Simple loops on 2-bridge spheres in 2-bridge link complements
The purpose of this note is to announce complete answers to the following questions. (1) For an essential simple loop on a 2-bridge sphere in a 2-bridge link complement, when is it null-homotopic in the link complement? (2) For two distinct essential simple loops on a 2-bridge sphere in a 2-bridge link complement, when are they homotopic in the link complement? We also announce applications of these results to character varieties and McShane’s identity.
2000 Mathematics Subject Classification:Primary 57M25, 20F06
The first author was supported by a 2-Year Research Grant of Pusan National University. The second author was supported by JSPS Grants-in-Aid 22340013 and 21654011.
Let be a knot or a link in and a punctured sphere in the complement obtained from a bridge sphere of . Then the following natural question arises.
(1) Which essential simple loops on are null-homotopic in ?
(2) For two distinct essential simple loops on , when are they homotopic in ?
A refined version of the first question for -bridge spheres of -bridge links was proposed in the second author’s joint work with Ohtsuki and Riley [20, Question 9.1(2)], in relation with epimorphisms between -bridge links. It may be regarded as a special variation of a question raised by Minsky [9, Question 5.4] on essential simple loops on Heegaard surfaces of -manifolds.
The purpose of this note is to announce a complete answer to Question 1.1 for -bridge spheres of -bridge links established by the series of papers [11, 12, 13, 14] and to explain its application to the study of character varieties and McShane’s identity .
The key tool for solving the question is small cancellation theory, applied to two-generator and one-relator presentations of -bridge link groups. We note that it has been proved by Weinbaum  and Appel and Schupp  that the word and conjugacy problems for prime alternating link groups are solvable, by using small cancellation theory (see also  and references in it). Moreover, it was shown by Sela  and Préaux  that the word and conjugacy problems for any link group are solvable. A characteristic feature of our work is that it gives complete answers to special (but also natural) word and conjugacy problems for the link groups of 2-bridge links, which form a special (but also important) family of prime alternating links. (See [1, 4] for the role of -bridge links in Kleinian group theory.)
This note is organized as follows. In Sections 2, 3 and 4, we describe the main results, applications to character varieties and McShane’s identity. The remaining sections are devoted to explanation of the idea of the proof of the main results. In Section 5, we describe the two-generator and one-relator presentation of the -bridge link group to which small cancellation theory is applied, and give a natural decomposition of the relator, which plays a key role in the proof. In Section 6, we introduce a certain finite sequence associated with the relator and state its key properties. In Section 7, we recall small cancellation theory and present a characterization of the “pieces” of the symmetrized subset arising from the relator. In Sections 8 and 9, we describe outlines of the proofs of the main results.
The authors would like to thank Norbert A’Campo, Hirotaka Akiyoshi, Brian Bowditch, Danny Calegari, Max Forester, Koji Fujiwara, Yair Minsky, Toshihiro Nakanishi, Caroline Series and Ser Peow Tan for stimulating conversations.
2. Main results
For a rational number , let be the -bridge link of slope , which is defined as the sum of rational tangles of slope and (see Figure 1). The common boundary of the rational tangles is identified with the Conway sphere , where is the group of isometries of the Euclidean plane generated by the -rotations around the points in the lattice . Let be the -punctured sphere in the link complement . Any essential simple loop in , up to isotopy, is obtained as the image of a line of slope in by the covering projection onto . The (unoriented) essential simple loop in so obtained is denoted by . We also denote by the conjugacy class of an element of represented by (a suitably oriented) . Then the link group is identified with , where denotes the normal closure.
Let be the Farey tessellation, whose ideal vertex set is identified with . For each , let be the group of automorphisms of generated by reflections in the edges of with an endpoint , and let be the group generated by and . Then the region, , bounded by a pair of Farey edges with an endpoint and a pair of Farey edges with an endpoint forms a fundamental domain of the action of on (see Figure 2). Let and be the closed intervals in obtained as the intersection with of the closure of . Suppose that is a rational number with . (We may always assume this except when we treat the trivial knot and the trivial -component link.) Write
where , , and . Then the above intervals are given by and , where
(1) If two elements and of belong to the same orbit -orbit, then the unoriented loops and are homotopic in .
