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

COMPLEX STRUCTURES VERSUS HYPERBOLIC STRUCTURES

By a Riemann surface X we mean a one–dimensional complex manifold. This means that X comes equipped with an atlas of charts to C that has biholomorphic transition maps, i.e., transition maps are holomorphic with holomorphic inverses. Two Riemann surfaces X and Y are said to be isomorphic if there is a biholomorphic homeomorphism between them.

The uniformization theorem gives that any Riemann surface of genus g ≥ 2 is the quotient of the unit disk by a group of biholomorphic automorphisms acting properly discontinuously and freely on ; see, e.g., [158, Chapter 9]. Any group of biholomorphic automorphisms of preserves the hyperbolic metric on . Thus / has an induced hyperbolic structure, and conversely, any such hyperbolic structure gives a complex structure on X. In other words, for g ≥ 2 there is a bijective correspondence:

Using isothermal coordinates, one can define a complex struc ture on any surface endowed with a Riemannian metric. It follows that Teich(Sg ) can also be identified with the set of conformal classes of Rieman nian metrics on Sg .

QUASICONFORMAL MAPS AND AN EXTREMAL PROBLEM

Let U and V be open subsets of C, and let f : U → V be a homeomorphism that is smooth outside of a finite number of points.

Using the usual notation for maps R2 → R2, we can write f as f (x, y) = (a(x, y), b(x, y)), where a, b : R2 → R. Where it is defined, the derivative df is then the real-linear map

We can also write

df = fx dx + fy dy where fx = (ax, bx) and fy = (ay , by ).

The dilatation of the map f is defined to be the number

Kf = sup Kf (p)

where the supremum is taken over all points p where f is differentiable. Thus 1 ≤ Kf ≤ ∞. If Kf < ∞ we say that f is a quasiconformal or Kf –quasiconformal map between the domains U and V of C. Note that biholomorphic maps are conformal with conformal inverses, hence are 1– quasiconformal. The notion of quasiconformal homeomorphism was first considered by Gr¨otzsch in 1928.

Quasiconformal maps. Let f : X → Y be a homeomorphism between Riemann surfaces that is smooth outside of a finite number of p oints. Assume further that f respects the orientations induced by the complex structures on X and Y and that f −1 is smooth outside of a finite number of points. Since the transition maps in any atlases for X and Y are biholomorphic (hence 1–quasiconformal), and since the local express ions for f are orientation preserving, there is a well-defined notion of th e dilatation Kf (p) of f at a point p X where f is smooth. Since f is smooth outside of a finite number of points we can define Kf = sup Kf (p) as above. We will say that f is quasiconformal or Kf –quasiconformal if Kf < ∞.

A map between Riemann surfaces is holomorphic if, in any chart, it is given by a holomorphic map from some domain in C to C. A bijective, holomorphic map between Riemann surfaces is called a conformal map. Conformal maps between Riemann surfaces are also biholomorphic, that is, they have holomorphic inverses. The last fact fact follows from the open mapping theorem and Theorem 10.34 in Rudin's book [149].

Lemma 11.1 Let f : X → Y be a homeomorphism between Riemann surfaces. Then f is a 1–quasiconformal homeomorphism if and only if it is a conformal map.

Proof. First of all, since f is a homeomorphism, its derivative must be nonzero at all points where it is defined. Indeed, if f ′ has a zero of order m − 1 at x X, then the open mapping theorem implies that f is m-to-1 in a small neighborhood of x [149, 10.32].

Suppose that f is conformal. In this case f ′ is defined at every point, and by the above argument we know that f ′ never vanishes. It follows that f takes circles in the tangent space of X to (nondegenerate) circles in the tangent space of Y [149, Theorem 14.2], and so f is 1–quasiconformal.

304 CHAPTER 11

Now suppose that f is 1–quasiconformal. This is the same as saying that fz¯ ≡ 0 wherever it is defined. Let A X be the set of points where f ′ is not defined. The restriction of f to X − A is then holomorphic. Since f |X−A is also bijective, it is conformal. As above, it follows that f |X−A is biholomorphic. Since f is a homeomorphism, its singularities at A must be removable [149, Theorem 10.20]. Since f is continuous it follows that f is

Let X be a Riemann surface. The following facts are easy to verify directly from the definitions.

1.If f and g are quasiconformal homeomorphisms of X then f ◦ g is quasiconformal and

Kf ◦g ≤ Kf Kg .

2.If f is a quasiconformal homeomorphism then so is f −1 and Kf −1 =

Kf .

3.If g is conformal automorphism then Kf ◦g = Kf = Kg◦f .

