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

u U we have q(u) = reiθ. Then Ω(q) Teich(Sg ) is obtained from X

Beltrami differentials versus quadratic differentials. Let X be a Riemann surface representing a point X Teich(Sg ). We already discussed that correspondence between QD(X) and the cotangent space to Teich(Sg ) at X. There is a natural pairing between a quadratic differentials and Beltrami differentials, that we can use to identify the tangent space of Teich(Sg ). Specifically, if a holomorphic quadratic differential q is given locally by φ(z) dz2 and a Beltrami differential µ is given locally by µ(z) dz/dz then

we set

Z

hq, µi = φµ |dz|.

X

This pairing allows us identify the tangent space to Teich(Sg ) at X as the space of Beltrami differentials on X modulo the subspace of “infinitesimally trivial” Beltrami differentials (both spaces are infinite d imensional, but the quotient has dimension 6g − 6). The infinitesimally trivial Beltrami differentials are the ones that are the derivatives of homeomorphisms of Sg ≈ X that are homotopic to the identity.

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11.7 THE TEICHMULLER METRIC

Let X, Y Teich(Sg ), and say that X and Y are represented by marked Riemann surfaces X and Y , respectively. Again, because of the fixed identifications of X and Y with Sg , it makes sense to say that a map X → Y is homotopic to the identity. Let f : X → Y be a Teichm ¨uller mapping homotopic to the identity, whose existence is guaranteed by Teichm ¨uller's existence theorem (Theorem 11.6). Let K = Kf be the dilatation of f . We define the Teichmuller¨ distance between X and Y to be

1

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

By Teichm ¨uller's uniqueness theorem (Theorem 11.7) the function dTeich is well-defined.

For the next proposition, recall that in Section 10.2 we defin ed a topology on Teich(Sg ), the algebraic topology.

Proposition 11.15 The Teichmuller¨ distance dTeich defines a complete metric on Teich(Sg ) whose topology is compatible with the algebraic topology on Teich(Sg ).

The metric defined by dTeich is called the Teichmuller¨ metric.

Proof. Teichm ¨uller's existence theorem (Theorem 11.6) implies that dTeich(X, Y) = 0 if and only if there is a Teichm ¨uller mapping h : X → Y of dilata-

tion 1 that is homotopic to the identity. By Lemma 11.1 the homeomorphism h is conformal. This is the same as saying that X = Y in Teich(Sg ). The fact that the inverse of a K–quasiconformal homeomorphism is a K– quasiconformal mapping gives that dTeich(X, Y) = dTeich(Y, X). The triangle inequality for dTeich comes from the elementary fact that the composition of a K–quasiconformal homeomorphism and a K′–quasiconformal homeomorphism is a KK′–quasiconformal homeomorphism. Thus dTeich is a metric.

Next we show completeness of (Teich(Sg ), dTeich ). Let X be a point of Teich(Sg ) represented by a marked Riemann surface X. Recall that in Section 11.6 we defined a map

Ω : QD1(X) → Teich(Sg )

and showed it was a homeomorphism. Under Ω−1, a point in Teich(Sg ) at distance log(K)/2 from the basepoint X maps to a point of QD1(X) whose norm is (K − 1)/(K + 1). For K ≥ 1, we have an inequality

(K − 1)/(K + 1) ≤ 12 log(K).

If K is bounded from above then (K −1)/(K + 1) is bounded away from 1. Thus Ω−1 takes closed balls about the basepoint X Teich(Sg ) to compact balls about the origin in QD1(X). Since Ω−1 is a homeomorphism, this implies that closed balls about the basepoint in Teich(Sg ) are compact, and

tor of 1/2 is included so that certain 2–dimensional subspaces of Teich(Sg ),

called “Teichm ¨uller disks,” are isometric to the hyperbol ic plane H2. Briefly, a Teichmuller¨ disk is obtained as follows: Start with a complex structure on a surface S coming from a polygon with parallel sides identified, e.g., t he Swiss cross example. If we apply an element of SL(2, R), acting as a linear transformation of R2, the images of the sides of the polygon are still parallel, and so we obtain a new complex structure on S. The stabilizer of a complex structure is the orthogonal group. Since SL(2, R)/SO(2, R) ≈ H2, it follows that the SL(2, R)–orbit of the original marked complex structure is

a copy of H2 in Teich(Sg ). It turns out that this inclusion H2 ֒→ Teich(Sg ) is an isometric embedding.

