The boundary of a fibered face of the magic -manifold and the asymptotic behavior of the minimal pseudo-Anosovs dilatations
Let be the minimal dilatation of pseudo-Anosovs defined on an orientable surface of genus with punctures. Tsai proved that for any fixed , the logarithm of the minimal dilatation is on the order of . We prove that if is relatively prime to or for each , then
Our examples of pseudo-Anosovs ’s which provide the upper bound above have the following property: The mapping torus of is a single hyperbolic -manifold called the magic manifold, or the fibration of comes from a fibration of by Dehn filling cusps along the boundary slopes of a fiber. The main tool in this paper is the boundary of a fibered face of .
Key words and phrases:mapping class group, pseudo-Anosov, dilatation, entropy, magic manifold
2000 Mathematics Subject Classification:Primary 57M27, 37E30, Secondary 37B40
Let be an orientable surface of genus with punctures and the mapping class group of . According to the work of Nielsen and Thurston, elements of are classified into three types: periodic, reducible, pseudo-Anosov, see . Pseudo-Anosov mapping classes have rich dynamical and geometric properties. The hyperbolization theorem by Thurston  relates the dynamics of pseudo-Anosovs and the geometry of hyperbolic fibered -manifolds. The theorem asserts that is pseudo-Anosov if and only if the mapping torus of is a hyperbolic -manifold with finite volume.
Each pseudo-Anosov element has a representative called a pseudo-Anosov homeomorphism. Such a homeomorphism is equipped with a constant called the dilatation of . If we let be the topological entropy of , then the identity holds, see [3, Exposé 10]. The dilatation of does not depend on the choice of a pseudo-Anosov homeomorphism , and hence the dilatation of is defined to be . We call the quantities and the entropy and normalized entropy of respectively, where is the Euler characteristic of .
If we fix , the set of entropies of pseudo-Anosovs defined on is a closed discrete subset of , see . In particular there exists a minimal entropy, and hence there exists a minimal dilatation. We denote by , the minimal dilatation of pseudo-Anosov elements in . The minimal dilatations are determined in only a few cases, see .
Let us set and . Penner proved in  111Let and be functions on . We write if there exists a constant , independent of , such that .that . This work by Penner was a starting point on the study of the asymptotic behavior of the minimal dilatations on surfaces varying topology. Later it was proved by Hironaka-Kin that , and by Tsai that . See also Valdivia. The following theorem, due to Tsai, is in contrast with the cases of genus or .
Theorem 1.1 ().
For any fixed , we have
We ask the following question which is motivated by Theorem 1.1.
Given , does exist? What is its value?
This is an analogous question, posed by McMullen, which is asking whether exists or not, see .
Given , there exists a sequence with such that
We define the polynomial for nonnegative integers and :
We shall see that there exists a unique real root greater than of such that
(Lemma 4.1). The root gives the following upper bound of the minimal dilatations.
For and , suppose that . Then
Suppose that satisfies
or for each .
For example, holds for since is relatively prime to and ; does not hold for because and . We point out that infinitely many ’s satisfy . In fact if is prime, then is relatively prime to for each . Such a enjoys , and this leads to
If is prime for , then
One can simplify in Theorem 1.5, since is relative prime to and . In the case , is equivalent to
or for each .
Our results are proved by using the theory on fibered faces of hyperbolic fibered -manifolds , developed by Thurston, Fried, Matsumoto and McMullen. (See Section 3.) Let be the Thurston norm, and let be a fibered face of . The work of Thurston tells us that if has the second Betti number more than , then it admits a family of fibrations on dominated by , where is the core over with the origin and is its interior. In other words, such a fibered -manifold provides infinitely many pseudo-Anosovs defined on surfaces with variable topology. By work of Fried, the entropy function defined on these fibrations admits a unique continuous extension . By the continuity of and , we have the continuous function
The normalized entropy function is constant on each ray in through the origin. It is shown by Fried that the restriction has the property such that goes to as goes to a point on the boundary of .
These properties give us the following observation: Fix a manifold as above. For any compact set , there exists a constant satisfying the following. Let be any integral class of and let be the monodromy of the fibration associated to . Then the normalized entropy is bounded by from above whenever , where is the projective class of .