Teichmuller's¨ extremal problem. In 1928 Gr¨otzsch considered the following natural extremal problem, at least in the case of rectangles. Because Teichm ¨uller later considered the case of general Riemann surfaces, this problem is sometimes referred to as Teichmuller's¨ extremal problem.

Fix a homeomorphism f : X → Y of Riemann surfaces, and consider the set of dilatations of quasiconformal homeomorphisms X → Y in the homotopy class of f . Is the infimum of this set realized? If so, is the minimizing map unique?

Teichm ¨uller's theorems (see below) give a positive solution to both questions (under the assumption of negative Euler characteristic). The minimizing map is called the Teichmuller¨ map.

In Section 11.7 we will use Teichm ¨uller's theorems to define a metric on Teichm ¨uller space called the Teichm ¨uller metric, as follows. Let g ≥ 2 and let X, Y Teich(Sg ). The points X and Y can be represented by Riemann surfaces X and Y , each with a fixed homeomorphic identification with the fixed surface Sg (the marking). Because of the identifications, there is a unique preferred homeomorphism of Riemann surfaces X → Y , namely,

the one that corresponds to the identity map of Sg . For abstract Riemann surfaces without markings, there is no way to choose a preferred map. For

Xand Y, we can ask a refined version of Teichm ¨uller's extremal prob lem, that is, we can ask for the infimum of the dilatations of quasic onformal homeomorphisms X → Y in the preferred homotopy class. Teichm ¨uller's theorems say that there exists a unique quasiconformal homeomorphism h :

X→ Y of minimal dilatation among all maps X → Y in the preferred homotopy class. We can then define a distance function

1

dTeich (X, Y) = 2 log(Kh).

In Section 11.7 we will prove that dTeich is a metric on Teich(Sg ).

As we will see below, the Teichm ¨uller map is smooth outside a finite set of points in Sg , but is not smooth at all points of Sg . This is precisely why we defined the notion of quasiconformality for homeomorphis ms that are smooth outside a finite set of points. Quasiconformality can be defined for homeomorphisms with significantly weaker smoothness condi tions than we have assumed. We chose smoothness outside a finite set of poin ts since this is easier to work with and avoids technical difficulties, but it is still general enough for all of our applications.

11.1 MEASURED FOLIATIONS

We will see that Teichm ¨uller maps, the maps that appear as solutions to Teichm ¨uller's extremal problem, are homeomorphisms of a surface that stretch along one foliation of the surface and shrink along a transverse foliation. In order to make this precise, we first need to give a careful disc ussion of measured foliations.

MEASURED FOLIATIONS ON THE TORUS

Before giving the general definition of a measured foliation , we restrict our attention to the case of the torus where (as usual) the situation is much simpler. We will also explain what it means for a linear map of the torus to stretch the torus along one foliation and shrink along a another.

Let ℓ be any line through the origin in R2. The line ℓ determines a foliation

eℓ of R2 consisting of the set of all lines in R2 parallel to . Translations of

F ℓ

shows that such surfaces do not admit foliations. This can be corrected by allowing foliations with a finite number of singularities of a specific type.

Singular foliations. A singular foliation F on a closed surface S is a decomposition of S into a disjoint union of subsets of S called the leaves of F, and a finite set of points of S, called singular points of F, such that the following two conditions hold.

1.For each nonsingular point p S there is a smooth chart from a neighborhood of p to R2 that takes leaves to horizontal line segments. The transition maps between any two of these charts are smooth maps of the form (x, y) 7→(f (x, y), g(y)). In other words the transition maps take horizontal lines to horizontal lines.

2.For each singular point p S there is a smooth chart from a neighborhood of p to R2 that takes leaves to the level sets of a k–pronged saddle, k ≥ 3; see Figure 11.1.

Figure 11.1 A foliation at a singular point.

We say that a singular foliation is orientable if the leaves can be consistently oriented, that is, if each leaf can be oriented so that nearby leaves are similarly oriented. It is not hard to see that a foliation is locally orientable if and only if each of its singularities has an even number of prongs. For instance, the foliation in Figure 11.1 is not orientable in a neighborhood of the singular point. However, there do exist foliations that are locally orientable but not (globally) orientable.

The Euler–Poincar e´ formula. The following proposition gives a topological contraint on the total number of prongs at all singularities of a measured foliation.

gular foliation. Let Ps denote the number of prongs at a singular point s. Then

X

2χ(S) = (2 − Ps)

where the sum is over all singular points of the foliation.

