Left-orderable fundamental groups and Dehn surgery
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 left-orderable fundamental group, the result of Dehn surgery is a closed 3-manifold that need not have this property. We take this as motivation for the study of left-orderability in the context of Dehn surgery, and establish a condition on peripheral elements that must hold whenever a given Dehn surgery yields a manifold with left-orderable fundamental group. This leads to a workable criterion used to determine when sufficiently positive Dehn surgery produces manifolds with non-left-orderable fundamental group. As examples we produce infinite families of hyperbolic knots – subsuming the -pretzel knots – for which sufficiently positive surgery always produces a manifold with non-left-orderable fundamental group. Our examples are consistent with the observation that many (indeed, all known) examples of L-spaces have non-left-orderable fundamental group, as the given families of knots are hyperbolic L-space knots. Moreover, the behaviour of the examples studied here is consistent with the property that sufficiently positive surgery on an L-space knot always yields an L-space.
This paper studies left-orderable groups in the context of 3-manifold topology, and in particular, the question of whether there exist left-orderings of a fundamental group that descend to certain quotients of the group. To begin, we recall the following:
A non-trivial group is left-orderable if there exists a strict total ordering of the elements of that is left-invariant: Whenever then , for any .
We adhere to the somewhat non-standard convention that the trivial group is not left-orderable. An equivalent definition of left-orderability may be given in terms of positive cones; our convention is consistent with requiring that the positive cone be non-empty.
Left-orderability may be viewed as a geometric property: A countable group is left-orderable if and only if it acts faithfully on by order-preserving homeomorphisms. In fact, this leads to a connection between left-orderability and -manifolds as Boyer, Rolfsen and Wiest show that compact, connected, orientable manifolds supporting -covered foliations111We suppose throughout that all foliations are co-orientable. For the relevant definitions of -covered and taut foliation see , for example. have left-orderable fundamental group. For Seifert fibred manifolds the connection is even stronger: a Seifert fibred space with base orbifold has left-orderable fundamental group if and only if it supports a taut foliation . The work of Calegari and Dunfield  and Roberts, Shareshian and Stein  provides related results in the hyperbolic setting. In particular, Calegari and Dunfield show that atoroidal rational homology spheres admitting a taut foliation have virtually left-orderable fundamental group (that is, there exists a left-orderable subgroup of finite index). Moreover, restricting to integer homology spheres, the existence of a taut foliation implies left-orderable fundamental group, so that in this setting establishing non-left-orderability of the fundamental group can be a useful obstruction to such a foliation of the manifold .
Burns and Hale  characterize left-orderability in terms of finitely generated subgroups.
Theorem 2 (Burns-Hale).
A group is left-orderable if and only if every nontrivial finitely generated subgroup surjects onto a left-orderable group.
While using this characterization on an arbitrary group is often an intractable problem, Boyer, Rolfsen and Wiest combine this result with Scott’s Compact Core Theorem  to give a natural characterization of left-orderability for fundamental groups of 3-manifolds .
Theorem 3 (Boyer-Rolfsen-Wiest).
Let be a compact, connected, irreducible, orientable 3-manifold, possibly with boundary. Then is left-orderable if and only if surjects onto a left-orderable group.
It follows immediately that whenever the group is left-orderable. On the other hand, this characterization can be quite subtle for rational homology spheres. In particular, while it follows from this result that the complement of any knot has left-orderable fundamental group, this need not be the case for a rational homology sphere resulting from Dehn surgery on the knot (an operation described below). As a result, the following question seems well motivated and is the principal focus of this work:
How does left-orderability of fundamental groups behave with respect to Dehn surgery?
