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

that is, if f ↔ [Φ] and X ↔ [ρ], then

f· X ↔ [Φ] · [ρ].

12.2MODULI SPACE OF THE TORUS

The moduli space of the torus, M(T 2) is also known as the modular surface.

We saw in Section 10.1 that Teich(T 2) can be identified with the familiar

sition 10.1. For any X Teich(T ) and any f Mod(T ), we have

η(f · X) = σ(f ) · η(X).

In other words, the proposition states that η semiconjugates the action of f Mod(T 2) on Teich(T 2) to the action of σ(f ) SL(2, Z) on H2.

Proof. It is enough to check the statement of the proposition on a set of generators of Mod(T 2), say

Say that α and β are based loops in T 2 representing generators for π1(T 2)

with ˆ (this makes sense if we identify and with their images i(α, β) = 1 α β

in H1(T 2; Z)). The isomorphism of Theorem 2.5 identifies M with the mapping class Tα−1 (thinking of α as an unoriented simple closed curve), and N with the order 4 mapping class (TαTβ Tα)−1; the latter can be described by cutting T 2 along α and β, rotating the square by π/2, and regluing.

Given [(X, φ)] Teich(T 2), we identify it with a marked lattice C ≈ R2, with the vector 1 corresponding to the oriented curve α, and some complex number τ in the upper-half plane corresponding to β (see the proof of Proposition 10.1). We know that

Tα−1 · [(X, φ)] = [(X, φ ◦ Tα)]

(where we appropriately regard Tα as either a mapping class or a homeomorphism), and, using the formula Tφ(α) = φ ◦ Tα ◦ φ−1 (Fact 3.6), we have

(φ ◦ Tα)(β) Tφ(α)(φ(β)) (φ ◦ Tα)(α) φ(α)

where denotes the isotopy relation. In other words, the effect of Tα−1 on the marked lattice is to keep 1 fixed, and to send τ to τ + 1. But this exactly describes the action of the corresponding M ¨obius transformation z 7→z + 1 on H2, and that is what we wanted to show.

By similar reasoning, the mapping class associated to N acts on the marked lattice (1, τ ) by sending it to the marked lattice (−τ, 1). To get the induced action on H2 we need to put the latter into “standard form” (rotate/flip so the first complex number is 1). If we write τ = reiθ, then the resulting lattice corresponds to

We thus have

M(S1) = Teich(T 2)/ Mod(T 2) ≈ H2/ SL(2, Z),

where the action is given by Proposition 12.1. The kernel of the SL(2, Z)

action on H2 is {±I} = Z(SL(2, Z)), and so M(S1) can also be written as

H2/ PSL(2, Z).

A fundamental domain for the SL(2, Z) action on H2 is shown in Figure 12.1. It is easy to see that the indicated region contains a fundamental domain: given τ H2, apply N (at most) once to get that |τ | ≥ 1, and then apply powers of M so that |Re(τ )| ≤ 1/2. Then, simply by writing down the equations defining the region, we find that there are no oth er identifications of interior points, so that it is indeed a fundamental domain.

Figure 12.1 The fundamental domain for the modular surface.

As in the higher genus case, fixed points of the action of Mod(T 2) on Teich(T 2) correspond to isotopy classes of isometries, or, finite orde r elements of SL(2, Z). Recall from Section 7.1 that, up to powers, there are only two conjugacy classes of finite order elements of SL(2, Z). The first is that of the matrix N , which fixes the point i, and rotates by an angle of π, thus identifying the two “halves” of the circular boundary o f the fundamental domain. This fixed point corresponds to the isometry of th e square torus obtained by rotating the square by an angle π/2. The second conjugacy

whose class in PSL(2, Z) has order 3 and whose unique fixed point in H2 is the point eiπ/3. This fixed point corresponds to the order 3 symmetry of the hexagonal torus (the relationship between the point eiπ/3 and the hexagonal torus is explained in Figure 12.2).

We can also see how SL(2, Z) identifies the sides of its fundamental domain: the left side is glued to the right side by the translation z 7→z + 1 corresponding to M , and the two halves of the bottom side are identified by a rotation of angle π about i, corresponding to N . Therefore, topologically, M(S1) is a punctured sphere. Taking into account the fixed points, w e see that M(S1) has the structure of an orbifold with signature (0; 2, 3, ∞), where ∞ signifies the puncture. That is, we can think of M(S1) as a punctured sphere with cone points of order 2 and 3.

