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

In other words, every element of Mod(S) has a power that is obtained by decomposing the surface along disjoint curves, doing Dehn twists along some of those curves, and doing pseudo-Anosov mapping classes on some of the complementary components. Ivanov showed that the power needed (i.e. the integer k in Corollary 13.4) is 1 in the case of the Torelli group, or, in fact, any congruence subgroup Mod(S)[m] with m ≥ 3; see [87, Corollary 1.8].

In general, the collection of curves {ci} in the statement of Corollary 13.4 can be taken to be the canonical reduction system for f (described above).

AN APPLICATION TO MAPPING TORI

We now explain a connection between the Nielsen–Thurston Cl assification and surface bundles over the circle. The mapping torus of f Mod(S) is defined as

S × [0, 1]

Mf = (x, 0) (φ(x), 1) ,

where φ is a homeomorphism that represents f . The homeomorphism type of the resulting manifold Mf is independent of the choice of representative φ. We have the following facts:

·Every surface bundle over S1 is a mapping torus.

·π1(Mf ) = π1(S) f Z.

·If f1 and f2 are conjugate then Mf1 is homeomorphic to

Mf2 .

Different conjugacy classes of mapping classes can give homeomorphic mapping tori. In fact, there are examples of 3-manifolds that fiber in infinitely many different ways, and indeed they are homeomorph ic to mapping tori where the genera of the fibers are arbitrarily large . However, the conjugacy class of the gluing map is an important piece of information: this conjugacy class, together with the “Thurston norm” of t he 3-manifold, determines the homeomorphism type of the 3-manifold. The following theorem says that the gluing map alone actually determines the geometry that the manifold admits.

Theorem 13.5 (Thurston) Let S = Sg , where g ≥ 2. Let Mf denote the mapping torus for f Mod(S). Then

1.f is periodic Mf admits a metric locally isometric to H2 × R.

2.f is reducible Mf contains an incompressible (π1-injective) torus.

3.f is pseudo-Anosov Mf admits a hyperbolic metric.

The first two forward implications are easy: if f is periodic, then Mf is

implication follows, since no hyperbolic manifold has a fini te cover locally isometric to H2 ×R, and no hyperbolic manifold has a subgroup isomorphic to Z × Z.

The first two reverse implications are not hard. The last forw ard implication is a deep theorem of Thurston; see [164, 139].

Torus case. Every orientable torus bundle over S1 is homeomorphic to a mapping torus Mf for some f Mod(T 2). In this case, the classification of mapping tori can be refined as follows.

Theorem 13.6 Let Mf denote the mapping torus for f Mod(T 2). Then

1.φ is periodic Mf is locally isometric to Euclidean 3-space.

2.φ is reducible Mf is locally isometric to Nil geometry.

3.φ is Anosov Mf is locally isometric to Sol geometry.

Note that in each case Mf admits a locally homogeneous metric. For descriptions of Nil and Sol geometries, see [167].

13.4 PROOF OF THE CLASSIFICATION THEOREM

We start by recalling the trichotomy for elements of Isom(H2). The first approach is to consider the translation length of an element φ of Isom(H2):

τ (φ) = inf {d(x, φ(x))}.

x H2

The isometry φ is called elliptic if τ (φ) is 0 and is realized, parabolic if τ (φ) is 0 and is not realized, and hyperbolic if τ (φ) is nonzero and is realized.

We will see that Bers' proof of the Nielsen–Thurston Classifi cation is based upon the first approach to the classification for Isom(H2), and that Thurston's proof—which we touch upon in Chapter 15, but do not give in ful l—is similar to the second. The classification of elements of Isom(H2) via Jordan canonical form does not generalize to Mod(S), since, except for a few exceptional surfaces, the mapping class group is not linear in any obvious way.1

In the case of the torus, we saw that the Nielsen–Thurston tri chotomy boiled

tries and then show that these correspond to periodic, reducible, and pseudoAnosov mapping classes, respectively.

CLASSIFICATION OF ISOMETRIES

Let d(·, ·) denote the Teichm ¨uller distance. Forh Isom(Teich(S)), define the translation distance of h to be:

τ (h) = inf d(X, h · X).

X Teich(S)

Any h Isom(Teich(S)) naturally falls into one of the following three categories.

Elliptic: τ (h) is achieved and equals zero.

Parabolic: τ (h) is not achieved.

Hyperbolic: τ (h) is achieved and is positive.

Since Mod(S) acts by isometries on Teich(S) with the Teichm ¨uller metric, this gives rise to a classification of mapping classes. T he point of Bers' proof is to show that this classification is exactly the Niels en–Thurston Classification.

1The mapping class groups of S0,0, S0,1, S0,2, S0,3, S0,4, S1,0, and S1,1 are linear. Bigelow and Budney proved that Mod(S2) is linear, but the representation is far from obvious. It is still unknown if Mod(Sg ) is linear for g ≥ 3.

Thus, N1 is strictly contained in P1. Similarly, N2 is also disjoint from a, and so N1 ∩ N2 = . By symmetry, N1 and N2 are both disjoint from N3,

Note that our proof of the Collar Lemma really shows something stronger than the statement of the lemma: we found annuli of the given size that are not only embedded, but are also disjoint from each other.

c2

a

c3

d

N1

c1

Figure 13.2 The picture for the Collar Lemma.

For our proof of the Nielsen–Thurston Classification, we wil l not need the precise statement of Lemma 13.7. Rather, we will only need the following much weaker statement, sometimes attributed to Margulis.

