Orderable 3-manifold groups

Boyer, Steven; Rolfsen, Dale; Wiest, Bert

doi:10.5802/aif.2098

Orderable 3-manifold groups [ Groupe de 3-variétés ordonnables ]

Boyer, Steven ; Rolfsen, Dale ; Wiest, Bert

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

Résumé
Abstract

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.

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.

MR 2141698 | Zbl 1068.57001 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2098
Classification : 57M05, 57M50, 20F60
Mots clés : 3 variétés, groupe ordonnable, groupe-LO

@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},
     pages = {243--288},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {1},
     year = {2005},
     doi = {10.5802/aif.2098},
     zbl = {1068.57001},
     mrnumber = {2141698},
     language = {en},
     url = {archive.numdam.org/item/AIF_2005__55_1_243_0/}
}

Boyer, Steven; Rolfsen, Dale; Wiest, Bert. Orderable 3-manifold groups. Annales de l'Institut Fourier, Tome 55 (2005) no. 1, pp. 243-288. doi : 10.5802/aif.2098. http://archive.numdam.org/item/AIF_2005__55_1_243_0/

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