To this end, suppose that is a compact, connected, irreducible, orientable 3-manifold, with . We recall the notion of Dehn filling, following the conventions of Boyer . Given a primitive class (referred to as a slope) we obtain a closed 3-manifold via Dehn filling by identifying the boundary of a solid torus to the boundary of in such a way that is identified with . In the case that is the exterior of a knot in , there is a canonical basis for the fundamental group of the boundary (the peripheral subgroup) given by the knot meridian and the Seifert longitude . Thus, every slope may be written in the form where the elements generate the group . In this setting, we generally denote by and refer to this manifold as the result of Dehn surgery (also referred to as -surgery) along the knot . On the level of the fundamental group Dehn filling corresponds to killing the surgery slope , in the sense that .
In this setting, Question 4 asks if there exists a left-ordering of that descends to a left-ordering of the quotient . For example, such left-orderings exist when is the complement of the figure eight knot and is a slope in the interval , or when is the complement of a positive torus knot and is a slope smaller than [3, 20]. The tools used in each of these examples are very specific to the geometry of the given situation, and it is not clear how either method might generalize to accommodate a larger class of knots. Moreover, while the question of preserving (or not preserving) left-orderability under quotients has been studied in the field of orderable groups, few of the general theorems available are applicable to the specialized set-up of Dehn surgery (see [10, Chapter 3 and Chapter 5] for a summary of some available results).
Our main result introduces a workable obstruction to the existence of a left-ordering of that descends to a left-ordering of . That is, we provide a condition on the left-orderings of the group which ensures that the quotient in question will not inherit any left-ordering.
Theorem 5 (see Corollary 10).
Let denote the exterior of a non-trivial knot in , and fix the canonical basis for the peripheral subgroup , as above. If the implication
holds for every left-ordering of then is not left-orderable for any .
While this criterion applies to a more general class of manifolds (see in particular Theorem 9), we are able to apply a variant of this criterion to certain classes of knots in . In particular, we show that sufficiently positive surgery on positive -torus knots gives rise to manifolds with fundamental group that cannot be left-ordered (see Theorem 24). Since the result of such a surgery is Seifert fibred , this recovers a known result (see discussion below) via different means. However, we are also able to extend our calculation to cover a large class of hyperbolic knots. We prove:
Let be a positively -twisted -torus knot. If (1) and , or (2) and , then is not left-orderable for sufficiently positive.
The notion of a positively twisted torus knot is introduced in Section 3. This class includes, for example, the -pretzel knots for odd (note that the cases correspond to torus knots). As an immediate consequence, the result of sufficiently positive surgery on any of the knots considered in this theorem is a hyperbolic -manifold that does not admit an -covered foliation. Indeed, the criterion applied to establish this fact may be a useful algebraic tool for obstructing such a foliation in other contexts. However, as an -covered foliation is an instance of a taut foliation, it is interesting to note that this fact may be obtained by other means for the knots in question. We describe this further in the next section.
1.1. Background and further motivation
While the main focus of this work is the interplay between left-orderability and Dehn surgery, part of our motivation for studying left-orderability of fundamental groups in this context comes from Heegaard Floer homology. Let be a closed, connected, oriented 3-manifold, and denote by the (‘hat’ version of) Heegaard Floer homology of [17, 16]. We will be particularly interested in a class of manifolds introduced in  for which the Heegaard Floer homology is a simple as possible.
A closed, connected, orientable 3-manifold is an L-space if it is a rational homology sphere satisfying .
Examples of L-spaces include lens spaces, as well as manifolds admitting elliptic geometry [18, Proposition 2.3].
An interesting topological property enjoyed by this class of manifolds is that they do not admit taut foliations, according to a result of Ozsváth and Szabó [15, Theorem 1.4]. Restricting to the class of Seifert fibred spaces with base orbifold , a converse to this result has been established by Lisca and Stipsicz [12, Theorem 1.1]. That is, non-L-spaces within this class are characterized by the existence of a taut foliation. Combined with a result of Boyer, Rolfsen and Wiest [4, Theorem 1.3(b)], this characterization may be restated in terms of left-orderability of the fundamental group, as observed by Peters  and Boyer, Gordon and Watson . In fact, with this observation as a point of departure, it has been established that a Seifert fibred space (without restriction on the base orbifold) is an L-space if and only if its fundamental group cannot be left-ordered  (see also ).