We can glean from the above discussion the important fact that the action of Mod(T 2) on Teich(T 2) is properly discontinuous. One of the main results in this chapter is the analogous result for higher genus surfaces. By the the-

Figure 12.2 The cut and paste operation from a regular hexagon to the parallelogram spanned by 1 and eiπ/3.

ory of orbifolds, the orbifold fundamental group π1orb(M(S1)) of M(S1)

is isomorphic to the group of covering transformations, namely PSL(2, Z). By the Van Kampen theorem for orbifolds, π1orb(M(S1)) is generated by

two loops, one around each cone point. Further, these generators have order 2 and 3, respectively, and there are no other relations. Thus, we have recovered the classical fact that PSL(2, Z) ≈ Z/2Z Z/3Z.

As is true in higher genus, M(S1) encodes oriented isometry classes of marked tori/lattices. Note that i + ǫ and i − ǫ (for ǫ R) correspond to isometric tori that are not oriented isometric. A fundamental domain for the quotient of Teich(T 2) by Mod±(T 2) would be, say, the left half of the fundamental domain for Mod(T 2).

12.3 PROPER DISCONTINUITY OF THE MAPPING CLASS GROUP AC-

¨

TION ON TEICHMULLER SPACE

Recall that the action of a group G on a topological space X is properly discontinuous if, for any compact set B X the set

{g G : g · B ∩ B 6= }

is finite. The main goal of this section is to prove the followi ng.

THEOREM 12.2 (Fricke) Let g ≥ 1. The action of Mod(Sg ) on Teich(Sg ) is properly discontinuous.

Theorem 12.2 (and its proof) applies to all hyperbolic surfaces. We make comments on the required modifications at the end of the proof .

Together with various properties of Mod(S) and its action on Teich(S), we can use Theorem 12.2 to deduce several basic properties of M(S).

Corollary 12.3 Let g ≥ 1. The quotient M(Sg ) is an aspherical orbifold, and is finitely covered by a manifold.

Proof. Any quotient of a manifold by a properly discontinuous action is an orbifold. As Teich(S) is contractible (Theorem 10.5), it follows from Theorem 12.2 that M(S) is an aspherical orbifold.

Recall that Mod(S) has a torsion-free subgroup of finite index (Corollary 8.9). Since the fixed points of the Mod(S) action on Teich(Sg ) all correspond to finite order elements of Mod(S), it follows that acts freely

It is in fact known that M(S) is a real-analytic orbifold, and that it has the structure of a complex orbifold (indeed it is a quasiprojective variety).

LENGTH SPECTRA OF HYPERBOLIC SURFACES

The proof of Fricke's theorem relies on three lemmas regarding the lengths of curves on a hyperbolic surface. In the end, Theorem 12.2 will come down to discreteness of the raw length spectrum for a point X in Teich(S), which is the set of positive real numbers

rls(X) = {ℓX(c)}

where c ranges over all isotopy classes of essential (or peripheral) simple closed curves in S (recall that ℓX(c) is the length of the X–geodesic in the class c).

Lemma 12.4 Let X Teich(Sg ). The set rls(X) is a closed, discrete subset of R. What is more, the set

{c : c an isotopy class of simple closed curves in Sg with ℓX(c) ≤ L}

is finite for any L.

Proof. Say that X is represented by (Sg , µ), where µ is a hyperbolic metric on Sg . Let p be some fixed basepoint for π1(Sg ), and let pe be some lift of p to the universal cover H2 of Sg . Fix some R > 0. Since the action of

In particular, for fixed R both sets are finite. This is almost the statement of the lemma, except that we want to replace based loops with unbased loops.

Since Sg is compact, the diameter D of (Sg , µ) is finite. It follows that any (unbased) simple closed curve in Sg of length L is homotopic to a closed curve in Sg that is based at p and that has length L+2D. Setting R = L+2D

The marked length spectrum for a point X in Teich(S) is the set of pairs (c, ℓX(c)), where c is an isotopy class of simple closed curves in S. We can think of the next lemma as saying that the marked length spectrum for a point in Teich(S)—in fact just a finite part of it—determines that point in

Teich(S).

