топология / Farb, Margalit, A primer on mapping class groups

Figure 14.7 An Anosov map of the torus.

Figure 14.8 An Anosov map demystified(?).

Figure 14.8 taken all at once gives another view of the mapping class f . But now, we can see how the combinatorial structure on the unstable foliation given by the rectangles translates into a purely combinatorial description of the homeomorphism f : the red rectangle is stretched twice over itself and once over the blue rectangle, and the blue rectangle is stretched once over each. If we turn that information into a matrix in the obvious way, we get nothing other than the original matrix:

2 1

1 1

And thus much of the information (in this case, all) of the original information about the mapping class is contained in this “transitio n matrix.” Note that we could have chosen a rectangle decomposition with more rectangles and gotten a larger matrix.

We remark that the two rectangles of the above Markov partition are similar. This is related to the fact that the transition matrix is equal to its transpose; in general the lengths and widths of the rectangles come from the transition matrix and its transpose, respectively.

One aspect of Thurston's approach to the classification of su rface homeomorphisms is that such a scheme as above is always possible. That is, given a pseudo-Anosov mapping class f , there is a particular rectangle decomposition of the surface, called a Markov partition, so that f has a combinatorial description as above; instead of cutting along a single stable arc, however, one typically needs to cut along several. From the resulting transition matrix, one can determine various properties of f . In particular, the dilatation of f is the largest real eigenvalue of the transition matrix. Therefore, Theorem 14.9 and one direction of Theorem 14.10 can be proven using this theory. The point is to show that any pseudo-Anosov mapping class of a fixed surface has a transition matrix whose size (number of ro ws) is uniformly bounded from above, and use the fact that the set of eigenvalues of integral n × n matrices is discrete (the other direction of Theorem 14.10 is by explicit construction). Also, Theorem 14.15, Theorem 14.21 and Corollary 14.22 can all be proven using the theory of Markov partitions. The idea for the latter two is that, as we iterate a pseudo-Anosov mapping class, any curve gets closer and closer to the horizontal foliation in each rectangle, and the number of horizontal components in each rectangle grows like λn, where here we are thinking of λ as the largest eigenvalue of the transition matrix. We refer the reader to [54] for the details.

In Chapter 15, we will delve more into Thurston's approach to the classification. We will employ a different, but equivalent, method for combinatorializing a pseudo-Anosov homeomorphism—the theory of train tracks. We recommend that the reader come back to this section and find th e train track hidden in Figure 14.8. It is also helpful to make the connection between the “famous example” in Chapter 15 with Corollary 14.22; it is si mply a matter of “blurring the vision”.

Chapter Fifteen

Thurston's proof

In this chapter we give some indication of how Thurston originally discovered the Nielsen–Thurston classification theorem. We begin with a concrete, accessible example that illustrates much of the general theory. We then provide a sketch of how that general theory works. Our goal is not to give a formal treatment, as per the rest of the text. Rather, we hope to convey to the reader that there is a beautiful circle of ideas surrounding the Nielsen– Thurston Classification, including Teichm ¨uller's theore ms, Markov partitions, train tracks, foliations, etc. We also hope the reader will be inspired to undertake a more serious study of the ideas presented in this chapter.

15.1 THE FAMOUS EXAMPLE

Figure 15.1 The mapping class f is σ1−1σ2.

A good place to start understanding surface homeomorphisms is on the thrice-punctured plane, as this is one of the simplest surfaces with infinite mapping class group. We consider the mapping class

f = σ1−1σ2

424 CHAPTER 15

(written in function notation), where σ2 and σ1−1 are the half-twists (see Chapter 9) indicated in Figure 15.1. To begin to understand this mapping class, we draw a simple closed curve c in the thrice-punctured plane, and keep track of what happens to its isotopy class as we iterate f . The first two iterations are shown in Figure 15.2.

Figure 15.2 The first two iterates of c under f .

At this point, the pictures become harder to draw, and it becomes increasingly difficult to perform further iterations. However, we m ake the following useful observation: the image f 2(c), as shown in Figure 15.2, can be represented by the data in Figure 15.3; we call this a train track for f 2(c). The basic idea is that we replace n “parallel strands” of f 2(c) by a single strand labelled n. The curve f 2(c) can be recovered from its train track by replacing an edge of the train track labelled n with n copies of itself. Upon doing this, we see that there is a unique way to glue all of these strands together at the switches (where the train track branches).