This suggests a correspondence between L-spaces and non-left-orderability of fundamental groups, a phenomenon that has been studied further in various settings [3, 8, 20]. In light of this correspondence, it is natural to ask whether properties enjoyed by L-spaces may be translated into statements regarding non-left-orderability of the fundamental group.
With respect to Dehn filling, L-spaces obey the following property (see [18, Proposition 2.1]). Fix a compact, connected, orientable 3-manifold with torus boundary. Given a pair of slopes and in with geometric intersection 1 and , if and are L-spaces then so is . In the setting of surgery on a knot in , this property may be restated a follows: If is an L-space for some integer , then is also an L-space for any rational number . A knot admitting -space surgeries will be referred to as an L-space knot; examples are provided by torus knots and, more generally, Berge knots – those knots known to admit lens space surgeries . Note that, up to taking mirrors, we may always assume that the integer is positive. We will restrict ourselves to considering positive surgeries in this work.
Given this property of -spaces with respect to Dehn surgery, the correspondence between L-spaces and the non-left-orderability of fundamental groups suggests that L-space knots should yield large families of 3-manifolds with fundamental group that cannot be left-ordered. In particular, one is led to ask:
Is the fundamental group of the manifold obtained by sufficiently positive surgery on an L-space knot non-left-orderable?
More generally, one might consider a manifold with torus boundary (as above) for which neither nor may be left-ordered, and ask whether is non-left-orderable. However, in the context of left-orderability questions pertaining to the relationship between three such quotients seem somewhat unnatural (or, at very least, unstudied). We leave this to future work.
The remainder of the paper is organized as follows. Section 2 contains the proof of our main criterion and its corollaries. In Section 3 we introduce a family of twisted torus knots, compute their fundamental groups, and fix a choice of generators for the peripheral subgroups. These serve as a class of examples to which we may apply our obstructions to left-orderable fundamental groups after Dehn filling, the focus of Section 4.
The authors thank Steve Boyer and Dale Rolfsen for their encouragement and input on this work, as well as Tye Lidman for helpful comments on an earlier version of the manuscript.
2. A criterion for obstructing left-orderability
The following criterion establishes a necessary condition on peripheral elements, given a left-orderable fundamental group arising from a Dehn filling.
Let be given, with and . Suppose that is compact, connected, orientable -manifold with incompressible torus boundary, and suppose that is not sent to under the quotient map . If is left-orderable, then there exists a left-ordering of relative to which the elements and have opposite signs.
Before turning to the proof of this result, we record some immediate corollaries that will be used in the sequel. These come in the form of sufficient conditions to conclude that certain Dehn surgeries will give rise to non-left-orderable fundamental groups. In Corollaries 10 and 11, denotes the exterior of a nontrivial knot in , with canonical generators and for the peripheral subgroup.
Let be given, with and . If implies for every left-ordering of , then is not left-orderable.
For contradiction, suppose that is left-orderable. Since is the complement of a nontrivial knot, is nontrivial  (note that we could alternatively appeal to our convention that the trivial group is not left-orderable), and hence is not sent to under the quotient map . Thus, we may apply Theorem 9 to conclude that there exists a left-ordering of such that and have opposite signs.
If , this contradicts the hypothesis that implies for every left-ordering of . If , then the opposite ordering of contradicts our hypothesis. ∎
Let be given, with and . If implies for all , then is not left-orderable.
Choose such that , and apply Corollary 10. ∎
2.1. The proof of Theorem 9
We now collect the requisite material for the proof of our main result.
Suppose that is a left-ordered group. A subgroup of is said to be convex relative to the left-ordering of if for every and the implication
The following propositions and definitions are standard, see [10, Propositions 2.1.1 – 2.1.3] for proofs. Together with the observations that follow, they establish the role of convex subgroups in the present context.
