ON HYPERBOLIC SURFACE BUNDLES OVER THE CIRCLE AS BRANCHED DOUBLE COVERS OF THE 3-SPHERE

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1 ON HYPERBOLIC SURFACE BUNDLES OVER THE CIRCLE AS BRANCHED DOUBLE COVERS OF THE 3-SPHERE SUSUMU HIROSE AND EIKO KIN Astract. The ranched virtual fiering theorem y Sakuma states that every closed orientale 3-manifold with a Heegaard surface of genus g has a ranched doule cover which is a genus g surface undle over the circle. It is proved y Brooks that such a surface undle can e chosen to e hyperolic. We prove that the minimal entropy over all hyperolic, genus g surface undles as ranched doule covers of the 3-sphere ehaves like 1/g. We also give an alternative construction of surface undles over the circle in Sakuma s theorem when closed 3-manifolds are ranched doule covers of the 3-sphere ranched over links. A feature of surface undles coming from our construction is that the monodromies can e read off the raids otained from the links as the ranched set. 1. Introduction This paper concerns the ranched virtual fiering theorem y Sakuma which relates Heegaard surfaces to fier surfaces. He proved the theorem in 1981 and it is a ranched version of the virtual fiering theorem y Agol and Wise which states that every hyperolic 3-manifold with finite volume has a finite cover which fiers over the circle. To state Sakuma s theorem let Σ = Σ g,p e an orientale, connected surface of genus g with p punctures possily p = 0, and let us set Σ g = Σ g,0. The mapping class group Mod(Σ) is the group of isotopy classes of orientation preserving selfhomeomorphisms on Σ which preserve the punctures setwise. By Nielsen-Thurston classification, elements in Mod(Σ) fall into three types: periodic, reducile, pseudo- Anosov [9]. To each pseudo-anosov element ϕ, there is an associated dilatation (stretch factor) λ(ϕ) > 1 (see [4] for example). We call the logarithm of the dilatation log(λ(ϕ)) the entropy of ϕ. Choosing a representative f : Σ Σ of ϕ we define the mapping torus T ϕ y T ϕ = Σ R/, where (x, t) (f(x), t + 1) for x Σ, t R. We call Σ the fier surface of T ϕ. The 3-manifold T ϕ is a Σ-undle over the circle with the monodromy ϕ. By Thurston [10] T ϕ admits a hyperolic structure of finite volume if and only if ϕ is pseudo-anosov. The following theorem is due to Sakuma [8, Addendum 1]. See also [3, Section 3]. Date: Feruary 6, Mathematics Suject Classification. Primary 57M27, 37E30, Secondary 37B40. Key words and phrases. pseudo-anosov, dilatation (stretch factor), 2-fold ranched cover of the 3-sphere, fiered 3-manifold, Heegaard surface, mapping class groups, raid groups. 1

2 2 S. HIROSE AND E. KIN 1 i i+1 n (2) skew( ) skew( ) (3) Figure 1. σ i B n. (2) Involution on the cylinder. Thick segment in the middle is the fixed point set of the involution. (3) = skew() is invariant under such an involution. Theorem 1 (Branched virtual fiering theorem). Let M e a closed orientale 3- manifold. Suppose that M admits a genus g Heegaard splitting. Then there is a 2-fold ranched cover M of M which is a Σ g -undle over the circle. It is proved y Brooks [3] that M in Theorem 1 can e chosen to e hyperolic if g max(2, g(m)), where g(m) is the Heegaard genus of M. See also [6] y Montesinos. Let D g (M) e a suset of Mod(Σ g ) consisting of elements ϕ such that T ϕ is homeomorphic to a 2-fold ranched cover of M ranched over some link. By Theorem 1 we have D g (M). By Brooks together with the stailization of Heegaard splittings, there is a pseudo-anosov element in D g (M) for each g max(2, g(m)). The set of fiered 3-manifolds T ϕ over all ϕ D g (M) possesses various properties inherited under ranched covers of M. It is natural to ask aout the dynamics of pseudo-anosov elements in D g (M). We are interested in the set of entropies of pseudo-anosov mapping classes. We fix a surface Σ and consider the set of entropies {log λ(ϕ) ϕ Mod(Σ) is pseudo-anosov} which is a closed, discrete suset of R ([1]). For any suset G Mod(Σ) let δ(g) denote the minimum of dilatations λ(ϕ) over all pseudo-anosov elements ϕ G. Then δ(g) δ(mod(σ)). For real valued functions f and h, we write f h if there is a universal constant c such that h/c f ch. It is proved y Penner [7] that log δ(mod(σ g )) 1 g. A question arises: what can we say aout the asymptotic ehavior of the minimal entropies log δ(d g (M)) s for each closed 3-manifold M? In this paper we consider this question when M is the 3-sphere S 3. Our main theorem is the following. Theorem A. We have log δ(d g (S 3 )) 1 g.