This observation enables us to investigate the asymptotic behavior of the minimal dilatations. The following asymptotic inequalities (which are the best known upper bounds) are proved by using a similar technique.
However for any fixed , the observation as above doesn’t work to investigate the asymptotic behavior varying because of Theorem 1.1. Theorem 1.1 implies that there exists no constant , independent of so that . Thus if there exists a sequence of integral classes with such that the fiber of the fibration associated to is a surface of genus having boundary components with , then the accumulation points of the sequence of projective classes must lie on the boundary of . (This is because there exists no constant , independent of , such that .)
Nevertheless we focus on a fibered face of a particular hyperbolic fibered -manifold, called the magic manifold . This manifold is the exterior of the chain link , see Figure 1. Our examples of pseudo-Anosovs ’s which provide the upper bounds in Theorems 1.3, 1.4 and 1.5 have the following property: The mapping torus of is homeomorphic to , or the fibration of comes from a fibration of by Dehn filling cusps along the boundary slopes of a fiber. We also point out that a family of the integral classes of is a main ingredient to prove the asymptotic inequalities (1)–(3) above, see .
We turn to the hyperbolic volume of hyperbolic -manifolds. The set of volumes of hyperbolic -manifolds is a well-ordered closed subset in of order type , see . In particular if we fix a surface , then there exists a minimum among volumes of hyperbolic -bundles over the circle.
Given , there exists a sequence with such that the minimal volume of -bundles over the circle is less than or equal to , the volume of the magic manifold . Furthermore if satisfies , then for any , the minimal volume of -bundles over the circle is less than or equal to .
We close the introduction by asking
Acknowledments. We would like to thank Eriko Hironaka for helpful conversations and comments.
2. Roots of polynomials
This section concerns the asymptotic behavior of roots of families of polynomials. Let
be a polynomial with real coefficients (), where is arranged in the order of descending powers of . Let be the number of variations in signs of the coefficients . For example if , then ; if , then . Descartes’s rule of signs (see ) says that the number of positive real roots of (counted with multiplicities) is equal to either or less than by an even integer.
Let , and be integers. Let
be a polynomial for each , where is a polynomial whose coefficients are positive integers. ( could be a positive constant.)
Suppose that is the leading term of . Then has a unique real root greater than .
Given and , we have
For any real numbers and , we have
(1) Under the assumption on , we have . By Descartes’s rule of signs, the number of positive real roots of is either or . Since and , the number of positive real roots of is exactly . Because goes to as does, has a unique real root .
(2) We have
We define and such that as follows.
We let for . Then
By Maclaurin expansion of , we have
Since goes to as goes to , we may assume that for some constant . Then
(The last inequality comes from for large.) Thus
The first equality above together with and tells us that
Recall that all coefficients of (appeared in ) are positive integers. If we write , where , then
Thus we obtain
For the proof of the claim (1), it is enough to prove that for and , we have and for large.
First, suppose that . Let us consider how the following four terms grow.
The first two terms are appeared in (2.1), and the last two are coming from (2.3). All four terms go to as does, since the last three terms have the positive powers of . Note that for any , we have for large. Keeping in mind of this, we observe that among the four terms in (2.4), is dominant. This is because
for large. These imply that holds for large, since is appeared in the numerator of , see (2.3).
Next, we suppose that . We can check that is still dominant among the four in (2.4). (The second and fourth terms are bounded as goes to .) Therefore we still have for large.
Finally we suppose that . Then the last three terms in (2.4) go to as goes to , because they have the negative powers of for large. Thus the numerator of , see (2.3), goes to as tends to . On the other hand, holds (see (2.2)), and hence the numerator of
goes to as does. Thus for large. This completes the proof of the first part of the claim (2).
Taking the logarithm of the both sides of yields
Since and are any numbers, we have the desired limit. This completes the proof of the second half of the claim (2).