There exists a collection of isotopy classes of simple closed } in Sg so that the map from Teich(Sg ) to R9g−9 given

X 7→ℓ{X(c1), . . . , ℓX(c9g−9)}

Proof. Let {γ1, . . . , γ3g−3} be a pants decomposition of Sg , and choose simple closed curves {β1, . . . , β3g−3} in Sg so that i(βi, γi) > 0 and i(βi, γj ) = 0 for i 6= j. We do not require the βi to be disjoint. Let αi = Tγi (βi).

Choose a set of Fenchel–Nielsen coordinates for Teich(Sg ) where the coordinate system of curves consists of the pants decomposition {γi} and any set

of seams. For X Teich(Sg ), we will show that the set {ℓX(αi), ℓX(βi), ℓX(γi)} determines the Fenchel–Nielsen coordinates of X.

The length parameters for X are exactly the ℓX(γi). It therefore remains to show that the twist parameters for X are uniquely determined by the {ℓX(αi), ℓX(βi), ℓX(γi)}. Let Xt be the point of Teich(S) with the same length parameters as X and with twist parameters t = (t1, . . . , t3g−3). Up

to a reparametrization of Teich(Sg ), we can assume that X = X0. We will

show that if ti 6= 0 for some i then either ℓXt (αi) 6= ℓX(αi) or ℓXt (βi) 6=

ℓX(βi).

Consider the functions A(t) = ℓXt (α1) and B(t) = ℓXt (β1). Since i(α1, γj ) = i(β1, γj ) = 0 for j 6= 1, both functions are simply functions of the param-

eter t1, which we denote by s. By Proposition 10.7, A(s) and B(s) are strictly convex, hence so is their sum (A + B)(s). Also, by definition, we have that A(s + 2π) = B(s).

Assume A(s) = A(0) for some s 6= 0. We will show that B(s) 6= B(0), i.e., A(2π) 6= A(2π + s). For concreteness say s > 0. Since A(s) = A(0), it follows from the strict convexity that A(t) < A(0) for t (0, s) and that A(t) is strictly increasing for t > s. If s < 2π, then s < 2π < 2π + s, and it follows that A(2π) < A(2π + s). If s > 2π, then 0 < 2π < s < 2π + s, so A(2π) < A(0) = A(s) < A(2π + s). Finally, if s = 2π, we certainly cannot have A(2π) = A(2π + s), for then A(t) would take the same value at 0, 2π, and 4π, violating strict convexity.

We have shown that if t = (t1, . . . , t3g−3) and ℓXt (α1) = ℓX(α1) and ℓXt (β1) = ℓX(β1) then t1 = 0. Since the same argument works for the

In light of Lemma 12.5, one might wonder if the raw length spectrum determines a point in M(S). It is a theorem of Vign´eras, however, that this is not the case, and in fact for certain S, Sunada proved that there is a postive dimensional subset of M(S) consisting of points with the same raw length spectrum [161].

The next lemma, due to Wolpert [174], gives the basic fact that any K– quasiconformal map distorts hyperbolic lengths of closed curves by a factor of at most K.

Lemma 12.6 (Wolpert's lemma) Let φ : X1 → X2 be a K–quasiconformal homeomorphism between hyperbolic surfaces X1 and X2. For any isotopy class c of simple closed curves in X1, we have

ℓX1 (c) ≤ ℓX2 (φ(c)) ≤ KℓX1 (c). K

We can rephrase Wolpert's lemma as follows. Let X1, X2 Teich(S) and suppose that dTeich(X1, X2) ≤ log(K)/2. For any isotopy class c of simple

+ H2 e ≈ e ≈ H2

Proof. Let γ1, γ2 Isom ( ) be isometries of X1 X2 corresponding to c and φ(c), respectively, and consider the annuli A1 and A2 obtained by taking the quotient of H2 by hγ1i ≈ Z and hγ2i ≈ Z, respectively. Since the map π1(Xi) → Isom+(H2) is only well-defined up to conjugacy in PSL(2, R), we can take γ1 to be the map z 7→eℓX1 (c)z and γ2

z, where we think of H2 as the upper-half plane in C.

We can put the annuli A1 and A2 in a standard form; that is, for each i, we can find the unique (open) Euclidean annulus Ami of circumference 1 and height mi, so that Ai is conformally equivalent to Ami . We call the number mi the modulus of the annulus Ai. To find the standard form of A1, note that we can choose a branch of the natural logarithm that takes the upper half-plane H2 to the infinite strip of points in C with imaginary part in (0, π). Under this identification, the group hγ1i corresponds to the infinite cyclic group of translations generated by z 7→z + ℓX1 (c). Since the natural logarithm is a conformal map, A1 is conformally equivalent to the annulus obtained by identifying vertical sides of a rectangle whose width is ℓX1 (c) and whose height is π; therefore, the modulus m1 is equal to π/ℓX1 (c). Similarly, m2 = π/ℓX2 (φ(c)).

