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.
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.
Keywords: 3-manifold, orderable group, LO-group
Mot clés : 3 variétés, groupe ordonnable, groupe-LO
Boyer, Steven 1; Rolfsen, Dale ; Wiest, Bert 
@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{\textquoteright}institut Fourier}, volume = {55}, number = {1}, year = {2005}, doi = {10.5802/aif.2098}, zbl = {1068.57001}, mrnumber = {2141698}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2098/} }
TY - JOUR AU - Boyer, Steven AU - Rolfsen, Dale AU - Wiest, Bert TI - Orderable 3-manifold groups JO - Annales de l'Institut Fourier PY - 2005 SP - 243 EP - 288 VL - 55 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2098/ DO - 10.5802/aif.2098 LA - en ID - AIF_2005__55_1_243_0 ER -
%0 Journal Article %A Boyer, Steven %A Rolfsen, Dale %A Wiest, Bert %T Orderable 3-manifold groups %J Annales de l'Institut Fourier %D 2005 %P 243-288 %V 55 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2098/ %R 10.5802/aif.2098 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.2098/
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