Suppose that is a nontrivial subgroup of the left-ordered group with ordering . Then is convex relative to the ordering if and only if the prescription
provides a well-defined left invariant ordering of the set of left cosets .
The left-ordering of the set of cosets in Proposition 13 is called the quotient ordering of arising from the left-ordering of , and will be denoted by .
Suppose that is a left-orderable group, and that is a nontrivial subgroup of . The restriction of to the subgroup will be denoted by , and is called the restriction ordering of arising from .
is a short exact sequence of nontrivial groups. Then is convex relative to the left-ordering of if and only if and are left-orderable. Moreover, the left-ordering of is related to the left-orderings and by the following rule: Given , if , if and only if ; otherwise, and if and only if .
The proof is routine. ∎
We remark that a version of this proposition holds even if does not have a group structure. That is, we may use a left-invariant ordering of any convex subgroup and a left-invariant ordering of the set of left cosets in order to create a left-invariant ordering of .
Let be given, with , and suppose that is a basis for . If and
then and have opposite signs.
Recall that the determinant is the signed area the parallelogram bounded by the vectors and . In particular, because the acute angle from to is swept out in a clockwise direction, the quantity is negative. We rewrite:
so the quantity is negative.
On the other hand, the determinant is positive, and we compute that the quantity is positive. Therefore, and have opposite signs. ∎
Let be given, with . Let be any ordering of relative to which the subgroup is convex. If then and have opposite signs in the ordering of .
Consider the short exact sequence
where the map is the quotient map. Since , neither nor lies in the kernel of the map . Thus, by Proposition 16 we must show that and have opposite signs in order to show that and have opposite signs relative to the left-ordering of .
Choose such that . Writing
we see that and . From Lemma 17, we know that and have opposite signs. ∎
Let be a nontrivial subgroup of the left-ordered group . If is convex relative to the left-ordering , then is convex relative to the restriction ordering .
The proof is routine. ∎
Suppose that is compact, connected, orientable -manifold with incompressible torus boundary, and let be a slope in , and suppose that is not sent to under the quotient map . If is left-orderable, then we may define a left-ordering of such that is convex relative to the restriction ordering of .
Suppose that is left-orderable, and denote by the normal closure of in . In particular, is a left-orderable group, because it is a subgroup of , which is left-orderable by Theorem 3 (see [4, Theorem 1.1]). By Proposition 16, we may use the short exact sequence
to create a left-ordering of relative to which is a convex subgroup.
A nontrivial subgroup of is either isomorphic to a copy of the integers or it is of rank two. In the case that is convex and of rank two, is the whole group. This follows from the observation that for convex groups, if for some then . Thus, if and generate and is any other element of , we may write for some integers , hence is in and .
We apply these observations to the subgroup , which is convex in relative to the left-ordering by Lemma 19. In our present setting, since is not sent to under the quotient map , we conclude that must be isomorphic to a copy of the integers. Since is primitive,
and this subgroup is convex relative to the ordering of . ∎
Proof of Theorem 9.
With the above results in place, we are now in a position to prove our main criterion. Recall that is compact, connected, orientable -manifold with incompressible torus boundary. By Proposition 20, we may create a left-ordering of such that is convex relative to the restriction ordering of . By Lemma 18, the elements and have opposite signs in the restriction ordering , and hence they must have opposite signs in the ordering of . This completes the proof. ∎
2.2. A remark on bi-orderability
Recall that a bi-ordering of a group is a left-ordering that is also invariant under right multiplication. Work of Clay and Rolfsen has shown that the knot group of an L-space knot cannot be bi-ordered . In light of Question 8 we have the following proposition, consistent with this observation.
Recall that an element in a group is primitive if it cannot be written as for some with . Every element in a knot group may be written as a power of some primitive element , and so we assume that where is primitive and .
Since is bi-orderable and is primitive, there exists a left-ordering of such that is the smallest positive element . If also satisfies , then theorem is established.
