Limits of sequences of pseudo-Anosov maps and of hyperbolic 3-manifolds
There are two objects naturally associated with a braid of pseudo-Anosov type: a (relative) pseudo-Anosov homeomorphism ; and the finite volume complete hyperbolic structure on the 3-manifold obtained by excising the braid closure of , together with its braid axis, from . We show the disconnect between these objects, by exhibiting a family of braids with the properties that: on the one hand, there is a fixed homeomorphism to which the (suitably normalized) homeomorphisms converge as ; while on the other hand, there are infinitely many distinct hyperbolic 3-manifolds which arise as geometric limits of the form , for sequences .
This article presents a somewhat surprising phenomenon on the interface between the theories of surface homeomorphisms and of 3-manifold geometry. Two theorems due to Thurston associate to certain mapping classes on a surface — the pseudo-Anosov mapping classes — two different types of canonical objects.
In this paper we consider mapping classes of marked spheres, represented by elements of Artin’s braid groups: an -braid defines a mapping class on the -marked disk, and hence on the -marked sphere. We say that is of pseudo-Anosov type if and only if the corresponding mapping class is, and in this case we can associate to it:
a homeomorphism , unique up to conjugacy, which is pseudo-Anosov relative to the marked points (that is, whose invariant foliations are permitted to have -pronged singularities at these points); and
the hyperbolic -manifold111All of the -manifolds in this paper are of the form for some pseudo-Anosov braid , and we consider them as hyperbolic -manifolds without further comment. — where is the closure of and is its braid axis — which is homeomorphic to the mapping torus of (acting on the sphere punctured at the marked points).
We will present a family of pseudo-Anosov braids , with , with the following properties:
The pseudo-Anosov homeomorphisms can be normalized in such a way that as , where is a fixed sphere homeomorphism (the tight horseshoe map, derived from Smale’s horseshoe map).
The hyperbolic -manifolds have the property that there are infinitely many distinct finite volume hyperbolic -manifolds which can be obtained as geometric limits for some sequence .
The braids are the NBT braids of : they are pseudo-Anosov braids for which the corresponding pseudo-Anosov homeomorphisms have particularly simple train tracks (see Remark 5). The fact that as is a straightforward consequence of results of : the main content of this paper is an analysis of possible geometric limits of sequences .
The principal technique which we will use is Dehn surgery, and we now briefly recap some key definitions and results, in order to fix conventions (which are taken from section 9 of Rolfsen’s book ). Let be a link in with components , and let be a closed tubular neighborhood of which is disjoint from the other components of . Pick a basis for such that the ‘meridian’ is contractible in and the ‘longitude’ has linking number with .
If is a homotopically non-trivial simple closed curve in , then we can construct a -manifold
where is a homeomorphism which takes onto . Writing , we say that is obtained from by Dehn filling with surgery coefficient : this definition is independent of the choices of orientations of , and . (This corresponds to Dehn filling coefficient in the notation used by SnapPy , where the coefficients and lead to the same surgery. We will always assume that and are coprime.) In particular, setting if and only if , i.e., and , then : filling with surgery coefficient is the same as erasing the component from the link .
Suppose now that we have assigned surgery coefficients to some of the components of , and that is an unknotted component of . Applying a positive meridional twist to the (solid torus) complement of a tubular neighborhood of is referred to as performing a twist on : if is a disk bounded by which the other components of intersect transversely, then the effect of this twist on the link is to replace each segment of which intersects with a helix which screws through a collar of in the right-handed sense. If , then performing a twist on means performing such twists if , or left-handed twists if .
The revised link after a twist on describes the same -manifold as provided that the surgery coefficients (on those components of which have them) are updated using the formulæ:
where and are the surgery coefficients on before and after the twist, and is the linking number of with .
In this paper, we will only perform twists in the case where is the closure of a braid together with its axis; and we will only perform them on either the braid axis or a fixed component of (one which corresponds to a single string of the braid). It will therefore be convenient to describe the effects of such twists directly on the braid.
A -twist on the braid axis replaces with , where is a full twist in the braid group.
Figure 1 (a) is a schematic representation of , where has a fixed string which links one of the other strings. The effect of a twist on the corresponding component of is shown in (b), which is followed by a conjugacy to obtain the braid of (c). Because this braid has the same structure as , the process can be repeated more times to obtain the braid of (d), which is the effect of applying a twist on the fixed string. It has more strands than .
