O. Box 8888, Centre-ville, Montréal, H3C 3P8, Québec (Canada), UBC, Department of Mathematics, Room 121, 1984 Mathematics Road, Vancouver V6T 1Z2 B.C. (Canada), Université de Rennes 1, Institut Mathématique, Campus de Beaulieu, 35042 Rennes Cedex (France)

Annales de l'Institut Fourier, Volume 55 (2005) no. 1, pp. 243-288.

Abstract
Résumé

We investigate the orderability properties of fundamental groups of 3-dimensional manifolds. Many 3-manifold groups support left-invariant orderings, including all compact $P^{2}$ -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.

On étudie l’ordonnabilité des groupes fondamentaux des variétés de dimension 3. Les groupes de nombreuses 3-variétés admettent un ordre invariant à gauche, y compris les groupes de toutes les variétés compactes $P^{2}$ -irréductibles dont le premier nombre de Betti est positif. Pour sept des huit géométries (toutes sauf l’hyperbolique) on caracterise exactement les variétés dont les groupes sont ordonnables à gauche, voire bi- ordonnables ; on démontre aussi qu’elles ont toutes des groupes virtuellement bi- ordonnables. L’ordonnabilité virtuelle en général, notamment pour les 3-variétés hyperboliques, est un problème qui reste ouvert et qui est lié à des conjectures de Waldhausen et d’autres auteurs.

MR Zbl

DOI: 10.5802/aif.2098

Classification: 57M05, 57M50, 20F60
Keywords: 3-manifold, orderable group, LO-group
Mot clés : 3 variétés, groupe ordonnable, groupe-LO

Author's affiliations:

Boyer, Steven ¹; Rolfsen, Dale ; Wiest, Bert

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     title = {Orderable 3-manifold groups},
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     pages = {243--288},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {55},
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Boyer, Steven; Rolfsen, Dale; Wiest, Bert. Orderable 3-manifold groups. Annales de l'Institut Fourier, Volume 55 (2005) no. 1, pp. 243-288. doi : 10.5802/aif.2098. https://aif.centre-mersenne.org/articles/10.5802/aif.2098/

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