3 ON HYPERBOLIC SURFACE BUNDLES OVER S 1 AS BRANCHED DOUBLE COVERS OF S g 2g+1 Figure 2. Simple closed curves on Σ g. Let q L : M L S 3 denote the 2-fold ranched covering map of S 3 ranched over a link L in S 3. Along the way in the proof of Theorem A we give an alternative proof of Theorem 1 when M = M L in Theorem B. A feature of surface undles M L coming from our construction is that their monodromies can e read off the raids otained from the links as the ranched set. To state Theorem B, we need 3 ingredients. 1. Involution skew : B n B n. Let B n e the (planar) raid group with n strands and let σ i denote the Artin generator of B n as in Figure 1. We define an involution skew : B n B n σ ϵ 1 n 1 σ ϵ 2 n 2 σ ϵ k nk σ ϵ k n nk σ ϵ 2 n n 2 σ ϵ 1 n n 1, ϵ i = ±1. We say that B n is skew-palindromic if skew() =. The raid skew() is skew-palindromic for any B n. (There is a skew-palindromic raid which can not e written y skew() for some, for example σ 1 σ 2 σ 3 B 4.) We write = skew(). Note that skew : B n B n is induced y the involution on the cylinder as shown in Figure 1(2) and skew-palindromic raids are invariant under such an involution. 2. Homomorphism t : B 2g+2 Mod(Σ g ). Let t i denote the right-handed Dehn twist aout the simple closed curve with the numer i in Figure 2. Then there is a homomorphism t : B 2g+2 Mod(Σ g ) which sends σ i to t i for i = 1,..., 2g + 1, since Mod(Σ g ) has the raid relation. (We apply elements of mapping class groups from right to left.) The hyperelliptic mapping class group H(Σ g ) is the sugroup of Mod(Σ g ) consisting of elements with representative homeomorphisms that commute with some fixed hyperelliptic involution. By Birman-Hilden [2], H(Σ g ) is generated y t i s. Thus H(Σ g ) = t(b 2g+2 ). 3. Circular plat closure C(). We use two types of links in S 3 otained from raids. One is the closure cl(β) of β B g+1 as in Figure 3. The other is the circular plat closure C() of B 2g+2 with even strands as in Figure 3(2)(3). We also use the link C() W, the union of C() and the trivial link W = O O with 2 components as shown in Figure 3(4). Any link in S 3 can e represented y C(β ) for some raid β. To see this, it is well-known that L is the closure cl(β) for some β B g+1 (g 1). The desired raid

4 4 S. HIROSE AND E. KIN g+1 g+1 O ' O' (2) (3) (4) Figure 3. cl(β) for β B g+1 and β B 2g+2. (2) C() for B 2g+2. (3) C(σ 3 σ 4 σ 5 ). (4) C() W, where W = O O. β B 2g+2 can e otained from β y adding g + 1 straight strands as in Figure 3. For a raid B 2g+2 let q = q C() : M C() S 3 e the 2-fold ranched covering map of S 3 ranched over C(). There is a (g + 1)-ridge sphere S for the link C() S 3. Hence M C() admits a genus g Heegaard splitting with the Heegaard surface q 1 (S). Then we have the following result. Theorem B. Let M C() e the 2-fold ranched cover of M C() ranched over the link