Since Ps ≥ 3, Proposition 11.2 implies that a surface S with χ(S) > 0 cannot carry a (singular or nonsingular) foliation. Proposition 11.2 also implies that any foliation on a surface S with χ(S) = 0 must have no singular points and that any foliation on a surface S with χ(S) < 0 must have at least one singular point. Because of this, we will unambiguously use the term “foliation” for foliations that have singularities as well as for those that do not.

The Euler–Poincar´e formula is a straightforward conseque nce of the Poincar´e– Hopf formula for vector fields applied to the context of line fi elds; see [54, Expos´e 5, §1.6].

Measured foliations. As in the case of foliations on the torus, we would like to equip foliations on higher genus surfaces with a transverse measure, that is, a “length function” defined on arcs transverse to the foliation. In order to do this precisely we will need some preliminaries.

Let F be a foliation on a surface S. A smooth arc α in S is transverse to F if α misses the singular points of F and is transverse to each leaf of F at each point in its interior. Let α, β : I → S be smooth arcs transverse to F. A leaf-preserving isotopy from α to β is a map H : I × I → S such that

·H(I × {0}) = α and H(I × {1}) = β

·H(I × {t}) is transverse to F for each t [0, 1]

·H({0} × I) and H({1} × I) are each contained in a single leaf

Note that the second and third conditions imply that H({s}×I) is contained in a single leaf for any s [0, 1].

A transverse measure µ on a foliation F is a function that assigns a positive real number to each smooth arc transverse to F, so that µ is invariant under

leaf-preserving isotopy, and µ is regular (i.e. absolutely continuous) with respect to Lebesgue measure. In other words, this last condition means that each point of S has a neighborhood U and a smooth chart U → R2 so that the measure µ is induced by |dy| on R2.

A measured foliation (F, µ) on a surface S is a foliation F of S equipped with a transverse measure µ.

Figure 11.2 Two transverse foliations near a singular point.

We say that two measured foliations are transverse if their leaves are transverse away from the singularities; see Figure 11.2. Note that transverse measured foliations must have the same set of singularities.

Natural charts. There is another way of defining a measured foliation on a surface S. Let {pi} be a finite set of points in S. Suppose we have an atlas for S − {pi} where all transition maps are of the form

(x, y) 7→(f (x, y), c ± y),

for some constant c depending on the transition map. Then it makes sense to pull back the horizontal foliation of R2, with its transverse measure |dy| (the absolute variation in the y–direction). After reinserting the pi, the result is a measured foliation on S.

Conversely, given any measured foliation, one can construct an atlas where the transition maps are given as above and where the transverse measure is given by |dy|. Any chart from such an atlas is called a set of natural coordinates for the measured foliation.

If we have an ordered pair of transverse measured foliations, and there is an atlas where, away from the singular points, the first foliati on is the pullback

of the horizontal foliation of R2 with the measure |dy| and the second foliation is the pullback of the vertical foliation with the measure |dx|, then we say that this atlas, and each of its charts, is natural with respect to the pair of measured foliations.

The action of Homeo(S). There is a natural action of Homeo(S) on the set of measured foliations of S. Namely, if φ Homeo(S) and if (F, µ) is a measured foliation of S, then the action of φ on (F, µ) is given by

φ · (F, µ) = (φ(F), φ (µ))

where φ (µ)(γ) is defined as µ(φ−1(γ)) for any arc γ transverse to φ(F). As a consequence, the mapping class group Mod(S) acts on the set of isotopy classes of measured foliations (the quotient of the set of measured foliations by Homeo0(S)).

Measured foliations as 1–forms. Any locally-orientable measured foliation (F, µ) can be described locally in terms of a closed 1–form, as follo ws. In any chart where F is orientable, there is a closed real-valued 1–form ω so that, away from the singular points of F, the leaves of F are precisely the integral submanifolds of the distribution given by the kernel of ω, and µ is given by the formula

Z

µ(γ) = ω

γ

for any arc γ transverse to F. One only needs to check this statement for neighborhoods of the two types of points of F, nonsingular and singular. A measured foliation is given by the kernel of a globally define d 1–form if and only if it is orientable.

FOUR CONSTRUCTIONS OF MEASURED FOLIATIONS

In this subsection we give four concrete ways of constructing measured foliations on a closed surface.

From a polygon. Given any closed surface S, we can realize S as the quotient of a polygon P in R2 by side identifications. We are using the Euclidean plane here and not the hyperbolic plane because we want to consider structures inherited from Euclidean geometry. We impose two additional conditions: (i) any time two edges of P are identified, they are parallel, and (ii) the total Euclidean angle around each point of S is greater than π (the second condition only needs to be checked at the vertices of P ). We do not