A construction of pseudo-Anosov braids with small normalized entropies
Let be a pseudo-Anosov braid whose permutation has a fixed point and let be the mapping torus by the pseudo-Anosov homeomorphism defined on the genus fiber associated with . We prove that there is a -dimensional subcone contained in the fibered cone of such that the fiber for each primitive integral class has genus . We also give a constructive description of the monodromy of the fibration on over the circle, and consequently provide a construction of many sequences of pseudo-Anosov braids with small normalized entropies. As an application we prove that the smallest entropy among skew-palindromic braids with strands is comparable to , and the smallest entropy among elements of the odd/even spin mapping class groups of genus is comparable to .
Key words and phrases:mapping class groups, pseudo-Anosov, dilatation, normalized entropy, fibered -manifolds, braid group
2010 Mathematics Subject Classification:57M99, 37E30
Let be an orientable surface of genus with punctures for . We set . By mapping class group , we mean the group of isotopy classes of orientation preserving self-homeomorphisms on preserving punctures setwise. By Nielsen-Thurston classification, elements in are classified into three types: periodic, reducible, pseudo-Anosov [29, 9]. For we choose a representative and consider the mapping torus , where identifies with for and . Then is a fiber of a fibration on over the circle and is called the monodromy. A theorem by Thurston  asserts that admits a hyperbolic structure of finite volume if and only if is pseudo-Anosov.
For a pseudo-Anosov element there is a representative of called a pseudo-Anosov homeomorphism with the following property: admits a pair of transverse measured foliations and and a constant depending on such that and are invariant under , and and are uniformly multiplied by and under . The constant is called the dilatation and and are called the unstable and stable foliation. We call the logarithm the entropy, and call
Penner  proved that if is pseudo-Anosov, then
See also . For a fixed surface , the set is a closed, discrete subset of (). For any subgroup or subset let denote the minimum of over all pseudo-Anosov elements . Then . We write if there is a universal constant such that . It is proved by Penner  that the minimal entropy among pseudo-Anosov elements in on the closed surface of genus satisfies
For any , consider the set consisting of all pseudo-Anosov homeomorphisms defined on any surface with the normalized entropy . This is an infinite set in general (take for example) and is well-understood in the context of hyperbolic fibered -manifolds. The universal finiteness theorem by Farb-Leininger-Margalit  states that the set of homeomorphism classes of mapping tori of pseudo-Anosov homeomprhisms is finite, where is the fully punctured pseudo-Anosov homeomprhism obtained from . (Clearly .) In other words such lives in some fibered cone for a -manifold in the finite list determined by . Thus -manifolds in the finite list govern all pseudo-Anosov elements in . It is natural to ask the dynamics and a constructive description of elements in . There are some results about this question by several authors [4, 15, 20, 22, 32], but it is not completely understood. In this paper we restrict our attention to the pseudo-Anosov elements in defined on the genus surfaces, and provide an approach for a concrete description of those elements.
Let be the braid group with strands. The group is generated by the braids as in Figure 1. Let be the symmetric group, the group of bijections of to itself. A permutation has a fixed point if for some . We have a surjective homomorphism which sends each to the transposition .
The closure of a braid is a knot or link in the -sphere . The braided link
is a link in obtained from with its braid axis (Figure 2). Let denote the exterior of which is a -manifold with boundary. It is easy to find an -holed sphere in (Figure 2(3)). Clearly is a fiber of a fibration on and its monodromy is determined by . We call the -surface for .
A braid is periodic (resp. reducible, pseudo-Anosov) if the associated mapping class is of the corresponding type (Section 2.3). If is pseudo-Anosov, then the dilatation is defined by and the normalized entropy is defined by . The following theorem is due to Hironaka-Kin [16, Proposition 3.36] together with the observation by Kin-Takasawa [22, Section 4.1].
There is a sequence of pseudo-Anosov braids such that , for each and as .
In particular admits an infinitely family of genus fibers of fibrations over .
Let be a pseudo-Anosov braid with strands. We say that a sequence has a small normalized entropy if and there is a constant which does not depend on such that . By (1.1) a sequence having a small normalized entropy means . One of the aims in this paper is to give a construction of many sequences of pseudo-Anosov braids with small normalized entropies. The following result generalizes Thereom 1.1.