q 1 (W ). Then M C() is homeomorphic to the mapping torus T t( ). Acknowledgments. We would like to thank Makoto Sakuma and Yuya Koda for helpful conversations and comments. The first author was supported y Grantin-Aid for Scientific Research (C) (No. 16K05156), JSPS. The second author was supported y Grant-in-Aid for Scientific Research (C) (No. 18K03299), JSPS. 2. Proof of Theorem B Proof of Theorem B. We construct the 3-sphere S 3 from two copies of the 3-all B 3 y gluing their oundaries together. Consider the link C() W so that C() is contained in one of the 3-alls, and W is given y the union of the four thick segments in the two 3-alls, see Figure 4(2). Let S e the sphere in S 3 which is the union of the two shaded disks in the same figure. The 2-sphere S is a (g + 1)-ridge sphere for C(), and the preimage q 1 (S) is a genus g Heegaard surface of M C(). Let q W : M W S 3 e the 2-fold ranched covering map of S 3 ranched over W (Figure 4). The preimage q 1 W (B3 ) is homeomorphic to the solid torus D 2 S 1. Then M W is otained from two copies of D 2 S 1 y gluing their oundaries together, and hence M W is homeomorphic to S 2 S 1. Oserve that the link q 1 W (C()) is a closure of the spherical raid = skew(), i.e. q 1 W (C()) = cl( ) S 2 S 1. Let p : N cl( ) S 2 S 1 e the 2-fold ranched covering map of S 2 S 1 ranched over cl( ). The 2-fold ranched ( cover of the level ) surface S 2 {u} for u S 1 ranched at the 2g + 2 points in (S 2 {u}) cl( ) is a closed surface of genus g. Thus

5 ON HYPERBOLIC SURFACE BUNDLES OVER S 1 AS BRANCHED DOUBLE COVERS OF S 3 5 T M p _~ ~ t( ~ ) M C() W _~ 2 1 S S gluing two solid tori q W (3) M C() S 3 q =q C() gluing two 3-alls q W 2 1 (2) S 3 S S Figure 4. cl( ) S 2 S 1. (2) C() W S 3. (3) Diagram: M C() M C() is the 2-fold ranched cover of M C() ranched over q 1 (W ). N cl( ) is a Σ g -undle over S 1 with the monodromy t( ), i.e. N cl( ) is homeomorphic to T t( ). Since C() and W are disjoint in S 3, the construction implies that N cl( ) T t( ) is homeomorphic to M C(). This completes the proof. 3. Proof of Theorem A Given a raid we first give a construction of a raid (with more strands than ) such that C() is amient isotopic to C( ). The ottom and top endpoints of a planar raid with n strands are denoted y l 1,..., l n and u 1,..., u n from left to right. For a raid B 2g+2 with even strands, we choose a raid B 2g+3 otained from y adding a strand, say (g + 2) connecting the middle of two points l g+1 and l g+2 with the middle of two points u g+1 and u g+2. Of course is not unique. For example when = 1 = σ 3 B 4, one can choose (= 1 ) = σ2 3 σ 4 B 5. See Figure 5. We consider = skew( ) B 2g+3 with ottom endpoints l 1,..., l 2g+3 and top endpoints u 1,..., u 2g+3. The raid has the strand (g + 2) with endpoints l g+2 and u g+2. If we remove this strand from, then we otain = skew().