TEICHMULLER¨ GEODESICS

In Section 11.3 we explained how any point X Teich(Sg ) and any holomorphic quadratic differential q QD(X) determine an embedded copy of R ֒→ Teich(Sg ) containing X, called a Teichmuller¨ line. By the definition of dTeich, this embedding is actually an isometric embedding.

Proposition 11.16 Let g ≥ 1. Teichmuller¨ lines in Teich(Sg ) are bi-infinite geodesics with respect to the Teichmuller¨ metric.

Even more is true: Teichm ¨uller lines account for all geodesics in (Teich(Sg ), dTeich).

THEOREM 11.17 Let g ≥ 1. Every geodesic segment in (Teich(Sg ), dTeich) is a subsegment of some Teichmuller¨ line. In particular, there is a unique geodesic in Teich(Sg ) between any two points.

Proof. Let X and Z be points of Teich(Sg ) and suppose Y Teich(Sg ) satisfies

d(X, Y) + d(Y, Z) = d(X, Z).

In other words, if KXY, KYZ, and KXZ are the stretch factors of the corresponding Teichm ¨uller maps, we have

log(KXYKYZ) = log(KXY) + log(KYZ) = log(KXZ),

and so

KXYKYZ = KXZ.

The composition of the two Teichm ¨uller maps

f : X → Y → Z

has dilatation at most KXYKYZ (this again is simply the general fact that the composition of a K–quasiconformal homeomorphism with a K′–quasiconformal homeomorphism is a KK′–quasiconformal homeomorphism). Since KXYKYZ = KXZ, Teichm ¨uller's uniqueness theorem (Theorem 11.7) gives that f must

in fact be the Teichm ¨uller map X → Z.

It now remains to prove that if the composition of two Teichmuller¨ maps X → X′ → X′′, of dilatation K and K′, respectively, is another Teichm ¨uller map of dilatation KK′, then it must be that X, X′, and X′′ lie on a Teichm ¨uller line. This is not hard to see: if at any point the terminal foliation for the first map is not aligned with the initial foliation for the second map, then the dilatation of the composition will be less than KK′.

The second statement of the theorem is an immediate consequence of the

THE TEICHMULLER¨ METRIC FOR THE TORUS

Some intuition for the Teichm ¨uller metric and Teichm ¨uller geodesics can be gleaned from understanding them in the special case when g = 1, that is for Teich(T 2). The situation in this case is simple enough that it can be worked out explicitly.

Recall that we exhibited a bijection between Teich(T 2) and H2 (Proposition 10.1). We now give a significant strengthening of this fa ct.

Theorem 11.18 We have

The bijection H2 → Teich(T 2) from Proposition 10.1 induces an isometry

(H2, dH2 ) → (Teich(T 2), 2 dTeich).

Proof. First, the action of SL(2, R) on equivalence classes of marked lattices in R2 is the same as the action of SL(2, R) on Teich(T 2) = H2 via M ¨obius transformations. Indeed, SL(2, R) is generated by matrices of the form

where t R. Therefore this is a slight generalization of the fact that Mod(T 2) acts on Teich(T 2) by M ¨obius transformations (see the proof of Proposition 12.1).

Let X, Y Teich(T 2) and let x and y be the corresponding points in H2. Say that dH2 (x, y) = δ. Let A be an element of SL(2, R) that corresponds to the hyperbolic element of Isom+(H2) with axis passing through x and y and with translation distance δ. The matrix A is unique up to sign. We can

for some C SL(2, R). As above, we can also think of A as acting by a linear transformation on R2. What is more, if X and Y are represented by a marked flat tori X and Y then A can be regarded as a map X → Y . Indeed, if we represent X and Y by marked lattices in R2, and the action of A on R2 takes any X–lattice to some Y–lattice.

