Left-orderable fundamental groups and Dehn surgery

@article{Clay2010LeftorderableFG,
  title={Left-orderable fundamental groups and Dehn surgery},
  author={Adam Clay and Liam Watson},
  journal={arXiv: Geometric Topology},
  year={2010}
}
There are various results that frame left-orderability of a group as a geometric property. Indeed, the fundamental group of a 3-manifold is left-orderable whenever the first Betti number is positive; in the case that the first Betti number is zero this property is closely tied to the existence of certain nice foliations. As a result, many large classes of 3-manifolds, including knot complements, are known to have left-orderable fundamental group. However, though the complement of a knot has… 

Figures from this paper

ORDERABLE GROUPS AND TOPOLOGY MINICOURSE NOTES
  • Mathematics
  • 2014
The goal of this minicourse is to study the orderability properties of fundamental groups of 3-manifolds, and when possible, explain orderability or non-orderability of the fundamental group via
On L-spaces and left-orderable fundamental groups
Examples suggest that there is a correspondence between L-spaces and three-manifolds whose fundamental groups cannot be left-ordered. In this paper we establish the equivalence of these conditions
Non-left-orderable surgeries on twisted torus knots
Boyer, Gordon, and Watson have conjectured that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable. Since large classes of L-spaces can
On cabled knots, Dehn surgery, and left-orderable fundamental groups
Previous work of the authors establishes a criterion on the fundamental group of a knot complement that determines when Dehn surgery on the knot will have a fundamental group that is not
Ordered Groups and Topology
This is a draft of a book submitted for publication by the AMS. Its theme is the remarkable interplay, accelerating in the last few decades, between topology and the theory of orderable groups, with
Left-orderable, non-L-space surgeries on knots
TLDR
This work introduces a way to provide knots with left-orderable, non-L-space surgeries and presents infinitely many hyperbolic knots on each of which every nontrivial surgery is ahyperbolic, left- orderable,non- L-space surgery.
Non-left-orderable surgeries on 1-bridge braids
Boyer, Gordon, and Watson have conjectured that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable. Since Dehn surgeries on knots in
Circular orderability of 3-manifold groups
This paper initiates the study of circular orderability of 3-manifold groups, motivated by the L-space conjecture. We show that a compact, connected, P-irreducible 3-manifold has a circularly
Infinite families of non-left-orderable L-spaces
For each connected alternating tangle, we provide an infinite family of non-left-orderable L-spaces. This gives further support for Conjecture [3] of Boyer, Gordon, and Watson that is a rational
Order-detection of slopes on the boundaries of knot manifolds
. Motivated by the L -space conjecture, we investigate various notions of order-detection of slopes on knot manifolds. These notions are designed to characterise when rational homology 3-spheres
...
...

References

SHOWING 1-10 OF 29 REFERENCES
On L-spaces and left-orderable fundamental groups
Examples suggest that there is a correspondence between L-spaces and three-manifolds whose fundamental groups cannot be left-ordered. In this paper we establish the equivalence of these conditions
On cabled knots, Dehn surgery, and left-orderable fundamental groups
Previous work of the authors establishes a criterion on the fundamental group of a knot complement that determines when Dehn surgery on the knot will have a fundamental group that is not
Ordered groups, eigenvalues, knots, surgery and L-spaces
  • Adam Clay, D. Rolfsen
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 2011
Abstract We establish a necessary condition that an automorphism of a nontrivial finitely generated bi-orderable group can preserve a bi-ordering: at least one of its eigenvalues, suitably defined,
Orderable 3-manifold groups
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
Laminations and groups of homeomorphisms of the circle
Abstract.If M is an atoroidal 3-manifold with a taut foliation, Thurston showed that π1(M) acts on a circle. Here, we show that some other classes of essential laminations also give rise to actions
On knot Floer homology and lens space surgeries
A Note on Group Rings of Certain Torsion-Free Groups
Abstract As a step towards characterizing ID-groups (i.e., groups G such that, for every ring R without zero-divisors, the group ring RG has no zero-divisors), Rudin and Schneider defined Ω-groups, a
Ozsvth-Szab invariants and tight contact 3-manifolds, III
We characterize L-spaces which are Seifert fibered over the 2-sphere in terms of taut foliations, transverse foliations and transverse contact structures. We give a sufficient condition for certain
Discretely ordered groups
We consider group orders and right-orders which are discrete, meaning there is a least element which is greater than the identity. We note that free groups cannot be given discrete orders, although
Holomorphic disks and three-manifold invariants: Properties and applications
In [27], we introduced Floer homology theories HF - (Y,s), HF∞(Y,s), HF + (Y, t), HF(Y,s),and HF red (Y, s) associated to closed, oriented three-manifolds Y equipped with a Spiny structures s ∈ Spin
...
...