6 6 S. HIROSE AND E. KIN u 1 u 2 u 3 u 4 u 5 u 1 u 2 u 3 u 4 u 1 u 2 u 3 u 4 u 5 add a strand l 1 l 2 l 3 l 4 l 1 l 2 l 3 l 4 l 5 (2) l 1 l 2 l 3 l 4 l 5 S {-1} [-1,1] (3) 1 (4) Figure 5. = σ 3. (2) = σ 2 3 σ 4. (3). ( = σ 1 σ 3 is otained from y removing the strand with endpoints l 3 and u 3.) (4) Braid on A. 1 S {1} [-1,1] glue glue S 1 {-1} [-1,1] (2) (3) Figure 6. Solid torus (D 2 S 1 ) 1. (2) S 1 S 1 [ 1, 1] otained from S 1 [ 1, 1] [ 1, 1] y ing two annuli S 1 {1} [ 1, 1] and S 1 { 1} [ 1, 1]. (3) Solid torus (D 2 S 1 ) +1. Now we construct S 2 S 1 from three pieces, two solid tori (D 2 S 1 ) ±1 and the product S 1 S 1 [ 1, 1] of a torus S 1 S 1 and the interval [ 1, 1] y gluing ( D 2 S 1 ) i to S 1 S 1 {i} together for i = ±1. See Figure 6. We think of S 1 as the quotient space [ 1, 1]/( 1 1), and consider the product S 1 S 1 [ 1, 1] = S 1 [ 1, 1]/( 1 1) [ 1, 1]. For the raided link r( ) = cl( ) A in S 3, we perform the 0-surgery along the raided axis A. Then we have a link in S 2 S 1 and we denote this link y cl( ) ausing the notation. We deform this link cl( ) in S 2 S 1 so that the knot cl( (g + 2)) ecomes the core of (D 2 S 1 ) 1 and cl( ) = cl( ) \ cl( (g + 2)) is contained in S 1 S 1 [ 1, 1]. One can regard as a raid on the annulus A := S 1 { 1} [ 1, 1] which is emedded in S 1 [ 1, 1] [ 1, 1], and one can

7 ON HYPERBOLIC SURFACE BUNDLES OVER S 1 AS BRANCHED DOUBLE COVERS OF S 3 7 think of the link cl( ) as the closure of the raid on A. See Figure 5(4). Let R : S 2 S 1 S 2 S 1 e the deck transformation of q W : S 2 S 1 S 3 as in the proof of Theorem B. Then R is an involution and q W sends the fixed point set of R to the trivial link W (Figure 4(2)). Let f : S 1 S 1 S 1 S 1 e any orientation preserving homeomorphism. We may assume that f commutes with the involution We consider the homeomorphism ι : S 1 S 1 S 1 S 1 (x, y) ( x, y). Φ f = f id [ 1,1] : S 1 S 1 [ 1, 1] S 1 S 1 [ 1, 1]. The image of cl( ) under Φ f may or may not e of the closure of some raid on A. We assume the former case ( ): ( ) Φ f (cl( )) = cl(γ) for some raid γ on A. Then R S 1 S 1 [ 1,1] = ι id [ 1,1] and the involution R S 1 S 1 [ 1,1] has a property such that R(cl(β)) = cl(skew(β)) for any raid β on A. cf. Figure 1(2). Since f commutes with ι, it follows that Φ f commutes with R S 1 S 1 [ 1,1]. Hence we have cl(γ) = Φ f (cl( )) = Φ f (cl(skew( ))) = Φ f R(cl( )) = RΦ f (cl( )) = R(cl(γ)). (The first and last equality come from the assumption ( ), and the second equality holds since is skew-palindromic.) Thus cl(γ) = R(cl(γ)). We further assume that ( ) ( Φ f (cl( )) = ) cl(γ) = cl( f ) for some raid f on A. Remark 2. Clearly ( ) implies cl(γ) = R(cl(γ)). It is likely the converse holds. Now, we think of the raid f on A as a planar raid as usual, and consider the link C( f ) in S 3. We have the following lemma. Lemma 3. Under the assumptions ( ) and ( ), C() and C( f ) are amient isotopic. Proof. Note that the quotient ( S 1 S 1 [ 1, 1] ) /R is homeomorphic to S 2 [ 1, 1]. Since Φ f commutes with R S 1 S 1 [ 1,1], Φ f induces a self-homeomorphism Φ f : S 2 [ 1, 1] S 2 [ 1, 1]. Since Φ f (cl( )) = cl( f ) we have Φ f (C()) = C( f ). Any orientation preserving selfhomeomorphism on S 2 is isotopic to the identity, and S 3 is a union of S 2 [ 1, 1] and two 3-alls y gluing the oundaries together. Thus Φ f extends to a selfhomeomorphism on S 3 which sends C() to C( f ). This completes the proof.