Since A has two real eigenvalues, there are two 1–dimensional eigen spaces on X along which A expands and contracts. Thus, A is a Teichm ¨uller map from X to Y . The holomorphic quadratic differential corresponding to this Teichm ¨uller map is the one whose horizontal foliation lies in the direction of the eigenspace for A corresponding to the leading eigenvalue.

As an isometry of H2, the matrix A is a hyperbolic isometry that translates along its axis a distance δ in the hyperbolic metric d2H. On the other hand, the dilatation of the action of A as a map X → Y is eδ , so that

dTeich(X, Y) = 12 log(eδ ) = δ/2.

It follows that the Teichm ¨uller metric on H2 ≈ Teich(T 2) is the hyperbolic

11.8 SURFACES WITH PUNCTURES AND BOUNDARY

In this chapter we have explained the basic theory of quasiconformal homeomorphisms, measured foliations, holomorphic quadratic differentials, Teichm ¨uller maps and the Teichm ¨uller metric. We assumed throughout that the surfaces in question are closed. Most of this theory carries over easily

to the case of surfaces with punctures, with some modificatio ns required. In this section we highlight some of these necessary changes.

Near a puncture, a foliation can take the form of a regular point or a k– pronged singularity with k ≥ 3, as in the case of foliations on closed surfaces. However, at a puncture we also allow 1–pronged singularities as in Figure 11.8.

Figure 11.8 A 1–pronged singularity on a surface with a punct ure.

Likewise, in the neighborhood of a puncture, holomorphic quadratic differentials must be of the form zk dz2, where k ≥ −1. That is, we allow simple poles at punctures.

Figure 11.9 Two different conformal types in the neighborhood of a puncture: a punctured disk (left) and an annulus (right).

In order to state Teichm ¨uller's existence and uniqueness theorems (Theorem 11.6 and Theorem 11.7) in this context, we need to distinguish two types of conformal structures near a topological puncture; see Figure 11.9. The puncture on the left hand side of the figure has a neighborh ood that is conformally equivalent to the unit disk in C minus 0, and the other puncture has a neighborhood that is conformally equivalent to an annulus {z C : r1 ≤ |z| ≤ r2}. A homeomorphism f : X → Y has a quasiconformal representative if and only if f takes punctures to punctures, and at each puncture these conformal types are preserved. In this case, both Teichm ¨uller existence and uniqueness theorems hold, assuming that the underlying topological surface has negative Euler characteristic. One way to prove this in the case of punctures of the first type is to find a double coveri ng of the Riemann surface in question where the complex structure can be extended over the punctures.

Since there is no quasiconformal homeomorphism homotopic to f : X → Y if the types of punctures are not preserved, one usually considers the Teichm ¨uller space of a surface where the conformal types of the punctures are part of the given data.

Boundary components are less convenient in Teichm ¨uller theory. However, we can extend most of the definitions and theorems. A foliatio n F on a surface S with nonempty boundary ∂S is locally the same as in the case when S is closed, but we now insist that:

1.each component of ∂S contain at least one singularity of F, and

2.any leaf of F not containing a singularity on ∂S but meeting a small tubular neighborhood of ∂S must be parallel to ∂S; that is, ∂S should be a union of leaves connecting singularities. See Figure 11.10.

Quadratic differentials on a punctured surface X are similarly constrained: each component of ∂X must contain at least one zero, and ∂X is part of the horizontal foliation away from the singularities. As a consequence, we see that a pair of foliations can be transverse on the interior of the surface, but must be parallel on at the boundary of X. So, for instance, the singular Euclidean metric degenerates at the boundary.

Figure 11.10 A measured foliation near the boundary of a surface.