Suppose that is a pseudo-Anosov braid whose permutation has a fixed point. There is a sequence of pseudo-Anosov braids with small normalized entropy such that as and for .
The proof of Theorem A is constructive. In fact one can describe braids explicitly. For a more general result see Theorems 5.1, 5.2. Let be the fibered cone containing . A theorem by Thurston  states that for each primitive integral class there is a connected fiber with the pseudo-Anosov monodromy of a fibration on over . The following theorem states a structure of .
Suppose that is a pseudo-Anosov braid whose permutation has a fixed point. Then there are a -dimensional subcone and an integer with the following properties.
The fiber for each primitive integral class has genus .
The monodromy for each primitive integral class is conjugate to
where depends on the class , is periodic and each is reducible. Moreover there are homeomorphisms on a surface for determined by and an embedding such that is the support of each and
Theorem B gives a constructive description of . Also it states that each is reducible supported on a uniformly bounded subsurface . It turns out from the proof that the type of the periodic homeomorphism does not depend on (Remark 3.3), see Figure 3(1).
We have a remark about Theorem A. Theorem 10.2 in  by McMullen also tells us the existence of a sequence of fibers and monodromies in such that as and . However one can not appeal his theorem for the genera of fibers . Theorem A says that has genus in fact.
As an application we will determine asymptotic behaviors of the minimal dilatations of a subsets of consisting of braids with a symmetry. A braid is palindromic if , where is a map such that if is a word of letters representing , then is the braid obtained from reversing the order of letters in . A braid is skew-palindromic if , where and is a half twist (Section 2.2). See Figure 4. We will prove that dilatations of palindromic braids have the following lower bound.
If is palindromic and pseudo-Anosov for , then
In contrast with palindromic braids we have the following result.
Let be the set of skew-palindromic elements in . We have
The hyperelliptic mapping class group is the subgroup of consisting of elements with representative homeomorphisms that commute with some fixed hyperelliptic involution as in Figure 5(1). It is shown in  that . See also [7, 15, 19] for other subgroups of . As an application we will determine the asymptotic behavior of the minimal dilatations of the odd/even spin mapping class groups of genus . To define these subgroups let be the mod- intersection form on . A map is a quadratic form if for . For a quadratic form , the spin mapping class group is the subgroup of consisting of elements such that . To define the two quadratic forms and we choose a basis of as in Figure 5(2). Let be the quadratic form such that for . Let be the quadratic form such that and for . A result of Dye  tells us that for any is conjugate to either or in . We call and the even spin and odd spin mapping class group respectively. It is known that attains the minimum index for a proper subgroup of and attains the secondary minimum, see Berrick-Gebhardt-Paris .
In particular for each quadratic form .
Acknowledgments. We would like to thank Mitsuhiko Takasawa for helpful conversations and comments.
Let be a link in the -sphere . Let denote a tubular neighborhood of and let denote the exterior of , i.e. .
Oriented links and in are equivalent, denoted by if there is an orientation preserving homeomorphism such that with respect to the orientations of the links. Furthermore for components of and of with if satisfies for each , then and are equivalent and we write
2.2. Braid groups and spherical braid groups
Let and . The half twist is given by . We often omit the subscript in , and when they are precisely -braids.
We put indices from left to right on the bottoms of strands, and give an orientation of strands from the bottom to the top (Figure 1). The closure is oriented by the strands. We think of as an oriented link in choosing an orientation of arbitrarily. (In Section 3 we assign an orientation of the braid axis for -monotonic braids).
If two braids are conjugate to each other, then their braided links are equivalent. Morton proved that the converse holds if their axises are preserved.
Theorem 2.1 (Morton ).
If is equivalent to for braids , then and are conjugate in .
Let us turn to the spherical braid group with strands. We also denote by , the element of as shown in Figure 1(1). The group is generated by . For a braid represented by a word of letters , let denote the element in represented by the same word as .
For a braid in or the degree of means the number of the strands, denoted by .
2.3. Mapping classes and mapping tori from braids
Let be the -punctured disk. Consider the mapping class group , the group of isotopy classes of orientation preserving self-homeomorphisms on preserving the boundary of the disk setwise. We have a surjective homomorphism
which sends each generator to the right-handed half twist between the th and st punctures. The kernel of is an infinite cyclic group generated by the full twist .
