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Orderability and Dehn filling
Geometric Topology · Submitted 11 Feb 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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 $\widetilde{\mathrm{PSL}_2 \mathbb{R}}$, which we organize into an infinite graph in $H^1(\partial M; \mathbb{R})$ called the translation extension locus. We include many plots of such loci which inform the proofs of our main results and suggest interesting avenues for future research.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.72 | 0.63 | 0.15 | asserted 0.85 |
| s1 | 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. | contribution 0.55 | 0.95 | 0.57 | asserted 0.99 |
| s2 | Our technique uses certain representations from $\pi_1(M)$ into $\widetilde{\mathrm{PSL}_2 \mathbb{R}}$, which we organize into an infinite graph in $H^1(\partial M; \mathbb{R})$ called the translation extension locus. | method 0.98 | 0.94 | 0.07 | asserted 1.00 |
| s3 | We include many plots of such loci which inform the proofs of our main results and suggest interesting avenues for future research. | contribution 0.46 | 0.95 | 0.04 | hedged possibility 0.63 |
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The paper
Original abstract
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 $\widetilde{\mathrm{PSL}_2 \mathbb{R}}$, which we organize into an infinite graph in $H^1(\partial M; \mathbb{R})$ called the translation extension locus. We include many plots of such loci which inform the proofs of our main results and suggest interesting avenues for future research.
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How the literature uses it
4 arXiv papers cite this in 8 sentences
Papers citing it · 5 sentences without a confident label
Cites it as support 1 paper
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Finally, one other piece of experimental evidence to support Conjecture [ref] is that Culler and Dunfield [ref] also remark that they were unable to find an $L$-space knot whose Alexander polynomial does not have a simple root on the unit circle.
Azumaya algebras and canonical components · $L$-space knots · 0.99
Cites it as background 2 papers
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A stronger conjecture states that for an irreducible $ℚ$-homology 3-sphere, being a non L-space, having left-orderable fundamental group and admitting a co-orientable taut foliation are the same (see e.g. Culler-Dunfield [ref] ).
Non L-space integral homology 3-spheres with no nice orderings · Introduction · 0.91
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Marc Culler and Nathan Dunfield [ref] defined the knot complement of $K$ to be lean, if every closed essential surface in the manifold $S^3_0(K)$ obtained by longitudinal Dehn surgery is a fiber in a fibration over circle.
Left-orderablity for surgeries on $(-2,3,2s+1)$-pretzel knots · Introduction · 0.93
What it cites · 29 sentences without a confident label
Uses the method or tool of 1 paper
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Following [ref] , we define the augmented representation variety $M$ to be the subalgebraic set of $R(M) ×^1$ consisting of all pairs $(ρ, x)$ where $x$ is a point of $$ that is fixed by the image of $π_1(∂M)$ under $ρ∘$.
math.NT/0308041 · Representation and character varieties · 0.80
Uses data or values from 1 paper
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For example, Goerner [ref] has calculated the trace fields of all 61,911 cusped manifolds that can be triangulated with at most 9 ideal tetrahedra [ref] .
The cusped hyperbolic census is complete · Applicability · 0.80
Compares with 3 papers
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Indeed, some of the results on branched covers in [ref] can be viewed as special cases of both the statement and the proof of Theorem [ref](a).
Left-orderability and cyclic branched coverings · Translation extension loci · 0.88
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Indeed, some of the results on branched covers in [ref] can be viewed as special cases of both the statement and the proof of Theorem [ref](a).
On left-orderability and cyclic branched coverings · Translation extension loci · 0.88
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Indeed, some of the results on branched covers in [ref] can be viewed as special cases of both the statement and the proof of Theorem [ref](a).
Riley's Conjecture on SL(2,R) Representations of 2-Bridge Knots · Translation extension loci · 0.88
Cites as support 1 paper
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By [ref] , in fact the set $X_(M)$ is the image of $R_2(M) ∪R_K(M)$ under $t$.
The variety of characters in PSL(2,C) · Real representations · 0.97
Cites as background 6 papers
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Recall from [ref] that an $L$-space is a $$-homology 3-sphere with minimal Heegaard Floer homology, specifically one where $(Y) = |H_1(Y;)|$.
On knot Floer homology and lens space surgeries · Introduction · 0.98
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We now review some basic facts about representation and character varieties with target group $$; for details, see e.g. [ref] .
The variety of characters in PSL(2,C) · Representation and character varieties · 1.00
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The equivalence of ([ref]) and ([ref]) was formulated as a question by Ozsváth and Szabó after they proved that ([ref]) implies ([ref]) [ref] .
Approximating $C^0$-foliations by contact structures · Introduction · 0.93
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