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

need to assume that P is connected. One example of this is the identification of Sg as the quotient of a regular (4g + 2)–gon in R2 with opposite sides identified. Another example is given in Figure 11.4.

Any foliation of R2 by parallel lines restricts to give a foliation of (the interior of) P . We claim that this foliation induces a foliation of S. It is easy to see that any point of S coming from a point of P that is not a vertex of P has a regular neighborhood that satisfies the definition of a r egular point of a foliation.

So what happens at a point p S corresponding to a vertex of P ? The first observation is that, since identified sides of P are parallel, the total angle around p is an integer multiple of π. In particular, there is some vertex pe of P in the preimage of p, and a vector v based at pe that points into P (possibly along an edge) and is parallel to the foliation of P . If we sweep out an angle of π starting with v, we find a closed Euclidean half-disk in S that is foliated by lines parallel to the diameter. If we continue to sweep out angles of π, we see that a neighborhood of p looks like some number of Euclidean half-disks, each foliated by lines parallel to the diameter, and glued along oriented radii. By our assumption on the total angle around each point of S coming from a vertex of P , we know that there are at least two half-disks glued at p. If there are exactly two half-disks, then p is a regular point. If there are k half-disks, where k ≥ 3, then p is a singularity with k prongs.

One measure on the induced foliation of S is the one given by the total variation of the Euclidean distance in the direction perpendicular to the foliation of P . The charts we described above are the natural charts for the nonsingular points.

Suppose that, in this construction, we orient each edge of P so that the identifications respect these orientations. If all side pai rings identify sides of P that are parallel in the oriented sense (as opposed to anti-parallel), then the resulting foliation of S is orientable. Indeed, either of the two orientations of the foliation on the interior of P extend to give an orientation of the entire foliation of S.

It is a fact that every measured foliation comes from this polygon construction. The idea is that the natural coordinates for a measured foliation pick out large rectangles in the surface that are foliated by horizontal lines. See Section 14.3 for further discussion.

Enlarging a simple closed curve. Let S be a closed surface of genus g.

We can realize S topologically as a Euclidean (4g + 2)–gon with opposite sides identified. We can straighten the sides of this polygon so that two opposite sides are horizontal and the other 4g sides are vertical. The result is a Euclidean rectangle R. If we retain the original identifications, we obtain a description of S as a quotient of R where the horizontal sides are identified and where segments of one vertical edge of the rectangle a re identified with segments of the other vertical edge. Let α be the nonseparating simple closed curve in S that is the image of the horizontal sides of the rectangle.

The foliation of R by horizontal lines induces, just as in the previous construction, a measured foliation of S. There is a one-parameter family of measures obtained by scaling the rectangle vertically. The special feature of this particular construction is that every closed leaf is isotopic to the curve α. We say that this measured foliation is obtained by enlarging the simple closed curve α. Note that, by change of coordinates, we can enlarge any nonseparating simple closed curve in a closed surface (it is also possible to extend the construction to separating curves).

Figure 11.3 A two-fold branched cover over the torus

From a branched cover. Let g ≥ 2, and let p : Sg → T 2 be a branched covering map. For our purposes, a branched cover of one topological surface over another is the quotient of one orientable surface by a finite group of orientation-preserving homeomorphisms. So, for instance, orbifold cover-

ings are branched coverings. One such example, with 2g − 2 branch points, is illustrated in Figure 11.3.

Any measured foliation (F, µ) of T 2 pulls back via p to a measured foliation (p (F), p (µ)) on Sg . The singularities of p (F) are precisely the branch points of the covering. The foliation p (F) has 2k separatrices at an order k branch point of the cover. Since the deck transformations of the cover Sg → T 2 are orientation-preserving, an orientation of the foliation on T 2 pulls back to an orientation on the foliation of Sg . Since every foliation of T 2 is orientable, every foliation of Sg obtained by this construction is orientable.

The same construction as above can be used to pull back measured foliations on any closed surface via any (branched or unbranched) cover.

From a pair of filling simple closed curves. Let α and β be two transverse simple closed curves that are in minimal position and that fill a closed surface S. Take, for instance, the example of Figure 1.7. We can think of α β as a 4–valent graph in S, where the vertices are the points of α ∩ β. In fact, by also considering the closures of the components of S − (α β) as 2–cells, we have a description of S as a 2–complex X.

We construct a dual complex X′. The complex X′ is formed by taking one vertex for each 2–cell of X, one edge transverse to each edge of X, and one 2–cell for each vertex of X. Since the vertices of X are 4–valent, it follows that X′ is a square complex, that is, each 2–cell of X′ is a square. What is more, each square of X′ has a segment of α running from one side to the opposite side.

