# Orderability and Dehn filling

@article{Culler2016OrderabilityAD, title={Orderability and Dehn filling}, author={M. Culler and N. Dunfield}, journal={arXiv: Geometric Topology}, year={2016} }

Motivated by conjectures relating group orderability, Floer homology, and taut foliations, we discuss a systematic and broadly applicable technique for constructing left-orders on the fundamental groups of rational homology 3-spheres. Specifically, for a compact 3-manifold $M$ with torus boundary, we give several criteria which imply that whole intervals of Dehn fillings of $M$ have left-orderable fundamental groups. Our technique uses certain representations from $\pi_1(M)$ into $\widetilde… Expand

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#### References

SHOWING 1-10 OF 63 REFERENCES

On left-orderable fundamental groups and Dehn surgeries on knots

- Mathematics
- 2013

We show that the resulting manifold by $r$-surgery on the hyperbolic twist knot $K_m, \, m \ge 2$, has left-orderable fundamental group if the slope $r$ satisfies the condition $r \in (-4,2m)$ if $m$… Expand

On L-spaces and left-orderable fundamental groups

- Mathematics
- 2011

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… Expand

On Culler-Shalen seminorms and Dehn filling

- Mathematics
- 1998

If F is a finitely generated discrete group and G a complex algebraic Lie group, the G-character variety of r is an affine algebraic variety whose points correspond to characters of representations… Expand

Foliations, orders, representations, L-spaces and graph manifolds

- Mathematics
- 2014

We show that the properties of admitting a co-oriented taut foliation and having a left-orderable fundamental group are equivalent for rational homology $3$-sphere graph manifolds and relate them to… Expand

Degenerations of ideal hyperbolic triangulations

- Mathematics
- 2005

Let M be a cusped 3-manifold, and let $${\mathcal{T}}$$ be an ideal triangulation of M. The deformation variety $${\mathfrak{D}(\mathcal{T})}$$ , a subset of which parameterises (incomplete)… Expand

Ordered Groups and Topology

- Mathematics
- 2016

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… Expand

Laminations and groups of homeomorphisms of the circle

- Mathematics
- 2003

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… Expand

Floer simple manifolds and L-space intervals

- Mathematics
- 2017

Abstract An oriented three-manifold with torus boundary admits either no L-space Dehn filling, a unique L-space filling, or an interval of L-space fillings. In the latter case, which we call “Floer… Expand

Holomorphic disks and genus bounds

- Mathematics
- 2004

We prove that, like the Seiberg-Witten monopole homology, the Heegaard Floer homology for a three-manifold determines its Thurston norm. As a consequence, we show that knot Floer homology detects the… Expand

TAUT FOLIATIONS IN PUNCTURED SURFACE BUNDLES, I

- Mathematics
- 2001

this model, a given fibration can be perturbed to give 'nearby' fibrations. In this paper, we describe an alternative way of perturbing fibrations with non- empty boundary. Instead of fibrations,… Expand