The map φ lifts to a K–quasiconformal mapping

e→

φ: A1 A2.

Note that since hγii < π1(Xi), this is formally weaker than saying that φ is a K–quasiconformal map from X1 = H2/π1(X1) to X2 = H2/π1(X2). The

solution to Gr¨otzsch's problem can be modified (slightly) t o prove that e

φ

changes the modulus by at most a multiplicative factor of K (in the solution to Gr¨otzsch's problem, think of thex–direction as the S1 direction). In other words, we have

1

K m2 ≤ m1 ≤ Km2.

FINISHING THE PROOF OF PROPER DISCONTINUITY

We now use Lemmas 12.4, 12.5, and 12.6 to prove the following theorem due to McKean [120, 121]. The proof we give is due to Wolpert [174]. Theorem 12.2 will follow rather directly.

THEOREM 12.7 Let X Teich(Sg ), and let BR(X) be the closed Teichmuller¨ R–ball about X. Then the set

{Y BR(X) : rls(X) = rls(Y)} is finite for any fixed R ≥ 0.

Since the Teichm ¨uller metric is complete, Theorem 12.7 is equivalent to the statement that the set

{Y Teich(Sg ) : rls(X) = rls(Y)}

is discrete.

Proof. As per Lemma 12.5, let {γ1, . . . , γ9g−9} be simple closed curves in Sg whose lengths specify a point in Teich(Sg ). By Lemma 12.6, for any

Y BR(X) we have ℓY(γi) ≤ KL for each i, where L = max{ℓX(γi)} and K = e2R. But if rls(Y) = rls(X) then ℓY(γi) rls(X). Lemma 12.4 says

that there are only finitely many points in rls(X) ∩ (0, KL]. Therefore, for any Y BR(X) with rls(Y) = rls(X) there are only finitely many choices for the values ℓY(γi), and hence finitely many such Y, by Lemma 12.5. 2

We are finally ready to demonstrate the proper discontinuity of the action of

Mod(Sg ) on Teich(Sg ).

Proof of Theorem 12.2. Let B be a fixed compact subset of Teich(Sg ), and let

A = {f Mod(Sg ) : f (B) ∩ B 6= }.

Our goal is to show that A is finite. Since any compact set in a metric space is contained in some closed ball, and since making B bigger makes A bigger, we may assume that B is equal to BR(X), the closed ball of radius

R about X Teich(Sg ).

360 CHAPTER 12

The union

[

f (B)

f A

is contained in B3R(X). For each f A (indeed for each f Mod(Sg )), we have rls(f · X) = rls(X). Thus by Theorem 12.7 applied to B3R(X), we have that the set {f · X : f A} is finite. But StabMod(Sg )(f · X) is finite

Punctures and boundary. Theorem 12.2 extends with little difficulty to the case of surfaces with punctures and/or boundary. Lemma 12.4 needs a slight modification in the noncompact case—delete disjoint horoball neighborhoods of the cusps and note that any essential geodesic must enter the complement. For Lemma 12.5, the only difference is that the constant 9g−9 must be adjusted.

The cohomology of moduli space. From the proper discontinuity of the Mod(Sg ) action on Teich(Sg ) (Theorem 12.2) and the contractibility of Teich(Sg ) (Theorem 10.5), we know that there is a continuous map

BDi +(Sg ) → M(Sg )

that induces the isomorphisms

H (M(Sg ); Q) ≈ H (BDi +(Sg ); Q) ≈ H (Mod(Sg ); Q).

In Section 6.6 we described elements of H (Mod(Sg ); Z) as characteristic classes of surface bundles. Thus the cohomology of moduli space is directly related to the theory of surface bundles.

A universal property for moduli space. Recall from Corollary 12.3 that

g has a finite manifold cover c g whose fundamental group is

M(S ) M(S ) is torsion free. It follows that, for any (Hausdorff, paracompact) space B, there is a bijection:

Recall from Section 6.6 that the set of conjugacy classes of representations π1(B) → can be identified with a subset of the set of isomorphism class es of oriented Sg –bundles over B.