Left-Orderable Fundamental Groups and Dehn Surgery

Clay, Adam; Watson, Liam

doi:10.1093/imrn/rns129

Cited by 32 publications

(49 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There are several known studies on 3-manifolds with a non left-orderable fundamental group by using Dehn surgery. For example, there are works by Clay and Watson [5] and Nakae [6]. In this paper, we show the following, which gives an extension of a result of Nakae.…”

Section: Introductionsupporting

confidence: 60%

“…Moreover by [5,Proposition 3.2], the element y v+1 x −1 represents a meridian of K. Thus we set a = y v+1 x −1 . Then, since a = y v+1 x −1 is equivalent to x = a −1 y v+1 , we have…”

Section: Applicationmentioning

confidence: 99%

“…Finally, as noted in the paragraph just below [5,Proposition 3.2], the presentation of the preferred longitude of K is obtained from the above by adding a …”

Section: Applicationmentioning

confidence: 99%

See 2 more Smart Citations

Non-left-orderable surgeries and generalized Baumslag–Solitar relators

Ichihara

Temma

2015

J. Knot Theory Ramifications

View full text Add to dashboard Cite

show abstract

Section: Introductionsupporting

confidence: 60%

Section: Applicationmentioning

confidence: 99%

See 1 more Smart Citation

Non-left-orderable surgeries and generalized Baumslag–Solitar relators

Ichihara

Temma

2015

J. Knot Theory Ramifications

View full text Add to dashboard Cite

show 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.…”

mentioning

confidence: 99%

“…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 Dehn surgery (that is, along a slope parametrized by a suitable large rational number) yields a manifold with non-left-orderable fundamental group [6]. To make this precise, consider the set of slopes…”

mentioning

confidence: 99%

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

Clay

Watson

2011

Mathematical Research Letters

Self Cite

View full text Add to dashboard Cite

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...

show abstract

On L-spaces and left-orderable fundamental groups

2012

Self Cite

View full text Add to dashboard Cite

Examples suggest that there is a correspondence between L-spaces and 3manifolds whose fundamental groups cannot be left-ordered. In this paper we establish the equivalence of these conditions for several large classes of such manifolds. In particular, we prove that they are equivalent for any closed, connected, orientable, geometric 3-manifold that is non-hyperbolic, a family which includes all closed, connected, orientable Seifert fibred spaces. We also show that they are equivalent for the 2-fold branched covers of non-split alternating links. To do this we prove that the fundamental group of the 2-fold branched cover of an alternating link is left-orderable if and only if it is a trivial link with two or more components. We also show that this places strong restrictions on the representations of the fundamental group of an alternating knot complement with values in Homeo+(S 1 ).While the trivial group obviously satisfies this criterion, in this paper we will adopt the convention that it is not left-orderable.The left-orderability of 3-manifold groups has been studied in work of Boyer, Rolfsen and Wiest [3]. An argument of Howie and Short [18, Lemma 2] shows that the fundamental group of an irreducible 3-manifold M with positive first Betti number is locally indicable, hence left-orderable [4]. More generally, such a group is left-orderable if it admits an epimorphism to a left-orderable group [3, Theorem 1.1(1)].The aim of this note is to establish a connection between L-spaces and the left-orderability of their fundamental groups. Given the results we obtain and those obtained elsewhere [6,7,8,40], we formalise a question which has received attention in the recent literature in the following conjecture. Conjecture 3. An irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable.It has been asked by Ozsváth and Szabó whether L-spaces can be characterized as those closed, connected 3-manifolds admitting no co-orientable taut foliations. Thus, in the context of Conjecture 3 it is interesting to consider the following open questions: Does the existence of a co-orientable taut foliation on an irreducible rational homology 3-sphere imply the manifold has a left-orderable fundamental group? Are the two conditions equivalent? Calegari and Dunfield have shown that the existence of a co-orientable taut foliation on an irreducible atoroidal rational homology 3-sphere Y implies that π 1 (Y ) has a left-orderable finite index subgroup [5, Corollary 7.6]. Of course, an affirmative answer to Conjecture 3, combined with [31, Theorem 1.4], would prove that the existence of a co-orientable taut foliation implies left-orderable fundamental group.Our first result verifies the conjecture in the case of Seifert fibred spaces.Theorem 4. Suppose Y is a closed, connected, Seifert fibred 3-manifold. Then Y is an L-space if and only if π 1 (Y ) is not left-orderable.The proof of this theorem in the case where the base orbifold of Y is orientable depends on results of Boyer, Rolfsen ...

show abstract

Left-Orderable Fundamental Groups and Dehn Surgery

Cited by 32 publications

References 17 publications

Non-left-orderable surgeries and generalized Baumslag–Solitar relators

Non-left-orderable surgeries and generalized Baumslag–Solitar relators

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

On L-spaces and left-orderable fundamental groups

Contact Info

Product

Resources

About