If , note that the subgroup is convex relative to the left-ordering , and thus descends to a left-invariant ordering of the cosets . Moreover, the coset is different from , since we are assuming . We may then define a left-ordering of relative to which is positive and is negative by reversing the ordering of the cosets, as in the following definition:
Given , if , declare if and only if ; otherwise, and if and only if for some (c.f. Proposition 16). It is easy to check that so defined provides a left-ordering satisfying , and the theorem follows. ∎
3. Torus knots and related constructions
3.1. Conventions for torus knots and their fundamental groups
Denote the -torus knot by , where and are relatively prime, positive integers. The knot , for example, is shown in Figure 1.
The knot group of the torus knot is given by the presentation , with generators and represented by the curves depicted in Figure 1. We recall that one may arrive at this presentation immediately from an application of the Seifert-Van Kampen Theorem applied to the obvious genus 1 Heegaard decomposition of , where and are generators the fundamental groups of the respective handlebodies. This point of view will be essential in what follows.
Denote the meridian and longitude curves by and . Relative to the generators we may write , where are integers satisfying , we may assume that and . Observe that by using the relation , we may rewrite the meridian as . The longitude is given by , since (equivalently, ) specify the surface framing of the torus knot (see Moser  for details).
These choices specify generators and for the peripheral subgroup up to conjugacy.
3.2. Preliminaries on twisted torus knots
We begin by recording some facts about twisted torus knots. We will focus on the positively twisted -torus knots, denoted where denotes the number of (positive) full twists added along a pair of strands. Further, we restrict our attention to the case when is congruent to modulo to streamline the discussion; the case when is congruent to 1 modulo 3 is similar. This family of knots may be constructed by adding a second handle to the the standard splitting torus for the knot , allowing two of the three strands to pass over the new handle, and finally adding positive full twists to the new handle. See Figure 2 for the case .
This is a simple construction giving rise to many familiar knot types. For example, it is an easy exercise to show that is the -pretzel knot, denoted . This class pretzel knots provides an infinite family of hyperbolic L-space knots , and includes, for example, the Berge knot .
3.3. Fundamental groups of twisted torus knots
We will use a genus two Heegaard decomposition of to compute the knot group of , which we will denote as . The computation follows the classical application of the Seifert-Van Kampen Theorem as in the case of the torus knots. We restrict ourselves to the case for .
Suppose that and for . Then the fundamental group of is
In particular, when we recover the torus knot group .
We begin by fixing notation. Let be the genus two Heegaard splitting of specified in Figure 3, so that is generated by and and are the free groups and respectively. We will use the Seifert-Van Kampen Theorem to express the knot group as a free product of and with amalgamation.
Therefore, we must first determine the image of the generators of in each of the groups and . The generators in each case are represented by the oriented blue, green and red curves illustrated in Figure 3.
Consulting Figure 3, we see that the generator represented by the green curve may be written as in terms of the generators of , or as in terms of the generators of , and so we have the relation . From the red generator, we read off , and from the blue generator, . Since and , we calculate
where , so that and both record (positive) full twists on two and three strands, respectively. ∎
3.4. Determining the peripheral subgroup
The peripheral subgroup of may be generated by the knot meridian and the surface framing of the knot, represented by those curves illustrated in Figure 3. Denote these elements by and respectively, in this section we will compute and in terms of the generators and . This done, we will fix a choice of peripheral subgroup generated by and , as these generators are easier to work with in Section 4.
First notice that it is immediate from Figure 3 that , where as in the previous section.
Next, we turn to determining in terms of the generators and . To this end, we consider the portion of the knot that lies in the lower handle of the handlebody depicted in Figure 3. This portion of the knot gives us a -braid as in Figure 4, the generators and are as shown. Recall that is the full twist on three strands, which generates the cyclic centre of .
We first conjugate by the generator , this isomorphism by conjugation has the effect of moving linking the top strand to linking the bottom strand of the associated 3-braid (see Figure 5).