We shall also consider twists on fixed strings which link a ribbon of other parallel strings of the braid. Figure 2 shows the effect of a twist in this case, determined analogously. If the ribbon consists of strings, then this increases the number of strings of by .
In order to carry out a twist on a fixed string, we will conjugate to take the form of the right hand side of Figure 2. The twist will then reduce it to the braid on the left hand side.
Let be a link in such that is a complete hyperbolic 3-manifold of finite volume, be a sequence of rationals with , and be the sequence of 3-manifolds obtained by Dehn filling with surgery coefficients . Then converges geometrically to , and the convergence is non-trivial in the sense that and are distinct for all , so that there are infinitely many distinct 3-manifolds .
2. The braids
Recall that the positive permutation braid defined by a permutation is the unique -braid which induces the permutation on its strings, and which has the properties that every pair of strings crosses at most once, and that every crossing is in the positive sense (we adopt the convention, following Birman , that a braid crossing is positive if the left string crosses over the right one). Thus a diagram of can be constructed by drawing the first to the strings in order, with the string going from position to position and passing underneath any intervening strings which have already been drawn.
The following definition is from theorem 2.1 of , and the fact that the braids defined are of pseudo-Anosov type is contained in the proof of theorem 2.3 of the same paper. (There the braids are also defined for , but this is done in a different way and, since we are only interested in limits as , is not relevant here.) Here and throughout the paper, when we write a positive rational number as , we will always assume that and are coprime and positive.
Definition 2 (The braids ).
Let . The braid is the positive permutation braid (see Figure 3) defined by the cyclic permutation
It is helpful to organize the strings of in ribbons of parallel strings: the 5 cases of (2) yield, in order:
A ribbon of width which moves places to the right.
A ribbon of width which moves places to the right, thus leaving the target in position unassigned.
A ribbon of width which is sent to the final target positions with a half twist.
A ‘rogue’ string, which ends at the unassigned target in position .
A ribbon of width , which is sent to the first target positions with a half twist.
Definition 3 (The braids ).
It will be convenient for us to conjugate the braids by a half twist of the final strings, thereby turning the half twist on the final ribbon into a full twist, and removing the half twist on the penultimate ribbon: these conjugated braids will be denoted (Figure 4). (The braids can be seen as circular braids, as shown on the right of the figure, with each string other than the rogue one rotating around the circle by either or positions. This point of view motivates constructions later in the paper — see Definitions 6 and 10.)
3. Pseudo-Anosov convergence to the tight horseshoe
The tight horseshoe map  is a 2-sphere homeomorphism which can be obtained by collapsing the horizontal and vertical gaps in the invariant Cantor set of Smale’s horseshoe map . In order to define it directly, we start with its sphere of definition, which is obtained by making identifications along the sides of a unit square as depicted in Figure 5. Infinitely many segments along the boundary of , two of length for each , are folded in half (so that the points of each segment, other than the center point, are identified in pairs). The top and right edges of are each a single folded segment, and the other segments are arranged on the left and bottom sides in decreasing order of length from the top left and bottom right vertices respectively. The fold segment endpoints, together with the bottom left corner, are identified to a single point . It can be shown (see for example ) that the space so obtained is a topological sphere (and, in fact, that the Euclidean structure on induces a well defined conformal structure on ).
To define the tight horseshoe map, let be the (discontinuous and non-injective) map defined by
That is, stretches by a factor 2 horizontally, contracts it by a factor vertically, and maps its left half to its bottom half, and its right half, with a flip, to its top half. The identifications on are precisely those needed to make continuous and injective, so that it defines a homeomorphism , the tight horseshoe map. (It is an example of a generalized pseudo-Anosov map : it has (horizontal and vertical) unstable and stable invariant foliations, but these foliations have infinitely many -pronged singularities — at the centers of the fold segments — accumulating on an ‘-pronged singularity’ corresponding to the fold segment endpoints and the bottom left vertex.)
For each , let be a pseudo-Anosov homeomorphism in the mapping class of the -marked sphere defined by . The convergence of to as is an immediate consequence of results from . The following statement is a summary of the relevant parts of theorems 5.19 and 5.31 of that paper.
There is a continuously varying family of homeomorphisms of the standard 2-sphere, with the properties that:
is topologically conjugate to ; and
there is a decreasing function , satisfying as , such that is topologically conjugate to for each .