Figure 15.3 Converting f 2(c) into a train track.

We also note that four of the labels in the train track are redundant. Indeed,

given the segments labelled 6 and 4, the rest of the numbers are determined by the following switch condition: the sum of labels going into a switch from one side equals the sum of the labels leaving the switch from the other side.

The nice thing about this setup is that we can simply iterate f on the train track to see what happens to c. For the sake of generality, we start at a train track with labels x and y. We notice that each of the curves c, f (c), and f 2(c) can be represented by the given train track. In (x, y) coordinates, we see that the curves c, f (c), and f 2(c) are given by (0, 2), (2, 2), and (6, 4), respectively.

In Figure 15.4, we apply f to this train track. We now notice the following magical phenomenon: the resulting train track can be represented by the original train track! Wherever we see parallel tracks, we replace them by a single track whose label is the sum of the labels from the parallel tracks; think of “zipping” the track together. The result is shown in Figure 15.4. We can describe the action of f on this train track by the matrix

The matrix M has eigenvalues

√√

426 CHAPTER 15

with eigenvectors

Since any essential simple closed curve intersects this train track, we see that the geometric intersection number of any isotopy class of simple closed curves b with f n(c) grows like

√ !n

3 + 5

2

i(b, c),

√

as promised by Theorem 14.21 (it is indeed the case that 3+ 5 is the dilata-

2

tion; in fact f is the image of the ( 21 11 ) map of the torus under the hyperelliptic involution, as in Section 9.4). Our discussion here suggests the proof of that theorem.

Not every curve is carried by the train track we have been considering, that is, even if we vary x and y over all integers, we will not obtain every isotopy class of simple closed curve—consider, for instanc e, the simplest curve surrounding the two punctures on the right. However, it is true that there is a finite collection of (four!) train tracks in the 3-punctured plane so that any isotopy class of curves is carried by at least one of these train tracks.

Figure 15.5 Any simple closed curve in the thrice punctured plane can be broken up into canonical pieces.

To see the last statement, note that, up to isotopy, any simple closed curve can be drawn in the union of the three squares shown in the picture at the top

of Figure 15.5. Further, a connected component of its intersection with one of the squares is one of the six types of arcs shown in the bottom picture of Figure 15.5. Now, an essential curve cannot use both types of dashed arcs, because then it follows that the curve is isotopic to the nonessential curve that surrounds all three punctures (it is nonessential because it is fixed by every mapping class). Since the other two types of arcs in the middle square intersect, a simple curve can use at most one of those. We therefore see that there are four kinds of curves, depending on which of each of the two pairs of arcs they use in the middle square. This information is exactly the same as saying that any simple closed curve is carried by one of the train tracks shown in Figure 15.6.

Figure 15.6 “Train track space” for the thrice-punctured pl ane.

The four train tracks in Figure 15.6 give four “coordinate ch arts” on the set of isotopy classes of simple closed curves in the thrice-punctured plane. If we allow the coordinates to be arbitrary positive real numbers, then we get a continuous space. As the action of f is projective, we may consider the quotient of this space by scaling. The resulting space, a topological circle, is shown in Figure 15.6. For lack of a better term at the moment, we call this projective train track space. Note that, the way we have set things up, the coordinates of the vertex at the intersection of two charts is not ambiguous.

The mapping class f acts on this projective train track space, and in Figure 15.7 we give a partial depiction of this action, using the coordinates and notation established in Figure 15.6. We see that there are source–sink dynamics. The sink is the projective train track discovered above. The source

428 CHAPTER 15

is the train track that one would find by applying the above pro cedure to f −1. This source lies in the “quadrant” opposite to the sink.

Because of the dynamics, we see that f certainly does not act simplicially on projective train track space—charts get stretched, or shru nk. For example, since f (u) = a, and since f fixes the source, it follows that, using the coordinates of the fourth quadrant, f (1 − ǫ, 1) lies in the third quadrant and f (1 + ǫ, 1) remains in the fourth quadrant, assuming ǫ is small enough. It may seem counterintuitive that different weights on the same train track can lead to combinatorially different tracks after applying f , but the sequences of pictures in Figures 15.8 and 15.9 explain this phenomenon.

unzip

Figure 15.8 Finding the image of the point (1, 1 + ǫ) from the bottom left quadrant.

