Orderable 3-manifold groups

By S. Boyer, D. Rolfsen, Bert Wiest. 2002

209 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/29064decf71ece87cd4ae63666fa9df8111dd769

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Foliations and the Geometry of 3-Manifolds (2007, background): “(locally indicable 3-manifold groups) The following example is due to Boyer-Rolfsen-Wiest [22].”
Foundations of Garside Theory (2013, background): “On the one hand, dee p relationships appeared between properties of 3-manifolds and the possible orderability of their fundamen tal group involving in particular L-spaces and co-oriented tau t foliations, see for instance Boyer–Rolfsen–Wiest [29] or Boyer–Gordon–Watson [28].”
Foliations, orders, representations, L-spaces and graph manifolds (2014, background): “.1] for a proof.1.) Theorem 1.2. If a graph manifold rational homology 3-sphere has a left-orderable fundamental group, then it is not an L-space. Theorem 1.1 is known to hold when W is Seifert ﬁbred [BRW], [BGW] (and that (1), (2), and (3) are equivalent to W not being an L-space [LS]). In this cas”
3-Manifold Groups (2012, background): “rderable. Note that left-orderability is not a ‘green property’, i.e., there exist compact 3-manifolds with non-left-orderable fundamental groups which admit left-orderable ﬁnite-index subgroups (see [BRW05] and [DPT05]). (C.17) Let N6= S1 ×D2 be a compact, orientable, irreducible 3-manifold. If Nha”
Floer simple manifolds and L-space intervals (2015, unknown): “Our idiosyncratic numbering owes to a result of Boyer, Rolfsen, and Wiest [8], which implies that (2.ρ) = (2.”