We can foliate each square of X′ by lines parallel to α. This gives rise to a foliation Fα on all of S. We declare the “width” of each square to be the same fixed number, and this gives a measure on Fα. The foliation associated to β is a measured foliation Fβ that is transverse to Fα.

This last construction is really just a special case of both the polygon construction and the branched cover construction. Indeed, we can think of X as a disconnected polygon with sides identified. Also, if we t hink of T 2 as the unit square with sides identified, then there is a branche d cover from S ≈ X → T 2 that takes each square of X to the unit square and takes the α–foliation to the foliation of the unit square by horizontal lines.

11.2 HOLOMORPHIC QUADRATIC DIFFERENTIALS

We now describe the complex-analytic counterparts to measured foliations, namely, holomorphic quadratic differentials. Since quasiconformal maps are most easily described via complex analysis, we will be able to exploit this point of view in proving Teichm ¨uller's theorems.

First, the holomorphic cotangent bundle to a Riemann surface X is the complex line bundle over X whose fiber above a point p X is the space of complex linear maps Tp(X) → C. A holomorphic 1–form is a holomorphic section of the holomorphic cotangent bundle of X. A holomorphic quadratic differential on X, which is the object of interest here, is a holomorphic section of the symmetric square of the holomorphic cotangent bundle of X. For instance the tensor square of a holomorphic 1–form on X is a holomorphic quadratic differential.

We can alternatively describe a holomorphic quadratic differential on X in terms of local coordinates, as follows. Let {zα : Uα → C} be an atlas for X. A holomorphic quadratic differential q on X is specified by a collection of expressions {φα(zα) dzα2 } with the following properties:

1.Each φα : zα(Uα) → C is a holomorphic function with a finite set of zeros.

2.For any two coordinate charts zα and zβ , we have

The second condition can be phrased as: “the collection {φα(zα) dzα2 } is invariant under change of local coordinates.” To say this ye t another way: if q is given in one chart by φα(z), and in another chart by φβ (z), and the change of coordinates from the first chart to the second chart is the holomorphic map ψ, then

φβ (z)(dψ)2 = φα(z).

It follows, for instance, that the order of a zero of a holomorphic quadratic differential is well-defined, independent of the chart.

More concretely, a holomorphic quadratic differential q on a Riemann surface X is a holomorphic map from the holomorphic tangent bundle of X to

C. What is more, for any p X and any v Tp(X) we have q(v) = q(−v). Say that in local coordinates z the holomorphic quadratic differential q is

given by φ(z) dz2. Then for v Tp(X) ≈ C, the image of v under the chart is a point dz(v) in Tz(p)(C) ≈ C and we have q(v) = φ(z(p))dz(v)2 .

Measured foliations from quadratic differentials. Given a holomorphic quadratic differential q on a Riemann surface X, we obtain a foliation by taking the union of the zeros of q with the set of smooth paths in X whose tangent vectors evaluate to positive real numbers under q. This foliation is called the horizontal foliation for q. If we instead take the paths in X whose tangent vectors evaluate to negative real numbers under q, the resulting foliation is called the vertical foliation for q.

Say that, within some chart, a holomorphic quadratic differential is given by the expression φ(z) dz2 . In any given chart, the function

induces a transverse measure µq on the horizontal foliation for q. By taking real parts instead of imaginary parts, we obtain a transverse measure on the vertical foliation for q. Below we will define natural coordinates for a holomorphic quadratic differential, where this formula always takes a standardized form. To check that µq really determines a transverse measure, one can either apply equation (11.1) directly or one can appeal to natural coordinates.

Consider for example a holomorphic quadratic differential q that in some coordinate chart has the form q(z) = dz2. A point in the tangent bundle of C can be written as a pair (z, v), where z C and v Tz (C) ≈ C. Then

q((z, v)) = v2.

Now v2 > 0 precisely when v is a nonzero real number, and v2 < 0 precisely when v is a purely imaginary number. Therefore, in the given chart, the horizontal foliation is the union of horizontal lines and the vertical foliation is the union of vertical lines. The measures for these foliations are the ones induced by |dy| and |dx|.

Now consider a holomorphic quadratic differential with local expression q(z) = zk dz2. In this case

q((z, v)) = zk v2.