8 8 S. HIROSE AND E. KIN Φ f (2) (3) Figure 7. Shaded region indicates annulus A with oundary {v} S 1 {±1}. (3) cl( 1 ) and (4) Φ f (cl( 1 )) = cl( 2 ) contained in S 1 S 1 [ 1, 1]. Let us consider the mapping class group Mod(D n ) of the n-punctured disk D n preserving the oundary D of the disk setwise. We have a surjective homomorphism Γ : B n Mod(D n ) which sends each generator σ i to the right-handed half twist etween the ith and (i + 1)st punctures. We say that β B n is pseudo-anosov if Γ(β) Mod(D n ) is of the pseudo-anosov type. When β is a pseudo-anosov raid, the dilatation λ(β) is defined y the dilatation of λ(γ(β)). We consider the aove homomorphism Γ : B 2g+2 Mod(D 2g+2 ) when n = 2g+2. Recall the homomorphism t : B 2g+2 Mod(Σ g ). The following lemma relates dilatations of β and t(β). Lemma 4. Let β B 2g+2 e pseudo-anosov and let Φ β : D 2g+2 D 2g+2 e a pseudo-anosov homeomorphism which represents Γ(β) Mod(D 2g+2 ). Suppose that the stale foliation F β for Φ β defined on D 2g+2 is not 1-pronged at the oundary D of the disk. Then t(β) Mod(Σ g ) is pseudo-anosov, and λ(t(β)) = λ(β) holds. Proof. Since F β is not 1-pronged at the oundary of the disk, Φ β : D 2g+2 D 2g+2 induces a pseudo-anosov homeomorphism Φ β : Σ 0,2g+2 Σ 0,2g+2 y filling D with a disk. By Birman-Hilden [2], we have a surjective homomorphism q : H(Σ g ) Mod(Σ 0,2g+2 ) sending the Dehn twist t i to the right-handed half twist h i etween the ith and (i + 1)st punctures for i = 1,, 2g + 1. Consider the 2-fold ranched cover Σ g Σ 0,2g+2 ranched at the 2g + 2 marked points (corresponding to the punctures of Σ 0,2g+2 ). Then there is a lift f β : Σ g Σ g of Φ β : Σ 0,2g+2 Σ 0,2g+2 which satisfies t(β) = [f β ] H(Σ g ). The stale foliation for the pseudo-anosov Φ β defined on Σ 0,2g+2 is lifted to the stale foliation for f β defined on Σ g. Thus f β is a pseudo- Anosov homeomorphism which represents t(β) = [f β ], and we have This completes the proof. λ([f β ]) = λ([φ β ]) = λ([φ β]) = λ(β).