A brief discussion of the ideas surrounding Theorem 4 may be helpful to the reader. Boyland  defined the braid type of a period orbit of an orientation-preserving disk homeomorphism to be the isotopy class of , up to conjugacy in the mapping class group of the -punctured disk: the braid type can therefore be described — although not uniquely — by a braid . He further defined the forcing relation, a partial order on the set of braid types: one braid type forces another if every homeomorphism which has a periodic orbit of the former braid type also has one of the latter. The forcing relation therefore describes constraints on the order in which periodic orbits can appear in parameterized families of homeomorphisms.
If is Smale’s horseshoe map, then standard symbolic techniques associate a code to a period orbit . This coding establishes a correspondence between the non-trivial periodic orbits of and those of the affine unimodal tent maps defined for by
whose periodic orbits are likewise coded in a standard way. The braids of Definition 2 — or, more accurately, the braids for alluded to before the definition — are precisely the pseudo-Anosov braids describing braid types of horseshoe periodic orbits which are quasi-one-dimensional, in the sense that the braid types that they force are exactly those corresponding to the periodic orbits of the tent map which has kneading sequence .
Another way to view the braids is as the braids of horseshoe periodic orbits whose mapping class is pseudo-Anosov and whose associated train tracks are the simplest possible: if the -gons about the orbit points are ignored, then the union of the remaining edges is an arc. This means that the only singularities of the invariant foliations of are 1-prongs at points of the orbit and an -prong at , where . This is what makes the orbits quasi-one-dimensional: the induced map on the train track (which is an interval) is a unimodal interval map.
One way to construct the pseudo-Anosov map in a mapping class is as a factor of the natural extension of a corresponding train track map. In , a similar method is used to construct a measurable pseudo-Anosov homeomorphism from the natural extension of each tent map with : these form the continuously varying family of Theorem 4. They are pseudo-Anosov maps if and only if the kneading sequence of the tent map is periodic and is the horseshoe code of one of the braids , i.e., if and only if for some , and in this case is topologically conjugate to .
Theorem 4 also provides limits of the pseudo-Anosov homeomorphisms as tends to an irrational , or to a rational either from above or from below (the image of is discrete). All such limits are generalized pseudo-Anosov homeomorphisms.
4. Convergence of mapping tori
Let , and consider the corresponding sequence of rationals defined by . By the description of the ribbon structure of the braids in Section 2, the braid is as depicted in Figure 4, with the first ribbon having width and the others having width .
In this section we will show that, for each , the mapping tori converge geometrically as to a hyperbolic manifold of finite volume. In the following section, we will prove that the set is infinite.
The crucial observation is that the sequence of mapping tori can be obtained from a single finite-volume hyperbolic 3-manifold by Dehn filling one of its cusps with a sequence of distinct surgery coefficients : it therefore follows from Theorem 1 that the sequence of mapping tori converges geometrically to .
The manifolds are themselves mapping tori, corresponding to braids which are obtained from by adding one additional string on the left.
Definition 6 (The braids ).
Let . The braid is obtained from by adding a fixed string on the left, which links with the final width ribbon of but not with the other strings, as depicted in Figure 6. (In the circular representation of Figure 4, this corresponds to adding a fixed string, not linking the rogue string, through the center of the circle.)
That is a pseudo-Anosov braid follows from the fact that is. (Any reducing curve would bound a disk containing at least two but not all of the punctures associated with the strings of . cannot contain the puncture associated to the fixed string, since then its image would also contain that puncture but a different set of the other punctures; it cannot contain a proper subset of the other punctures, since then would be reducible; and it cannot contain all of the other punctures since the associated strings link with the fixed string.) Therefore (where is the braid axis) is a finite volume hyperbolic -manifold with cusps.
Let and . Dehn filling the cusp of corresponding to the fixed string of with surgery coefficient yields .
It is immediate from Figure 2 that performing a twist on the component of corresponding to the fixed string increases the width of the first ribbon of from to . By (1), this changes the surgery coefficient on to , so that it can be erased, yielding the closure of the braid (see Figure 4). That is, Dehn filling with surgery coefficient yields as required. ∎
The following corollary is now immediate from Theorem 1.
For each the sequence converges geometrically to .
