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 -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 -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.
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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