On hyperbolic surface bundles over the circle as branched double covers of the -sphere
The branched virtual fibering theorem by Sakuma states that every closed orientable -manifold with a Heegaard surface of genus has a branched double cover which is a genus surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen to be hyperbolic. We prove that the minimal entropy over all hyperbolic, genus surface bundles as branched double covers of the -sphere behaves like 1/. We also give an alternative construction of surface bundles over the circle in Sakuma’s theorem when closed -manifolds are branched double covers of the -sphere branched over links. A feature of surface bundles coming from our construction is that the monodromies can be read off the braids obtained from the links as the branched set.
Key words and phrases:pseudo-Anosov, dilatation, stretch factor, entropy, -fold branched cover of the -sphere, fibered -manifold, fiber surface, Heegaard surface, branched virtual fibering theorem
2000 Mathematics Subject Classification:Primary 57M27, 37E30, Secondary 37B40
This paper concerns the branched virtual fibering theorem by Sakuma which relates Heegaard surfaces to fiber surfaces. He proved the theorem in 1981 and it is a branched version of the virtual fibering theorem by Agol and Wise which states that every hyperbolic -manifold with finite volume has a finite cover which fibers over the circle.
To state Sakuma’s theorem let be an orientable, connected surface of genus with punctures possibly , and let us set . The mapping class group is the group of isotopy classes of orientation preserving self-homeomorphisms on which preserve the punctures setwise. By Nielsen-Thurston classification, elements in fall into three types: periodic, reducible, pseudo-Anosov . To each pseudo-Anosov element , there is an associated dilatation (stretch factor) (see  for example). We call the logarithm of the dilatation the entropy of .
Choosing a representative of we define the mapping torus by
where for , . We call the fiber surface of . The -manifold is a -bundle over the circle with the monodromy . By Thurston  admits a hyperbolic structure of finite volume if and only if is pseudo-Anosov.
Theorem 1 (Branched virtual fibering theorem).
Let be a closed orientable -manifold and let be a genus Heegaard surface of . Then there is a -fold branched cover of which is a -bundle over the circle.
Let be a subset of consisting of elements such that is homeomorphic to a -fold branched cover of branched over some link. By Theorem 1 we have . By Brooks together with the stabilization of Heegaard splittings, there is a pseudo-Anosov element in for each . The set of fibered -manifolds over all possesses various properties inherited under branched covers of . It is natural to ask about the dynamics of pseudo-Anosov elements in . We are interested in the set of entropies of pseudo-Anosov mapping classes.
We fix a surface and consider the set of entropies
which is a closed, discrete subset of (). For any subset let denote the minimum of dilatations over all pseudo-Anosov elements . Then . For real valued functions and , we write if there is a universal constant such that . It is proved by Penner  that
A question arises: what can we say about the asymptotic behavior of the minimal entropies ’s for each closed -manifold ? In this paper we consider this question when is the -sphere . We prove that the asymptotic behavior of the minimal entropies ’s is the same as that of the ambient mapping class group.
We have .
Let denote the -fold branched covering map of branched over a link in . Along the way in the proof of Theorem A we give an alternative proof of Theorem 1 when -manifolds are of the form . A feature of surface bundles coming from our construction is that their monodromies can be read off the braids obtained from the links as the branched set. More precisely let be the braid group with strands and let denote an element of as in Figure 1(1). We define an involution
We say that is skew-palindromic if . Notice that is skew-palindromic for any . (There is a skew-palindromic braid which can not be written by for some , for example .) We write
We may assume that braids are contained in the cylinder. Then is induced by the involution on the cylinder as shown in Figure 1(2) and skew-palindromic braids are invariant under such an involution. See Figure 1(3).
Let denote the right-handed Dehn twist about the simple closed curve with the number in Figure 2. Then there is a homomorphism
which sends to for , since has the braid relation. (We apply elements of mapping class groups from right to left.) The hyperelliptic mapping class group is the subgroup of consisting of elements with representative homeomorphisms that commute with some fixed hyperelliptic involution. By Birman-Hilden , the subgroup is generated by ’s (), and hence we have .
The link obtained from and the trivial link with components is shown in Figure 3(4). We have the following result.
For a braid let be the -fold branched covering map of branched over . Let be the -fold branched cover of branched over the link . Then is a -bundle over the circle with the monodromy , i.e. is homeomorphic to .