(2) For any , there is a unique rational number such that is contained in the -orbit of . In particular, is homotopic to in . Thus if , then is null-homotopic in .
Thus the following question naturally arises (see [20, Question 9.1(2)]).
(1) Does the converse to Proposition 2.1(2) hold? Namely, is it true that is null-homotopic in if and only if belongs to the -orbit of or ?
(2) For two distinct rational numbers , when are the unoriented loops and homotopic in ?
The loop is null-homotopic in if and only if belongs to the -orbit of or . In other words, if , then is not null-homotopic in .
This has the following application to the study of epimorphisms between -bridge link groups (see [11, Section 2] for precise meaning).
There is an upper-meridian-pair-preserving epimorphism from to if and only if or belongs to the -orbit of or .
Suppose that is a rational number such that . For distinct , the unoriented loops and are homotopic in if and only if one of the following holds.
, where is an integer, and and satisfy and , where is a pair of relatively prime positive integers.
, namely is the Whitehead link, and the set equals either or .
The proof of Theorem 2.5 reveals the structure of the normalizer of an element of represented by . This enables us to show the following.
Let be a rational number such that . Suppose is hyperbolic, i.e., and , and let be a rational number contained in .
(1) The loop is peripheral if and only if one of the following holds.
and or .
for some integer , and .
for some integers , and .
(2) The conjugacy class is primitive in with the following exceptions.
and or . In this case is the third power of some primitive element in .
and . In this case is the second power of some primitive element in .
and . In this case is the second power of some primitive element in .
At the end of this section, we describe a relation of Theorem 2.3 with the question raised by Minsky in [9, Question 5.4]. Let be a Heegaard splitting of a -manifold . Let be the mapping class group of , and let be the kernel of the map . Identify with a subgroup of , and consider the subgroup of . Now let be the set of (isotopy classes of) simple loops in which bound a disk in . Let be the set of essential simple loops in which are null-homotopic in . Note that contains and invariant under . In particular, the orbit is a subset of . Then Minsky posed the following question.
When is equal to the orbit ?
The above question makes sense not only for Heegaard splittings but also bridge decompositions of knots and links. In particular, for -bridge links, the groups and in our setting correspond to the groups and , and hence the group corresponds to the group . To make this precise, recall the bridge decomposition , and let (resp. ) be the mapping class group of the pair (resp. ), and let be the kernel of the natural map (resp. ). Identify with a subgroup of the mapping class group of the -times punctured sphere . Recall that the Farey tessellation is identified with the curve complex of and there is a natural epimorphism from to the automorphism group of , whose kernel is isomorphic to . Then the groups and , respectively, are identified with the images of and by this epimorphism. Moreover, the sets and , respectively, correspond to the sets and . Theorem 2.3 says that the set of simple loops in which are null-homotopic in is equal to the orbit . Thus Theorem 2.3 may be regarded as an answer to the special variation of Question 2.7.
3. Application to character varieties
In this section and the next section, we assume , where and are relatively prime positive integers such that . This is equivalent to the condition that is hyperbolic, namely the link complement admits a complete hyperbolic structure of finite volume. Let be the -representation of obtained as the composition
where the last homomorphism is the holonomy representation associated with the complete hyperbolic structure.
Now, let be the once-punctured torus obtained as the quotient , and let be the orbifold where is the group generated by -rotations around the points in . Note that is the orbifold with underlying space a once-punctured sphere and with three cone points of cone angle . The surfaces and , respectively, are -covering and -covering of , and hence their fundamental groups are identified with subgroups of the orbifold fundamental group of indices and , respectively. The -representation of extends, in a unique way, to that of (see [4, Proposition 2.2]), and so we obtain, in a unique way, a -representation of by restriction. We continue to denote it by . Note that is type-preserving, i.e., it satisfies the following conditions.
is irreducible, i.e., its image does not have a common fixed point on .
maps a peripheral element of to a parabolic transformation.