9 ON HYPERBOLIC SURFACE BUNDLES OVER S 1 AS BRANCHED DOUBLE COVERS OF S g g+2 (2) Figure 8. g B 2g+2. (2) g B 2g+2. Proof of Theorem A. For g 1, we consider g = σ 3 σ 4 σ 2g+1 B 2g+2 and g = σ 1 σ 2 σ 2g 1 σ 3 σ 4 σ 2g+1 B 2g+2, see Figure 8. By Penner s result it is enough to prove that t( g ) is a pseudo-anosov element in D g (S 3 ) for large g and log λ(t( g )) 1/g holds. Applying Theorem B for the raid g we have the 2-fold ranched cover T t( g) M C( g ) ranched over q 1 (W ). We first prove that M C(g) S 3 for g 1. Clearly C( 1 ) = C(σ 3 ) is a trivial knot. The 2-fold ranched cover of S 3 ranched over a trivial knot is S 3, and hence M C(1 ) S 3. We add a strand to 1 = σ 3 B 4 so that 1 = σ2 3 σ 4 B 5, and think of 1 as a raid on A, see Figure 5. Choose any v S1 and consider the annulus A in S 1 S 1 [ 1, 1] with oundary {v} S 1 {±1}, see Figure 7. As a self-homeomorphism f on S 1 S 1, we take a Dehn twist aout {v} S 1. Then the self-homeomorphism Φ f on S 1 S 1 [ 1, 1] is an annulus twist aout A, see Figure 7(2)(3). Oserve that Φ f (cl( 1 )) = cl( 2 ) satisfying assumptions ( ) and ( ). By repeating this process it is not hard to see that Φ f (cl( j 1 )) = cl( j ) for each j 2. Thus Φ f (cl( j 1 )) satisfies ( ) and ( ) for each j. Lemma 3 tells us that C( g ) is a trivial knot for g 1 since so is C( 1 ). Thus M C(g ) S 3, and t( g ) D g (S 3 ) for g 1 y Theorem B. The proof of Theorem D in [5] says that Γ( g ) Mod(D 2g+2 ) is pseudo-anosov for g 2 and log λ( g ) 1 holds. Moreover the stale foliation of the pseudog Anosov reprerentative for Γ( g ) satisfies the assumption of Lemma 4, see the proof of Step 2 in [5, Theorem D]. Thus t( g ) is pseudo-anosov with λ(t( g )) = λ( g ) y Lemma 4, and we otain the desired claim log λ(t( g )) = log λ( g ) 1 g. This completes the proof. We end this paper with a question.

10 10 S. HIROSE AND E. KIN Question 5. Let M e a closed 3-manifold M which is the 2-fold ranched cover of S 3 ranched over some link. Then does it hold log δ(d g (M)) 1 g? References [1] P. Arnoux and J-P. Yoccoz, Construction de difféomorphismes pseudo-anosov, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, [2] J. Birman and H. Hilden, On mapping class groups of closed surfaces as cover spaces, Advances in the theory of Riemann surfaces, Annals of Math Studies 66, Princeton University Press (1971), [3] R. Brooks, On ranched covers of 3-manifolds which fier over the circle, J. Reine Angew. Math. 362 (1985), [4] B. Far and D. Margalit, A primer on mapping class groups, Princeton Mathematical Series 49, Princeton University Press, Princeton, NJ (2012). [5] S. Hirose and E. Kin, A construction of pseudo-anosov raids with small normalized entropies, preprint (2018). [6] J. M. Montesinos, On 3-manifolds having surface undles as ranched covers, Proc. Amer. Math. Soc. 101 (1987), no. 3, [7] R. C. Penner, Bounds on least dilatations, Proceedings of the American Mathematical Society 113 (1991), [8] M. Sakuma, Surface undles over S 1 which are 2-fold ranched cyclic covers of S 3, Math. Sem. Notes, Koe Univ., 9 (1981) [9] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, [10] W. Thurston, Hyperolic structures on 3-manifolds II: Surface groups and 3-manifolds which fier over the circle, preprint, arxiv:math/ Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chia, , Japan address: hirose susumu@ma.noda.tus.ac.jp Department of Mathematics, Graduate School of Science, Osaka University Toyonaka, Osaka , JAPAN address: kin@math.sci.osaka-u.ac.jp