(3) By the claim (2),
Let us set . We substitute for the inequality above:
We multiply all sides above by (for large). Then
Note that goes to as (and hence ) goes to . Since and are any numbers, it follows that
3. Thurston norm and fibered -manifolds
Let be an oriented hyperbolic -manifold with boundary (possibly ). The Thurston norm is defined on an integral class as follows.
where the minimum is taken over all oriented surface embedded in , satisfying , with no components of non-negative Euler characteristic. The surface which realizes this minimum is called the minimal representative of , denoted by . The norm defined on integral classes admits a unique continuous extension which is linear on the ray through the origin. The unit ball with respect to the Thurston norm is a compact, convex polyhedron. See  for more details.
Suppose that is a surface bundle over the circle and let be its fiber. The fibration determines a cohomology class , and hence a homology class by Poincaré duality. Thurston proved in  that there exists a top dimensional face on such that is an integral class of . On the other hand, the minimal representative for any integral class in becomes a fiber of the fibration associated to . Such a face is called a fibered face, and an integral class is called a fibered class.
The set of integral and rational classes of are denoted by and respectively. When is primitive, the associated fibration on has a connected fiber represented by . Let be the monodromy. Since is hyperbolic, is pseudo-Anosov. The dilatation and entropy are defined as the dilatation and entropy of respectively.
The entropy defined on primitive fibered classes is extended to rational classes as follows: For a rational number and a primitive fibered class , the entropy is defined by . It is shown by Fried in  that is concave, and in particular admits a unique continuous extension
Moreover Fried proved that the restriction of to the open fibered face has the property such that goes to as goes to a point on . Thus we have a continuous function
which is constant on each ray in through the origin. Thus the function
has a minimum, denoted by . Matsumoto refined the result by Fried. (See also McMullen.) He proved that is strictly concave. This implies that is achieved by a unique point in . The quantity is a significant invariant on the pairs , but we do not discuss this invariant in the present paper.
Teichmüller polynomial , developed by McMullen organizes the dilatations for all . Once one computes , the largest real root of the polynomial determined by and a given fibered class gives us the dilatation .
4. The magic -manifold
Monodromies of fibrations on have been studied in [9, 10, 11]. (See also a survey .) In Sections 4.1 and 4.2, we recall some results which tell us that the topology of fibered classes and the actual value of . In Section 4.3, we define a family of fibered classes of with two variables and , and we shall prove that it is a suitable family to prove theorems in Section 1 (cf. Remark 4.4).
Recall that is an orientable surface of genus with punctures. Abusing the notation, we sometimes denote by , an orientable surface of genus with boundary components.
4.1. Fibered face
Let , and be the components of the chain link . They bound the oriented disks , and with holes, see Figure 1. Let , , . The set is a basis of . Figure 1 illustrates the Thurston norm ball for which is the parallelepiped with vertices , , , ([18, Example 3 in Section 2]). Because of the symmetry of , every top dimensional face of is a fibered face.
We denote a class by . We pick a fibered face with vertices , , and , see Figure 1. The open face is written by
A class is an element of if and only if , , and . In this case, we have .
Let be a fibered class in . The minimal representative of this class is denoted by or . We recall the formula which tells us that the number of the boundary components of . We denote the tori , , by , , respectively, where be a regular neighborhood of a knot in . Let us set which consists of the parallel simple closed curves on . We define the subsets , in the same manner. By [10, Lemma 3.1], the number of the boundary components is given by
where is defined by .
4.2. Dilatations and the stable foliation of fibered classes
Teichmüller polynomial on the fibered face is computed in [10, Section 3.2], and it tells us that the dilatation of a fibered class is the largest real root of
see [10, Theorem 3.1]. (In fact, is a unique real root greater than of by Descartes’s rule of signs.)
Let be the monodromy of the fibration associated to a primitive class . Let be the stable foliation of the pseudo-Anosov . The components of (resp. , ) are permuted cyclically by . In particular the number of prongs of at a component of (resp. , ) is independent of the choice of the component. By [11, Proposition 3.3], the stable foliation has the property such that:
each component of has prongs,
each component of has prongs, and
each component of has prongs.
does not have singularities in the interior of .
4.3. Proofs of theorems
For and , define a fibered class as follows.
The class is primitive if and only if and are relatively prime. One can check the identity
We fix . Given and , we have
Apply Lemma 2.1 to the polynomial . ∎
Suppose that is primitive. Then the minimal representative