Collapsing to a puncture in the sphere we have a homomorphism
We say that is periodic (resp. reducible, pseudo-Anosov) if is of the corresponding Nielsen-Thurston type. The braids are periodic since some power of each braid is the full twist: .
We also have a surjective homomorphism
sending each generator to the right-handed half twist . We say that is pseudo-Anosov if is pseudo-Anosov. In this case is defined by the dilatation of .
2.4. Stable foliations for pseudo-Anosov braids
Recall the surjective homomorphism . We write for . Consider a pseudo-Anosov braid with . Removing the th strand from , we get a braid . Taking its spherical element, we have . Note that and are not necessarily pseudo-Anosov. A well-known criterion uses the stable foliation for the monodromy of a fibration on as we recall now. Such a fibration on extends naturally to a fibration on the manifold obtained from by Dehn filling a cusp along the boundary slope of the fiber which lies on the torus . Also extends to the monodromy defined on of the extended fibration, where is obtained from by filling in the boundary component of which lies on with a disk. Then is the corresponding braid for the extended monodromy defined on . Suppose that is not -pronged at the boundary component in question. (See Figure 6 in the case where is -pronged at a boundary component.) Then extends to the stable foliation for , and hence is pseudo-Anosov with the same dilatation as . Furthermore if is not -pronged at the boundary component of which lies on , then is still pseudo-Anosov with the same dilatation as .
2.5. Thurston norm
Let be a -manifold with boundary (possibly ). If is hyperbolic, i.e. the interior of possess a complete hyperbolic structure of finite volume, then there is a norm on , now called the Thurston norm . The norm has the property such that for any integral class , , where the minimum is taken over all oriented surface embedded in with and with no components of non-negative Euler characteristic. The surface realizing this minimum is called a norm-minimizing surface of .
Theorem 2.2 (Thurston ).
The norm on has the following properties.
There are a set of maximal open cones in and a bijection between the set of isotopy classes of connected fibers of fibrations and the set of primitive integral classes in the union .
The restriction of to is linear for each .
If we let be a fiber of a fibration associated with a primitive integral class in each , then .
We call the open cones fibered cones and call integral classes in fibered classes.
Theorem 2.3 (Fried ).
For a fibered cone of a hyperbolic -manifold , there is a continuous function with the following properties.
For the monodromy of a fibration associated with a primitive integral class , we have .
is a continuous function which becomes constant on each ray through the origin.
If a sequence tends to a point in the boundary as tends to , then . In particular .
We call and the entropy and normalized entropy of the class .
For a pseudo-Anosov element we consider the mapping torus . The vector field on induces a flow on called the suspension flow.
Theorem 2.4 (Fried ).
Let be a pseudo-Anosov mapping class defined on with stable and unstable foliations and . Let and denote the suspensions of and by . If is a fibered cone containing the fibered class , then we can modify a norm-minimizing surface associated with each primitive integral class by an isotopy on with the following properties.
is transverse to the suspension flow , and the first return map is precisely the pseudo-Anosov monodromy of the fibration on associated with . Moreover is unique up to isotopy along flow lines.
The stable and unstable foliations for are given by and .
2.6. Disk twist
Let be a link in . Suppose an unknot is a component of . Then the exterior (resp. ) is a solid torus (resp. torus). We take a disk bounded by the longitude of a tubular neighborhood of . We define a mapping class defined on as follows. We cut along . We have resulting two sides obtained from , and reglue two sides by twisting either of the sides degrees so that the mapping class defined on is the right-handed Dehn twist about . Such a mapping class on is called the disk twist about . For simplicity we also call a self-homeomorphism representing the mapping class the disk twist about , and denote it by the same notation
Clearly equals the identity map outside a neighborhood of in . We observe that if segments of pass through for , then is obtained from by adding the full twist near . In the case , see Figure 7. We may assume that fixes one of these segments, since any point in becomes the center of the twisting about .
For any integer , consider a homeomorphism
Observe that converts into a link such that is homeomorphic to . Then induces a homeomorphism between the exteriors of links
We use the homeomorphism in (2.1) in later section.
