On L-spaces and left-orderable fundamental groups

Boyer, Steven; Gordon, C. McA.; Watson, Liam

doi:10.1007/s00208-012-0852-7

Cited by 165 publications

(234 citation statements)

References 42 publications

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“…This result follows from [BGW,Proposition 7]. Compare with the proof of Theorem 7 of that paper and the comments which follow it.…”

Section: Proposition 72 ([Bgw])mentioning

confidence: 61%

“…Further, it follows from [BRW,Theorems 1.3 and 1.7] that when W is a non-hyperbolic geometric manifold, W has a left-orderable fundamental group if and only if it admits a co-oriented taut foliation. On the other hand, [BGW,Theorem 1 and Corollary 1] imply that for such manifolds, the latter is equivalent to the condition that W not be an L-space. Thus understanding the relationship between the three conditions reduces to the case when W is a rational homology 3-sphere which is either hyperbolic or has a non-trivial JSJ decomposition.…”

Section: Introductionmentioning

confidence: 99%

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

Boyer

Clay

2017

Advances in Mathematics

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Abstract. We show that the properties of admitting a co-oriented taut foliation and having a leftorderable fundamental group are equivalent for rational homology 3-sphere graph manifolds and relate them to the property of not being a Heegaard-Floer L-space. This is accomplished in several steps. First we show how to detect families of slopes on the boundary of a Seifert fibred manifold in four different fashions-using representations, using left-orders, using foliations, and using Heegaard-Floer homology. Then we show that each method of detection determines the same family of detected slopes. Next we provide necessary and sufficient conditions for the existence of a co-oriented taut foliation on a graph manifold rational homology 3-sphere, respectively a left-order on its fundamental group, which depend solely on families of detected slopes on the boundaries of its pieces. The fact that HeegaardFloer methods can be used to detect families of slopes on the boundary of a Seifert fibred manifold combines with certain conjectures in the literature to suggest an L-space gluing theorem for rational homology 3-sphere graph manifolds as well as other interesting problems in Heegaard-Floer theory.

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“…This result follows from [BGW,Proposition 7]. Compare with the proof of Theorem 7 of that paper and the comments which follow it.…”

Section: Proposition 72 ([Bgw])mentioning

confidence: 61%

Section: Introductionmentioning

confidence: 99%

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

Boyer

Clay

2017

Advances in Mathematics

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“…Then ρ sn (a) has order n. It follows from [H,Theorem 3.1] (see also [BGW,Theorem 6]) that π 1 (Σ n (K)) is leftorderable.…”

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confidence: 95%

“…By contrast, there are 2-bridge knots K such that Σ n (K) has non-left-orderable fundamental group for all n, by [Te,Proof of Theorem 2] and [BGW,Theorem 4].…”

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confidence: 99%

Riley's conjecture on SL(2,R) representations of 2-bridge knots

Gordon

2017

J. Knot Theory Ramifications

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“…(see [1,5,6,16,21]). Recall that an L-space is a rational homology sphere with Heegaard Floer homology that is as simple as possible, in the sense that rk HF(Y ) = |H 1 (Y ; Z)| (see [15]).…”

Section: Definition 1 a Group G Is Left-orderable If There Exists A mentioning

confidence: 99%

On cabled knots, Dehn surgery, and left-orderable fundamental groups

Clay

Watson

2011

Mathematical Research Letters

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Abstract. 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 left-orderable [6]. We provide a refinement of this criterion by introducing the notion of a decayed knot; it is shown that Dehn surgery on decayed knots produces surgery manifolds that have non-left-orderable fundamental group for all sufficiently positive surgeries. As an application, we prove that sufficiently positive cables of decayed knots are always decayed knots. These results mirror properties of L-space surgeries in the context of Heegaard Floer homology. Definition 1. A group G is left-orderable if there exists a partition of the group elementssatisfying P · P ⊆ P and P = ∅. The subset P is called a positive cone. This is equivalent to G admitting a left-invariant strict total ordering. For background on left-orderable groups relevant to this paper see [2,6]; a standard reference for the theory of left-orderable groups is [12]. As established by Boyer, Rolfsen and Wiest [2] (compare [11]), the fundamental group π 1 (K) of the complement of a knot K in S 3 is always leftorderable. Indeed, this follows from the fact that any compact, connected, irreducible, orientable 3-manifold with positive first Betti number has left-orderable fundamental group [2, Theorem 1.1]. However, the question of left-orderability for fundamental groups of rational homology 3-spheres is considerably more subtle (see [2,6]) and seems closely tied to certain codimension one structures on the 3-manifold (see [2,3,17]). Continuing along the lines of [6] this paper focuses on Dehn surgery, an operation on knots that produces rational homology 3-spheres. We recall this construction in order to fix notation and conventions.For any knot K in S 3 there is a preferred generating set for the peripheral subgroup Z⊕Z ⊂ π 1 (K) provided by the knot meridian µ and the Seifert longitude λ. The latter is uniquely determined (up to orientation) by the existence of a Seifert surface for K. We orient µ so that it links positively with K, and orient λ so that µ · λ = 1. For any rational number r with reduced form p q we denote the peripheral element µ p λ q by α r . At the level of the fundamental group, the result of Dehn surgery along α r is summarized by the short exact sequenceDate: March 11, 2011. Both authors partially supported by NSERC postdoctoral fellowships. Here α r denotes the normal closure of α r , and S 3 r (K) is the 3-manifold obtained by attaching a solid torus to the boundary of S 3 ν(K), sending the meridian of the torus to a simple closed curve representing the classWe will blur the distinction between α r as an element of the fundamental group or as a primitive class in the (projective) first homology of the boundary, and refer to these peripheral elements as slopes.While many examples of rational homology 3-spheres have left-orderable fundamental group [2], there exist infinite families of knots for which sufficiently positive Deh...

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On L-spaces and left-orderable fundamental groups

Cited by 165 publications

References 42 publications

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

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

Riley's conjecture on SL(2,R) representations of 2-bridge knots

On cabled knots, Dehn surgery, and left-orderable fundamental groups

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