By extending the concept of a geometrically infinite end of a Kleinian group, Bowditch  introduced the notion of the end invariants of a type-preserving -representation of . Tan, Wong and Zhang  (cf. ) extended this notion (with slight modification) to an arbitrary -representation of . (To be precise,  treats -representations. However, the arguments work for -representations.)
To recall the notion of end invariants, let be the set of free homotopy classes of essential simple loops on . Then is identified with , the vertex set of the Farey tessellation by the following rule. For each , let be the essential simple loop on obtained as the image of a line of slope in . Then the correspondence gives the desired identification . The projective lamination space is then identified with and contains as the dense subset of rational points.
Let be a -representation of .
(1) An element is an end invariant of if there exists a sequence of distinct elements such that and such that is bounded from above.
(2) denotes the set of end invariants of .
In the above definition, it should be noted that is well-defined though is defined only up to sign. Note also that the condition that is bounded from above is equivalent to the condition that the hyperbolic translation lengths of the isometries of are bounded from above.
Tan, Wong and Zhang [26, 30] showed that is a closed subset of and proved various interesting properties of , including a characterization of those representations with or , generalizing a result of Bowditch . They also proposed an interesting conjecture [30, Conjecture 1.8] concerning possible homeomorphism types of . The following is a modified version of the conjecture of which Tan  informed the authors.
Suppose has at least two accumulation points. Then either or a Cantor set of .
They constructed a family of representations which have Cantor sets as , and proved the following supporting evidence to the conjecture.
Let be discrete in the sense that the set is discrete in . Then if has at least three elements, then is either a Cantor set of or all of .
The above theorem implies that the end invariants of the representation induced by the holonomy representation of a hyperbolic -bridge link is a Cantor set. But it does not give us the exact description of . By using the main results stated in Section 2, we can explicitly determine the end invariants . To state the theorem, recall that the limit set of the group is the set of accumulation points in the closure of of the -orbit of a point in .
For a hyperbolic -bridge link , the set is equal to the limit set of the group .
We would like to propose the following conjecture.
Let be a type-preserving representation such that . Then is conjugate to the representation .
4. Application to McShane’s identity
In his Ph.D. thesis , McShane proved the following surprising theorem.
Let be a type-preserving fuchsian representation. Then
In the above identity, denotes the translation length of the orientation-preserving isometry of the hyperbolic plane. This identity has been generalized to cusped hyperbolic surfaces by McShane himself , to hyperbolic surfaces with cusps and geodesic boundary by Mirzakhani , and to hyperbolic surfaces with cusps, geodesic boundary and conical singularities by Tan, Wong and Zhang . A wonderful application to the Weil-Petersson volume of the moduli spaces of bordered hyperbolic surface was found by Mirzakhani . Bowditch  (cf. ) showed that the identity in Theorem 4.1 is also valid for all quasifuchsian representations of , where is regarded as the complex translation length of the orientation-preserving isometry of the hyperbolic -space. Moreover, he gave a nice variation of the identity for hyperbolic once-punctured torus bundles, which describes the cusp shape in terms of the complex translation lengths of essential simple loops on the fiber torus . Other -dimensional variations have been obtained by [2, 3, 26, 27, 28, 29, 30, 31].
As an application of the main results stated in Section 2, we can obtain yet another -dimensional variation of McShane’s identity, which describes the cusp shape of a hyperbolic -bridge link in terms of the complex translation lengths of essential simple loops on the bridge sphere. This proves a conjecture proposed by the first author in .