Some lines of the horizontal foliation are easy to spot, namely, vectors of the form (z, z), where zk+2 = 1. It is not hard, then, to see that the horizontal

foliation has the form of a (k + 2)–pronged singular point, as in the theory of measured foliations. The vertical foliation is the transverse foliation obtained by rotating the picture for the horizontal foliation by an angle of

π/(k + 2).

Quadratic differentials versus 1–forms. Why do we consider holomorphic quadratic differentials as opposed to holomorphic 1–f orms, which are differentials of the form φ(z) dz? The reason is that, for a holomorphic quadratic differential q, we have that q((z, v)) > 0 if and only if q((z, −v)) > 0, and so the associated horizontal and vertical foliations are not necessarily oriented. On the other hand, the (analogously defined) horiz ontal and vertical foliations for a holomorphic 1–form are automaticall y oriented. However, in our study of mapping class groups and Teichm ¨uller space we will be forced to deal with both oriented foliations and unoriented foliations.

Natural coordinates. Let q be a holomorphic quadratic differential on a compact Riemann surface X. We will now show that every point of X has local coordinates, called natural coordinates, so that in these local coordinates q(z) = zk dz2 for some k ≥ 0. Since we just showed that the horizontal and vertical foliations for zk dz2 satisfy the definition of a measured foliation, it will follow that the horizontal and vertical foliations for q really are transverse measured foliations, as defined above .

First consider a regular point p of q, that is, assume q(p) 6= 0. Let z : U → C be a local coordinate with z(p) = 0, and write q(z) = φ(z) dz2 in this chart. Since q is assumed to have finitely many zeroes, we can pick the chart small enough that φ(z) 6= 0 anywhere in this chart. Our goal is to show that there is a local coordinate ζ at p so that q(ζ) = dζ2. Such coordinates are obtained by composing z with the change of coordinates

Z z p

η(z) = φ(ω) dω

0

where some branch of the square root function is chosen (this is possible since φ 6= 0). Of course, when we integrate from 0 to z, really we mean to integrate along some (any) path from 0 to z.

The natural coordinates are ζ = η ◦ z : U → C. We can check that q has the desired form in the ζ coordinates. First, by the fundamental theorem of calculus, we have

p

dη = φ(z).

Now let q(ζ) = φ′(ζ) dζ2 in the ζ–coordinates. By equation (11.1) we have

φ′(ζ)(dη)2 = φ(z),

or

φ′(ζ)φ(z) = φ(z),

and so φ′(ζ) ≡ 1, as desired.

The natural coordinates ζ are unique up to translation and sign. It is therefore possible to associate measures to the horizontal and vertical foliations. Locally, the measures are given by |dy| and |dx|, respectively.

Now suppose that p X is a zero of q of order k ≥ 1. By a variation of the above argument, there is a local coordinate ζ so that, in this coordinate, q is given by q(z) = zk dz2. As above, these coordinates are called the natural coordinates. For the details of this argument, see [160, §6].

We have shown that the horizontal and vertical foliations of a holomorphic quadratic differential give a pair of measured foliations that are transverse to each other. It is a deep theorem of Hubbard–Masur that, given any measured foliation (F, µ), one can build a holomorphic quadratic differential whose corresponding horizontal foliation is, up to a certain equivalence, equal to (F, µ) [81].

The Euler–Poincar e´ formula revisited. From the Euler–Poincar´e formula (Proposition 11.2) and the correspondence between the order of a zero of a holomorphic quadratic differential and the number of prongs of the associated foliation, we deduce that a holomorphic quadratic differential must vanish at exactly 4g − 4 points, where points are counted with multiplicity.

Euclidean areas and lengths. The natural coordinates for a holomorphic quadratic differential q on a Riemann surface X endow X with a singular Euclidean metric. A singular Euclidean metric on a surface S is a flat metric outside of a finite number of points, around each of which the m etric is modelled on gluing flat rectangles together in the same way as is done to give a singular point of a measured foliation. Locally, the area form of this metric is given by

21i |φ(z)| dz dz = |φ(z)|dx dy,

where φ(z) dz2 is the local expression for q.

We can also talk about the Euclidean length of a path in X with respect to q. This length form is given by

|φ(z)|1/2 |dz| = |φ(z)|1/2 pdx2 + dy2.

The singular Euclidean metric induced by a holomorphic quadratic differential is nonpositively curved in the sense that it is “local ly CAT(0).” It follows that, given any arc α in X, there is a unique shortest path among all paths homotopic to α with endpoints fixed (see the proof of [31, Chapter II, Corollary 4.7]).