For details on extending Teichm ¨uller theory to the non-closed case, see, for example, [1, Ch. II].

Chapter Twelve

Moduli space

Moduli space is one of the fundamental objects of mathematics. It is ubiquitous, appearing as a basic object in fields from low-dimens ional topology to algebraic geometry to mathematical physics. The moduli space M(S) parametrizes, among other things, isometry classes of hyperbolic structures on S, conformal classes of Riemannian metrics on S, complex structures on S, and smooth algebraic structures on S.

Moduli space is defined to be the quotient of Teichm ¨uller spa ce by the action of the mapping class group. The main result in this chapter is that this action is properly discontinuous. This is a theorem of Fricke, which in particular implies that moduli space has the structure of a smooth orbifold.

Moduli space is not compact. We will prove Mumford's compactness criterion, which explains what it means to “go to infinity” in modul i space. We will also see that M(S) has only one end, i.e., it is connected at infinity.

In the next chapter we will use Fricke's theorem and Mumford's compactness criterion in order to prove the Nielsen–Thurston class ification for mapping classes.

One primary reason for the importance of M(S) is that it plays a fundamental role in the classification of surface bundles. In this chapter we will also explain the connection between the cohomology of moduli space and characteristic classes of surface bundles.

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12.1 MODULI SPACE AS A QUOTIENT OF TEICHMULLER SPACE

The action of Mod(S) on Teich(S) is defined as follows. Let X = [(Sg , µ)] be a point of Teich(S) where µ HypMet(S), let ψ Di +(S), and let

Suppose h lies in the kernel. If ψ is a representative of h, then ψ is isotopic to an isometry of S with respect to every hyperbolic metric on S. It follows that h fixes the length of every isotopy class of simple closed close d curve in S. Since for any pair of distinct isotopy classes of essential simple closed curve in S one can find a metric on S where their lengths are different, it follows that h in fact fixes the isotopy class of each simple closed curve in S. Conversely, it follows from Lemma 12.5 below (and its extension to the non-closed case) that if h Mod±(S) fixes the isotopy class of every simple closed curve in S then h lies in the kernel.

The geometric point of view. It follows easily from the definition of the Teichm ¨uller metric that the action of the extended mapping class group on Teichm ¨uller space is an isometric action. That is, we have a map

Υ : Mod±(S) → Isom(Teich(S)).

It is a theorem of Royden that, aside from the cases of S1,1 and S0,4 the map Υ is actually surjective [148]. In other words, Teichm ¨uller space has no other isometries besides the ones coming from the extended mapping class group. In the previous paragraph we described the kernel of Υ. In particular, for g ≥ 3, we have Mod±(Sg ) ≈ Isom(Teich(Sg )). For g = 2, we have Mod±(S2)/Z(Mod(S2)) ≈ Isom(Teich(S2)). Finally, for g = 1, since Teich(T 2) is isometric to H2 (up to scale), it follows that

Mod±(T 2)/ ker Υ ≈ PGL(2, Z) has infinite index in Isom(Teich(T 2)) ≈ PGL(2, R).

Stabilizers of points. Let S be a hyperbolic surface, let X Teich(S), and let h Mod(S). It follows from the definitions that h · X = X if and only if h has a representative that is an isometry of S wth respect to some hyperbolic metric. In particular h is finite order (Proposition 7.7), and so the failure of Mod(S) to act freely on Teich(S) is exactly due to torsion.

The algebraic point of view. Recall that the Dehn–Nielsen–Baer theorem states that Mod±(Sg ) is isomorphic to Out(π1(Sg )). There is a natural action

Out(π1(Sg )) DF(π1(Sg ), PSL(2, R))/ PSL(2, R).

Namely, given Φ Aut(π1(Sg )) and ρ DF(π1(Sg ), PSL(2, R)), we have

[Φ] · [ρ] = [ρ ◦ Φ−1].

This action coincides with the action

Mod±(S) Teich(Sg ) ≈ DF(π1(Sg ), PSL(2, R))/ PSL(2, R),