Orderable 3-manifold groups

Boyer, Steven; Rolfsen, Dale; Wiest, Bert

doi:10.5802/aif.2098

Boyer, Steven; Rolfsen, Dale; Wiest, Bert

Annales de l'Institut Fourier, Volume 55 (2005) no. 1, p. 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 2141698 | Zbl 1068.57001 | 1 citation in Numdam

DOI : https://doi.org/10.5802/aif.2098
Classification: 57M05, 57M50, 20F60
Keywords: 3-manifold, orderable group, LO-group

@article{AIF_2005__55_1_243_0,
     author = {Boyer, Steven and Rolfsen, Dale and Wiest, Bert},
     title = {Orderable 3-manifold groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {1},
     year = {2005},
     pages = {243-288},
     doi = {10.5802/aif.2098},
     zbl = {1068.57001},
     mrnumber = {2141698},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2005__55_1_243_0}
}

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. http://www.numdam.org/item/AIF_2005__55_1_243_0/

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