We proceed to deduce a formula describing by induction on the number of full twists, . Considering the case , Figure 6 indicates a homotopy between and the meridian .
With this case understood, we turn to adding full twists (as indexed by in Figure 3). Adding a copy of the full twist to our knot is accomplished by inserting the twist, appearing in Figure 7, at the dashed line in the base case illustrated in Figure 6. Note that the generator , as it appears in Figure 7, is homotopic to the core of the handle. This observation allows us to see that for each copy of added to our knot, our meridian (linking the left bottom strand) is modified by prefixing the base case with a copy of the generator .
Thus we arrive at the general formula . By taking the conjugates and , we arrive at:
Suppose , , and let . We may take as generators of the peripheral subgroup of the elements and .
Note that this formula agrees with the formula for the meridian of the -torus knot. This is not a coincidence, as there exists a homotopy (as shown in the above discussion) between and the loop in Figure 5 which is supported away from the twists added to the torus knot to create the knot .
For use in the next section, we record some identities that now follow immediately from the relation in . We have that
and refer to these as the meridian and surface framing respectively for the knot . Notice that as an immediate consequence of the construction, the canonical longitude is given by . This is a direct generalization of the surface framing for the torus knot in the case .
4. Examples and applications
We now turn to applications of our criteria to produce infinite families of rational homology spheres with non-left-orderable fundamental groups. With the obvious exception of surgery on torus knots, the constructions given here produce hyperbolic examples. All of the examples given are L-spaces.
4.1. Surgery on torus knots
The work of Moser establishes that surgery on a torus knot always yields a Seifert fibred space, or a connect sum of lens spaces . As a result, these surgery manifolds are completely understood in terms of L-spaces and non-left-orderability (compare [3, 20]). However, our interest is in establishing that sufficiently large positive surgery on a torus knot gives rise to a manifold with non-left-orderable fundamental group by applying of Corollary 11, directly from the presentation .
Let be a positive -torus knot. Then is not left-orderable whenever .
Suppose that in some left-ordering of . Then either or .
Observe that , so . Assuming both and , their product must also be negative, so we compute
a contradiction. ∎
Suppose that in some left-ordering of . Then for all .
Assuming that we deduce that both and are positive elements as well. We use this fact throughout the proof. By Lemma 25, there are two cases to consider.
Case 1: .
Suppose that is the smallest integer such that . Then we may write:
As is the smallest integer for which , we have , and hence .
Write , where and is positive. This allows us to rewrite the above expression, and use the fact that is central, to arrive at:
We conclude that is positive, as it is a product of positive elements.
Case 2: .
As above, suppose that is the smallest integer such that . Then we may write:
Since we have assumed minimal, both and are positive. Write , where and is positive. Then we use the fact that is central to arrive at
which is a product of positive terms as in Case 1, and the proof is complete. ∎
Suppose that in some left-ordering of . Then .
If , then both and are positive. Hence where . Therefore is a product of positive elements, and so is positive. ∎
Lastly, we observe that the quotient is not left-orderable either, as both generators of are mapped to torsion elements in the quotient (in fact, Moser shows that the quotient is finite ). Combining these observations, we have that surgery coefficients in the interval give rise to non-left-orderable fundamental groups, concluding the proof of Theorem 24.
4.2. Surgery on pretzel knots
We consider the L-space knots for . The fundamental groups of these knots are given by
together with the peripheral elements
If and , then -surgery on the -pretzel knot gives rise to a manifold with non-left-orderable fundamental group.
If is any left-ordering of , then implies for all integers .
Suppose that , and let be any positive integer. If , then the conclusion follows trivially, as an arbitrary product of positive elements is always positive. Assume then, without loss of generality, that is negative (and ).
We proceed by considering two cases, depending on whether the word is a positive or negative element.
Case 1: .
Notice that in this case we have
so that as a product of negative elements, unless . Since implies that , and by assumption, we need only consider .
Note that in this setting implies that and write