In the calculation, we are forced to use an “unzipping” proce dure. If we read the arrows backwards, we see that this is just the opposite of the zipping procedure used earlier. The key point as to why we get different combinatorial train tracks in Figures 15.8 and 15.9 is that, when we unzip, we are forced

to “peel off” the track of smaller weight. There's not enough track to peel off the one of larger weight; thus, we get different unzipping sequences and hence different combinatorial tracks at the end.

1 + ǫ 1 + ǫ

1

unzip

ǫ

1 − ǫ

unzip

Figure 15.9 Finding the image of the point (1 + ǫ, 1) from the bottom left quadrant. We warn the reader that it requires a clever isotopy to realize the second arrow.

15.2 A SKETCH OF THE GENERAL THEORY

In this section we give a sketch of how the ideas introduced in Section 15.1 can be used to analyze mapping classes of arbitrary surfaces. The full details of this approach are given in [54].

Fix a surface S = Sg,n; so that we can give concrete numbers for the dimensions of the associated spaces, we assume g ≥ 2. Two spaces associated to S are the Teichm ¨uller space Teich(S), and the space of measured foliations MF = MF(S). We consider the latter space as the space of equivalence classes of measured foliations, where the equivalence is generated by isotopy and by Whitehead moves; see Figure 15.12. Let S denote the set of isotopy classes of simple closed curves in S. The key idea is that both Teich(S) and MF map (disjointly) into RS≥0 −0, the space of nonzero func-

tions S → R≥0. When we pass to the projective space P (RS), the image of Teich(S) is still homeomorphic to an open ball of dimension 6g − 6 + 2n (no two points are projectively equivalent) and the image of MF, denoted PMF, is homeomorphic to a sphere of dimension 6g −6 + 2n −1. What is more, the union Teich(S) PMF is naturally endowed with the topology of a closed ball of dimension 6g−6+2n (in the notation, we implicitly identity Teich(S) with its image in the projective space). An element of Mod(S)

acts continuously on this ball, and so the Brouwer Fixed Point Theorem implies the existence of a fixed point. The Nielsen–Thurston cl assification is then obtained by analyzing the various possibilities for this fixed point.

In the remainder, we explain more of the details of this idea, in particular explaining how to describe PMF as a sphere. We begin with a basic idea of Thurston that allows one to model measured foliations combinatorially.

We define a train track τ on S to be a branched 1-manifold embedded in S; that is, τ is a graph embedded in S, endowed with a choice of (unsigned) tangent direction at each vertex. This choice partitions the edges incident to that vertex into “incoming edges” and “outgoing edges” (t he assignment of these names to the two sets is arbitrary). We call the vertex a switch of τ . A (transverse) measure µ on a train track τ is a choice of positive real number µ(ei) for each edge ei of τ , satisfying the switch conditions: for each switch of τ , the sum of the µ-weights of the incoming edges equals the sum of the µ-weights of the outgoing edges. The train track τ equipped with the measure µ is called a measured train track (τ, µ).

Given a measured train track (τ, µ), we can build a foliation F as follows: for each edge ei we take a rectangle Ri of width 1 and height µ(ei), and equip it with the measured foliation whose leaves are horizontal lines and whose transverse measure is dy. Thus the “total mass” of an arc crossing all horizontal leaves of Ri is µ(ei). Now, for each switch we have incoming edges and outgoing edges, and we glue the vertical sides of the rectangles accordingly; see Figure 15.10. The fact that µ satisfies the switch conditions ensures that the foliations on the rectangles match up and give a well-defined transverse measure on their union. We now have a union of glued rectangles, equipped with a transverse measured foliation, sitting inside our surface S. Assuming the complementary regions are contractible (or perhaps have at most one puncture), we can collapse these regions to get a foliation F on the whole surface (if the complementary regions are not contractible, one can still collapse onto a spine, but there are problems if complementary regions are bigons or “monogons”). We say that F is carried by τ .

1

3

2

Figure 15.10 The first step in converting a train track into a f oliation.