QUADRATIC DIFFERENTIALS ON THE TORUS

Let X be a closed Riemann surface of genus 1. There is a lattice Λ < C so that X ≈ C/Λ. Let π : C → X denote the quotient map. For a small

All of the transition maps for this atlas are translations. For any point x X the set of images of x under all charts is π−1(x).

Let q be a holomorphic quadratic differential on X. From equation (11.1) and the fact that all transition maps are translations, it follows that q can be written as a doubly-periodic holomorphic function φ : C → C. A doublyperiodic function C → C is bounded, and so by Liouville's theorem φ is constant. We therefore have that the set of holomorphic quadratic differentials on X is in bijection with C. Under this bijection, the horizontal foliation for the differential corresponding to z C has leaves consisting of straight lines that meet the x–axis with angle − arg(z)/2.

CONSTRUCTIONS OF QUADRATIC DIFFERENTIALS

Having explained some of the basics of holomorphic quadratic differentials, the question remains: how does one actually construct a holomorphic quadratic differential? Since we know how to derive a measured foliation from a holomorphic quadratic differential, it makes sense to generalize our two main constructions of measured foliations, namely, the construction via polygons and the construction via branched covers.

Quadratic differentials via polygons: the Swiss cross example. Just as we were able to construct measured foliations from certain polygons, we can

a

Figure 11.4 The Swiss cross.

also construct holomorphic quadratic differentials from the same polygons. Instead of explaining the construction in generality, we will explain one particular case, the so-called “Swiss cross example,” in de tail. This example exhibits all of the subtleties of the general case.

Consider the closed polygonal region P in C in Figure 11.4. Let S be the topological surface obtained by identifying sides of P by Euclidean translation, as indicated in the figure. We start by describing an a tlas for S that gives S the structure of a Riemann surface. The first chart is self-ev ident: the subset of S corresponding to the interior of P is already identified with an open subset of C. Now consider a point p in S corresponding to a point in the interior of an edge of P . Let U be an open neighborhood of p corresponding to a union of two half-disks in P (each half-disk is the intersection of an open disk in C with P ). To define the chart for U , we say that one “half” of U maps to the corresponding half-disk in P , and the other halfdisk maps to the image of its corresponding half-disk in P under a translation, where the translation is chosen so that the image of U under the chart is an open disk in C.

We proceed similarly at the corners. The 8 corners with angle π/2 glue together to form two disks in S, and so we can use the same method as above to glue the pieces together. Consider the other 4 corners. We see that, in S, these four vertices of P are identified to a single point, which we call s. We also see that the total Euclidean angle around the 4 copies of s in P is 4(3π/2) = 6π. Thus we cannot simply glue these pieces together by translations and expect to get an open disk in C. The solution

is as follows: one by one, translate each of the corresponding vertices of P to 0 C, apply the map z 7→z1/3, and then apply rotations so that the 4 “corners” glue together to form an open disk about 0 C; we take care so that the image of the kth corner lies between the rays with argument kπ/3 and (k + 1)π/3.

The above-defined charts indeed define a complex structure on S, that is, transition maps are biholomorphic. The only place where there could possibly be an issue is at the point s (since z1/3 is not differentiable at 0). However, this point only appears in one chart, so there is nothing to check.

We can now use the atlas we have constructed in order to define a n explicit holomorphic quadratic differential q on S, chart by chart. The notion of holomorphic here is taken with respect to the complex structure on S that we just constructed.

We define q on every chart except the last one constructed above by setting q(z) = dz2. On the last chart we let q(z) = 9z4 dz2. Note that each chart except for the last gives natural coordinates, since the local expression for q in these cases is dz2.

We can see that the point s will be a singularity of the horizontal foliation for q. The prongs at s come from 8 segments of the horizontal foliation for q, two at each preimage of s. In S, two of these pairs get identified (the ones labelled c and f ), and so in the end the singularity at s has 6 prongs (cf. Proposition 11.2). This agrees with the fact that we gave q an order 4 zero at s.

Let us now check that equation (11.1) holds for all transition maps. Consider a point near s that lies in the “big chart” (the first one we defined) and the “special chart” (the last one we defined) Call the big coordin ate z and the special coordinate w. Equation (11.1) demands that

(dz/dw)2 = 9w4.

But this is the same as saying that

(3w2)2 = 9w4,

which is obviously true. Checking equation (11.1) for the other transition maps is similar. Therefore q really is a holomorphic quadratic differential. The area and arc length forms are exactly the ones coming from the Euclidean metric in the big chart.

One way to get other holomorphic quadratic differentials on the Swiss cross