To describe the result, note that each cusp of the hyperbolic manifold carries a Euclidean structure, well-defined up to similarity, and hence it is identified with the quotient of (with the natural Euclidean metric) by the lattice , generated by the translations and corresponding to the meridian and (suitably chosen) longitude respectively. This does not depend on the choice of the cusp, because when is a two-component link there is an isometry of interchanging the two cusps. We call the modulus of the cusp and denote it by .
For a hyperbolic -bridge link with , the following identity holds:
Further the modulus of the cusp torus of the cusped hyperbolic manifold with respect to a suitable choice of a longitude is given by the following formula:
The main results stated in Section 2 are used to establish the absolute convergence of the infinite series.
5. Presentations of 2-bridge link groups
In the remainder of this note, and denote relatively prime positive integers such that and . Theorems 2.3 and 2.5 are proved by applying the small cancellation theory to a two-generator and one-relator presentation of the link group . To recall the presentation, let and , respectively, be the elements of represented by the oriented loops and based on as illustrated in Figure 3. Then is identified with the free group . Note that intersects the disk, , in bounded by a component of and the essential arc, , on of slope , in Figure 3.
To obtain an element, , of represented by the simple loop (with a suitable choice of an orientation and a path joining to the base point ), note that the inverse image of (resp. ) in is the union of the single arrowed (resp. double arrowed) vertical edges in Figure 4. Let be the line in of slope passing through the origin, and let be the line in obtained by slightly modifying near each of the lattice points in so that takes an upper circuitous route around it, as illustrated in Figure 4. Pick a base point from the intersection of with the second quadrant, and consider the sub-line-segment of bounded by and . Then the image of the sub-line-segment in is homotopic to the loop . Let be the word in obtained by reading the intersection of the line-segment with the vertical lattice lines (= the inverse images of and ) as in Figure 4. Then is represented by the simple loop , and we obtain the following two-generator one-relator presentation.
To describe the explicit formula for , set where is the greatest integer not exceeding . Then we have the following (cf. [22, Proposition 1]). Let
where is the greatest integer not exceeding .
If is odd, then
If is even, then
In the above formula, is obtained from the open interval of bounded by and .
We now describe a natural decomposition of the word , which plays a key role in the proof of the main results. Let () be the rational number introduced in Section 2. Then and the parallelogram in spanned by , , and does not contain lattice points in its interior. Consider the infinite broken line, , obtained by joining the lattice points
which is invariant by the translation . Let be the topological line obtained by slightly modifying near each of the lattice points in so that takes an upper or lower circuitous route around it according as the lattice point is of the form or for some , as illustrated in Figure 5. We may assume the base points and in also lie in . Then the sub-arcs of and bounded by and are homotopic in by a homotopy fixing the end points. Moreover, the word is also obtained by reading the intersection of the sub-path of with the vertical lattice lines. Pick a point whose -coordinate is , and set and . Let be the sub-path of bounded by and (), and consider the subword, , of corresponding to . Then we have the decomposition
where the lengths of the subwords are given by and . This decomposition plays a key role in the following section.
6. Sequences associated with the simple loop
We begin with the following observation.
The word is reduced, i.e., it does not contain or for any . It is also cyclically reduced, i.e., all its cyclic permutations are reduced.
The word is alternating, i.e., and appear in alternately, to be precise, neither nor appears in . It is also cyclically alternating, i.e., all its cyclic permutations are alternating.
This observation implies that the word is determined by the -sequence defined below and the initial letter (with exponent).
(1) Let be a nonempty reduced word in . Decompose into
where, for each , all letters in have positive (resp. negative) exponents, and all letters in have negative (resp. positive) exponents. (Here the symbol means that the two words are not only equal as elements of the free group but also visibly equal, i.e., equal without cancellation.) Then the sequence of positive integers is called the -sequence of .
(2) Let be a nonempty reduced cyclic word in represented by a word . Decompose into
where all letters in have positive (resp. negative) exponents, and all letters in have negative (resp. positive) exponents (taking subindices modulo ). Then the cyclic sequence of positive integers is called the cyclic -sequence of . Here the double parentheses denote that the sequence is considered modulo cyclic permutations.