Acknowledgments. We would like to thank Makoto Sakuma and Yuya Koda for helpful conversations and comments. The first author was supported by Grant-in-Aid for Scientific Research (C) (No. 16K05156), JSPS. The second author was supported by Grant-in-Aid for Scientific Research (C) (No. 18K03299), JSPS.
We now claim that a link in is ambient isotopic to for some braid . Alexander’s theorem says that is the closure for some . The desired braid can be obtained from by adding straight strands as in Figure 3(1).
Let () be a -dimensional handlebody of genus , and be an orientation reversing homeomorphism. We consider a closed -manifold obtained from the handlebodies by identifying the boundaries via . Then is called a genus Heegaard splitting, and is called a Heegaard surface of .
Proof of Theorem B.
We construct the -sphere from two copies of the -ball by gluing their boundaries together. Consider the link so that is contained in one of the -balls, and is given by the union of the four thick segments in the two -balls, see Figure 4(2). Let be the equator of which is the union of the two shaded disks in the same figure. The -sphere is a genus Heegaard surface of , and the preimage is a genus Heegaard surface of since .
Let be the -fold branched covering map of branched over (Figure 4(1)). The preimage of each -ball is homeomorphic to the solid torus . Then is obtained from two copies of by gluing their boundaries in an obvious way, and hence is homeomorphic to . Observe that the link is contained in one of the solid tori, and it is of the closure of the skew-palindromic braid , i.e.
Let be the -fold branched covering map of branched over . The -fold branched cover of for branched at the set of points is a closed surface of genus . Thus is a -bundle over . The monodromy of the -bundle in question is given by . Since and are disjoint in , the construction implies that the -bundle coincides with up to homeomorphism. See Figure 4(3). This completes the proof. ∎
Given a braid we give a construction of a braid (with more strands than ) such that is ambient isotopic to . The bottom and top endpoints of a braid with strands are denoted by and from left to right. For a braid we choose a braid obtained from by adding a strand connecting the central bottom endpoint with the central top endpoint . Of course is not unique. For example if , then one can choose . Clearly has the strand connecting with , and if we remove this strand from , then we obtain .
Now we construct (in the proof of Theorem B) from three pieces, two copies of a solid torus and a copy of by gluing to in an obvious way (). We put into so that the knot becomes the the core of and satisfies . See Figure 5(3). Let
be any orientation preserving homeomorphism. We consider the homeomorphism
The image of under may or may not be of the closure of some braid contained in . We assume that the former occurs:
is the closure of some braid , i.e. .
be an involution. Then and the involution has a property such that
See also Figure 1(2). Since any self-homeomorphism on commutes with , it follows that commutes with . Hence we have
(The first and last equality come from the assumption , and the second equality holds since is skew-palindromic.) Then and this means that is invariant under the involution on the cylinder as in Figure 1(2) and hence is skew-palindromic. We further assume that is of the form for some braid , i.e.
for some braid .
Then we have the following lemma.
and are ambient isotopic.
Note that the quotient is homeomorphic to . Since commutes with , induces a self-homeomorphism
Since we have . Any orientation preserving self-homeomorphism on is isotopic to the identity, and is a union of and two -balls by gluing the boundaries together. Thus extends to a self-homeomorphism on which sends to . This completes the proof. ∎
Proof of Theorem A.
Consider . Then
By Penner’s result it is enough to prove that is a pseudo-Anosov element of for large and its entropy behaves like , namely .
Applying Theorem B for we have the -fold branched cover of which is homeomorphic to . We first prove that for . Clearly is a trivial knot and hence . We add a strand to so that . Figure 5(2) illustrates under gluing top and bottom annuli together. Choose any and consider another annulus in with boundary , see the shaded region in Figure 5(2). As a self-homeomorphism on , we take a Dehn twist about . Then the self-homeomorphism on is an annulus twist about , see Figure 5(3)(4). Observe that . By repeating this process we have
be the homomorphism sending to the right-handed half twist between the th and st punctures. By Theorem D in  together with the proof (see Step 2 in the proof of Theorem D), one sees that is pseudo-Anosov for and its entropy behaves like . By Birman-Hilden , we have a surjective homomorphism sending to for . Then the identity holds. The construction of pseudo-Anosov mapping classes from using branched covers (see [4, Section 14.1.1] for example) tells us that is also pseudo-Anosov with the same entropy as . This completes the proof. ∎
We end this paper with a question.
Let be a closed -manifold which is the -fold branched cover of branched over some link. Then does it hold ?
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