3. -increasing braids and Theorem 3.2
Definitions of -increasing braids, signs and intersection numbers
Let be an oriented link in with a trivial component . We take an oriented disk bounded by the longitude of so that the orientation of agrees with the orientation of . For each component of such that and intersect transversally with , we assign each point of intersection or as shown in Figure 8.
Let be a braid with . We consider an oriented disk bounded by the longitude of . Such a disk is unique up to isotopy on . We say that a braid with is -increasing (resp. -decreasing) if there is a disk as above with the following conditions.
There is at least one component of such that .
Each component of and intersect with each other transversally, and every point of intersection has the sign (resp. ).
We set (resp. ), and call it the sign of the pair . We also call the associated disk of the pair . We say that is -monotonic if is -increasing or -decreasing. Then we set
and let be the cardinality of . We call the intersection number of the pair . If the pair is specified, then we simply denote and by and respectively. For example is -increasing with .
A braid is positive if is represented by a word in letters , but not . A braid is irreducible if the Nielsen-Thurston type of is not reducible.
Let be a positive braid with . Then is -increasing if is irreducible.
Suppose that a positive braid with is irreducible. Since is positive, there is a disk with the condition . Assume that fails in . Let be the boundary of the disk containing punctures. Consider a neighborhood of in which is an annulus. One of the boundary components of this annulus is an essential simple closed curve in preserved by . This means that is reducible, a contradiction. Thus satisfies , and is -increasing. ∎
Orientation of the axis
Let be -monotonic with and . Consider the braided link . The associated disk has a unique point of intersection with , and the cardinality of is . To deal with as an oriented link, we consider an orientation of as we described before, and assign an orientation of so that the sign of the intersection between and coincides with . See Figure 2(2).
Recall that is the exterior of which is a surface bundle over . We consider an orientation of the -surface which agrees with the orientation of .
We now define an oriented surface of genus embedded in . Consider small disks in the oriented disk whose centers are points of . Then is a sphere with boundary components obtained from by removing the interiors of those small disks. We choose the orientation of so that it agrees with the orientation of . We call the -surface for . For example, the -increasing braid has the -surface homeomorphic to a -holed sphere.
Let us consider the -dimensional subcone of spanned by and (Figure 9):
Let denote the closure of . We write . We prove the following theorem in Section 4.
For a pseudo-Anosov, -increasing braid with , let be the fibered cone containing . We have the following.
The fiber for each primitive integral class has genus .
The monodromy for each primitive integral class is conjugate to
where depends on , is periodic and each is reducible. Moreover there are homeomorphisms for on a surface determined by and an embedding such that is the support of each and
Proof of Theorem B.
Suppose that Theorem 3.2 holds. Let be a pseudo-Anosov braid such that . We consider the braid for . The full twist is an element in the center and holds for each , where is positive. Such properties imply that is positive for large. We fix such large . Since in , the braid is certainly pseudo-Anosov. Hence it is -increasing by Lemma 3.1. One can apply Theorem 3.2 for this braid, and obtains the subcone . Consider the th power of the disk twist about the disk bounded by the longitude of :
Since , we have . Let us set
where is the homeomorphism in (2.1). The isomorphism
sends to . (Here we note that the above is suppose to be large, but the homeomorphism makes sense for all integer .) The pullback of the subcone into is a desired subcone contained in . ∎
4. Proof of Theorem 3.2
We fix integers and . Throughout Section 4, we assume that is pseudo-Anosov and -increasing with . We now choose an associated disk about the pair suitably. Let denote the unit disk with the center in the plane . Let be the interval and let be a point in . We denote by , the disk with equally spaced points in . Let us denote these points by from left to right. We take a point between and so that the Euclidean distance is sufficiently small (e.g. ). Let denote the closed interval in with endpoints and . (Figure 10(1).) We regard as a braid contained in the cylinder and is based at points . Since , one can take a representative of such that is an interval in the cylinder:
Furthermore we may assume that of an associated disk of is a union of the following four segments as a set (Figure 10):
For a regular neighborhood of in , we have .
This is because every point , where is a component of , one can slide along so that the resulting point on is in . Said differently, preserving pointwise, we can modify a small neighborhood of near so that the resulting associated disk satisfies .
Under the conditions we have the following. For each