In the above definition, by a cyclic word, we mean the set of all cyclic permutations of a cyclically reduced word. By , we denote the cyclic word associated with a cyclically reduced word .
For a rational number with , let be the word in defined in Section 5. Then the symbol (resp. ) denotes the -sequence of (resp. cyclic -sequence of ), which is called the S-sequence of slope (resp. the cyclic S-sequence of slope ).
We can easily observe the following.
where is the natural decomposition of obtained at the end of the last section. It is also not difficult to observe and . By setting and , we have the following key propositions.
The decomposition satisfies the following.
Each is symmetric, i.e., the sequence obtained from by reversing the order is equal to . (Here, is empty if .)
Each occurs only twice in the cyclic sequence .
Set . Then consists of only and , and begins and ends with , whereas begins and ends with .
Let be as in Proposition 6.3. For a rational number with , suppose that the cyclic -sequence contains both and as a subsequence. Then .
7. Small cancellation conditions for 2-bridge link groups
A subset of the free group is called symmetrized, if all elements of are cyclically reduced and, for each , all cyclic permutations of and also belong to .
Suppose that is a symmetrized subset of . A nonempty word is called a piece if there exist distinct such that and . Small cancellation conditions and , where and are integers such that and , are defined as follows (see ).
Condition : If is a product of pieces, then .
Condition : For with no successive elements an inverse pair mod , if , then at least one of the products , is freely reduced without cancellation.
The following proposition enables us to apply small cancellation theory to the group presentation of .
Let be a rational number such that , and let be the symmetrized subset of generated by the single relator of the group presentation . Then satisfies and .
This proposition follows from the following characterization of pieces, which in turn is proved by using Proposition 6.3.
(1) A subword of the cyclic word is a piece if and only if does not contain as a subsequence and does not contain in its interior, i.e., does not contain a subsequence for some .
(2) For a subword of the cyclic word , is not a product of two pieces if and only if either contains as a proper initial subsequence or contains as a proper terminal subsequence.
8. Outline of the proof of Theorem 2.3
Let be the symmetrized subset of generated by the single relator of the group presentation . Suppose on the contrary that is null-homotopic in , i.e., in , for some . Then there is a van Kampen diagram over such that the boundary label is . Here is a simply connected -dimensional complex embedded in , together with a function assigning to each oriented edge of , as a label, a reduced word in such that the following hold.
If is an oriented edge of and is the oppositely oriented edge, then .
For any boundary cycle of any face of , is a cyclically reduced word representing an element of . (If is a path in , we define .)
We may assume is reduced, namely it satisfies the following condition: Let and be faces (not necessarily distinct) of with an edge , and let and be boundary cycles of and , respectively. Set and . Then we have . Moreover, we may assume the following conditions:
for every vertex .
For every edge of , the label is a piece.
For a path in of length such that the vertex has degree for , cannot be expressed as a product of less than pieces.
Since satisfies the conditions and by Proposition 7.2, is a -map, i.e.,
for every vertex ;
for every face .
Here, , the degree of , denotes the number of oriented edges in having as initial vertex, and , the degree of , denotes the number of oriented edges in a boundary cycle of .
Now, for simplicity, assume that is homeomorphic to a disk. (In general, we need to consider an extremal disk of .) Then by the Curvature Formula of Lyndon and Schupp (see [16, Corollary V.3.4]), we have
By using this formula, we see that there are three edges , and in such that and , where for each . Since is not expressed as a product of two pieces, we see by Proposition 7.3 that the boundary label of contains a subword, , with or . This in turn implies that the -sequence of the boundary label contains both and as subsequences. Hence, by Proposition 6.4, we have , a contradiction.
9. Outline of the proof of Theorem 2.5
Suppose, for two distinct , the unoriented loops