Corollary 13.8 There is a constant δ = δ(S) such that for X Teich(S), any two closed geodesics of length less than δ are disjoint.

PROOF OF THE CLASSIFICATION

We are now ready to give Bers' proof of the Nielsen–Thurston C lassification.

Proof of Theorem 13.2. Let S = Sg,n and let f Mod(S) (on a first pass, the reader should imagine n = 0). We will show the following:

1.f is elliptic in Isom(Teich(S)) f is periodic in Mod(S).

2.f is parabolic in Isom(Teich(S)) f is reducible in Mod(S).

3.f is hyperbolic in Isom(Teich(S)) f is pseudo-Anosov in Mod(S).

Case 1. f is elliptic in Isom(Teich(S)).

In this case, there is a point X = [(X, φ)] Teich(S) with f · X = X. By the definition of Teichm ¨uller space, φ ◦ f ◦ φ−1 is isotopic to an isometry of X, and so φ ◦ f ◦ φ−1 is periodic (Proposition 7.7). It follows that f is periodic.

Case 2. f is parabolic in Isom(Teich(S)).

Let {Xn} be a sequence in Teich(S) with the property that d(Xn, f ·Xn) → τ (f ). We first show that the projection of {Xn} to M(S) leaves every compact set in M(S). Suppose to the contrary that these projections lie in a fixed compact set in M(S). Then for some choice of hi Mod(S), the sequence {Yn}, where Yn = hn · Xn, stays in a fixed compact region of

Teich(S).

By compactness, there is a subsequence of {Yn} that converges to a point Y Teich(S). Since Mod(S) acts on Teich(S) by isometries we have

d(Yn, hnf h−n 1 · Yn) = d(h−n 1 · Yn, f · (h−n 1 · Yn)) = d(Xn, f · Xn),

Claim: d(Y, hk f h−k 1 · Y) = τ (f ) for some k.

Let n be fixed. Applying the triangle inequality to the four points

Y, Yn, hnf h−n 1 · Yn, and hnf h−n 1 · Y

we obtain

d(Y, hn f h−n 1·Y) ≤ d(Y, Yn)+d(Yn, hnf h−n 1·Yn)+d(hnf h−n 1·Yn, hnf h−n 1·Y).

are at most 3g −3+n disjoint isotopy classes of simple closed curves in XM (cf. Section 4.2). Thus it must be that two of the ci are in the same homotopy class. It follows that f k(c0) = c0 for some k > 0, so that f permutes the collection of isotopy classes {c0, c1, . . . ck−1}, and so f is reducible.

Case 3. f is hyperbolic in Isom(Teich(S)).

Let X = [(X, ψ)] be a point of Teich(S) that satisfies d(X, f · X) = τ (f ), and let γ be the geodesic in Teich(S) passing through X and f (X). By Theorem 11.17, γ is unique, and it is generated by a Teichm ¨uller map in the sense of Theorem 11.17.

Recall that our goal is to show that f is pseudo-Anosov. Intuitively, it is because Teichm ¨uller maps “look like” pseudo-Anosov homeo morphisms. As we shall see, going from this idea to a real proof is surprisingly involved.

We first show that f leaves γ invariant. Let Y γ be the midpoint of the segment from X to f · X. We have

d(Y, f · Y) ≤ d(Y, f · X) + d(f · X, f · Y)

=12 d(X, f · X) + 12 d(X, f · X)

=τ (f ).

By the minimality of τ (f ), we have d(Y, f · Y) = τ (f ). It follows that f · Y lies on γ. Thus, f 2 · X lies on γ, and further f i · X lies on γ for all i Z. It follows that f (γ) = γ. The geodesic γ is called an axis for f .

X

f 2(X)

Y

f (Y)

f (X)

Figure 13.3 When f is hyperbolic and τ (f ) is realized at X, then X, f (X), and f 2(X) must be colinear.

Natural charts for φ · (F, µ) and φ · (F′, µ′) are given by

By the commutativity of the diagram above, we obtain another set of natural charts for φ · (F, µ) and φ · (F′, µ′) as follows:

The reason for this is that, in natural coordinates, fK has the form AK . We also used the fact that h identifies Q0 with Q1 so that we could interpret the commutative diagram on the level of natural coordinates. We have just given two natural charts for the same pair of foliations. Any transition map is a translation possibly composed with negation. Since the natural coordinates for a quadratic differential are only well-defined up t o translation and negation, we may assume without loss of generality that, after appropriately restricting to overlapping charts, we have

ζj ◦ φ−1 = AK ◦ ζi

and so

φ−1 = ζj−1 ◦ AK ◦ ζi.

This is exactly saying that φ−1 locally looks like a pseudo-Anosov homeomorphism, and so f is a pseudo-Anosov mapping class.

Therefore, in the end, we see that (F, µ) is the stable foliation for f , and (F′, µ′) is the unstable foliation for f .

Finally, we prove the exclusivity statement of the theorem, namely that pseudo-Anosov mapping classes are neither periodic nor reducible. As part of Theorem 14.20 below, we will prove that, if f Mod(S) is pseudoAnosov, and α is a simple closed curve in S, and we endow S with the singular Euclidean metric induced by the stable and unstable foliations, then the length of the geodesic isotopic to f n(α) tends to infinity as n tends to infinity. On the other hand, this is false for periodic and red ucible mapping classes because, in either case, there is at least one isotopy class of simple closed curves that is fixed by a power of f . We emphasize that the proof of Theorem 14.20 only relies on the definition of a pseudo-Anoso v mapping