5. Infinitely many limit manifolds
Figure 7 is a plot of the volumes of the limit manifolds against , generated by SnapPy . The points in red are those for which is of the form . In this section we show how all of the corresponding manifolds can be obtained by Dehn filling a cusp of another hyperbolic -manifold with a sequence of distinct surgery coefficients, so that, again by Theorem 1, there are infinitely many distinct limit manifolds (which converge geometrically to as ).
Other apparently convergent sequences in Figure 7 correspond to similar sequences , such as and .
The volume of suggests that it may be the magic manifold. To see that this is indeed the case, consider the braids depicted in Figure 8, each representing the 3-manifold obtained by removing the braid closure together with its axis from . The braid on the left is , representing , while the one on the right represents the magic manifold (see for example Figure 3 of ). The operations converting each braid to the next are either twists on components of the associated links or braid conjugacies, and therefore leave the 3-manifolds unchanged. Specifically, these operations are, in order: conjugacy by ; a twist on the red component (see Figure 1 (d) and (a)); conjugacy by ; a twist on the braid axis; and conjugacy by .
The manifold is obtained from the -braid of the following definition (see Figure 9), whose closure is a three-component link. Note that the blue and green strings in the figure form a braid conjugate to (the conjugacy moves the green string from the left to the right of the braid diagram), and to this braid has been added a -string braid which ‘shadows’ the blue strings. It is not obvious a priori — at least, not to the authors — that Dehn filling the ‘black’ cusp of the resulting hyperbolic -manifold should yield the manifolds : rather, the braid was found experimentally using SnapPy .
It can be checked (for example, using a train track algorithm such as the one due to Bestvina and Handel ) that is pseudo-Anosov: the corresponding relative pseudo-Anosov homeomorphism has 1-pronged singularities at the marked points corresponding to the blue and green strings of Figure 9, a -pronged singularity at , and regular points at the black marked points. Therefore (where is the braid axis) is a finite volume hyperbolic 3-manifold with 4 cusps.
Let . Dehn filling the cusp of corresponding to the black strings of Figure 9 with surgery coefficient yields .
The left hand side of Figure 10 depicts a braid , which is together with an extra fixed string shown in red. We write and for the black and red components of , which are unknotted. We need to show that filling with coefficient and with coefficient (i.e. erasing from the link ) yields the -manifold .
The braid on the right hand side of the figure is obtained by conjugating by . Referring to Figure 1, performing a twist on yields the braid on the left of Figure 11, and a conjugacy by gives the braid on the right hand side of the figure. By (1), the updated surgery coefficients are:
Performing a twist on the braid axis yields the braid on the left hand side of Figure 12, which a further conjugacy by — to pull the black string around — reduces to the right hand side of the figure. (Here and in Figure 13, the parts of the blue strings which participate in the full twist have not been drawn, to clarify the diagrams.) The red component and the black component are now unlinked. The revised surgery coefficients are
We can now carry out the surgery on . Performing a twist on yields the braid of Figure 13 (in which the ribbon contains parallel strings). The surgery coefficient of is
so that it can be removed (and is not shown in Figure 13). Because and are unlinked, the surgery coefficient of is unchanged: .
We next perform a twist on , which produces the braid on the left hand side of Figure 14, and changes the surgery coefficient of to . A twist on therefore changes its coefficient to , so that it can be erased: this results in the braid on the right hand side of Figure 14, in which each of the four ribbons contains parallel strings.
To complete the proof, we exhibit a braid conjugacy between the braid on the right hand side of Figure 14 and the braid — that is, the braid of Figure 6 with all four ribbons containing parallel strings. (This conjugacy was discovered computationally, using sliding circuit set methods [13, 14] for small values of and extrapolating: the braids have small sliding circuit sets but large ultra summit sets .) Two successive conjugacies are shown in Figure 15. Here the first, second, and fourth ribbons have been enlarged by incorporating an additional parallel string, so that they each contain parallel strings.
Simplifying the braid on the right hand side of Figure 15 by isotopy of the strings yields the braid on the left hand side of Figure 16. Again, we have incorporated additional parallel strings into ribbons, so that the first two ribbons contain parallel strings, and the other two contain parallel strings. A final conjugacy which moves the green string to the left, underneath all of the other strings, gives the braid on the right hand side of the figure, and incorporating additional parallel strings into the rightmost two ribbons yields as required.
The sequence converges geometrically to as , and there are infinitely many distinct hyperbolic 3-manifolds .
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