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

Chapter Thirteen

The Nielsen–Thurston Classification

The main goal for this chapter is to understand the Nielsen–T hurston classification for mapping classes, which states that every elem ent of the mapping class group is either finite order, reducible, or pseudo -Anosov. Nielsen wrote a series of papers on surface homeomorphisms in the 1920s–1940s, spanning over 400 pages [134, 135, 136, 138]. His approach was to consider the induced action of Mod(S) on ∂H2. Because Nielsen's work is lengthy and lacked sufficient organizing perspective, this work was largely ignored by topologists for many years.

In 1974, Thurston developed the theory of measured foliations on surfaces and used this to prove the classification as stated above. We d iscuss his approach in Chapter 15. A posteriori, it became clear that all of the required tools for the classification were already discovered by Niel sen. A paper of Miller explains how to understand the pseudo-Anosov case (the most important case) of the classification from the Nielsen point of view [127].

Thurston never wrote down his proof of the classification, al though he did distribute an announcement of his results, which appeared years later in print [166]. This announcement is remarkable for being both extremely brief and extremely rich with insight. The first complete pub lished proof of the classification is due to Bers in 1978 [12], who proved the t heorem from the point of view of Teichm ¨uller theory. Around the same time a group at l'Universit´ Orsay, led by Albert Fathi, Franc¸ois Lauden bach, and Valentin Po´enaru, worked out the full details to Thurston's proof. T he result is a 284 page monograph known as “FLP” [54]. The relationship betwee n the works of Thurston, Bers, and Nielsen is explained in a paper by Gilman [62]. Other points of view on Nielsen–Thurston theory are contained in t he writings of Handel and Thurston [70], Casson and Bleiler [40], and Bonahon [27]. The key objects in these works are geodesic laminations, which are implicit in Nielsen's work.

In this chapter we present Bers' proof of the Nielsen–Thurst on classification. We begin by giving the classification in the special cas e of the torus, which is derived from the classical trichotomy for elements of SL(2, R).

13.1 THE CLASSIFICATION FOR THE TORUS

Recall that in Chapter 1 we classified the nontrivial element s of Isom+(H2) into three types: elliptic, parabolic, and hyperbolic. Since Mod(T 2) ≈ SL(2, Z) (Theorem 2.5) and since PSL(2, R) ≈ Isom+(H2), we automatically obtain a classification of elements of Mod(T 2), namely, by considering the type of the corresponding element of Isom+(H2).

What we would really like, though, is a classification of elem ents of Mod(T 2) that is intrinsic to the torus. As we now show, the three types of hyperbolic

isometries correspond to three qualitatively different types of elements of

Mod(T 2).

to the M ¨obius transformation

z 7→az + b cz + d

acting on the upper half-plane.

If τ is elliptic, then this means that τ fixes a point of H2 and is hence a rotation. By the proper discontinuity of the action of SL(2, Z) on H2, we see that τ must be a finite order rotation. Thus A, hence f , is finite order.

If τ is parabolic, then τ fixes a unique point in ∂H2. This is the same as saying that A has a unique real eigenvector. It follows that A has exactly one real eigenvalue, and that this eigenvalue has multiplicity 2. Since the product of the eigenvalues of A is equal to the determinant, which is 1, the eigenvalue for A is 1 −1. This means that A fixes a vector in R2. Since A is an integer matrix, it follows that A fixes a rational vector in R2. But from

this it follows that f fixes the corresponding isotopy class of simple closed curves in T 2. In the case we say that f is reducible.

If τ is hyperbolic, then τ fixes two points in ∂H2. This is equivalent to the statement that A has two real eigenvectors, or, that A has two distinct real eigenvalues. Since the determinant of A is 1, it follows that its two eigenvalues are inverses, say λ and 1/λ, where λ > 1. Therefore, A has two eigenspaces in R2, one of which is stretched by a factor of λ and one of which is contracted by a factor of λ. Thus, on the torus, we obtain what we refer to as an Anosov package. By this we mean that there is on T 2 a pair of foliations Fs and Fu, called the stable and unstable foliations, that satisfy the following properties.

1.Each leaf of Fs and of Fu is the image of an injective map R → T 2.

2.The foliations Fs and Fu are transverse at all points.

3.There is a natural transverse measure µs (resp. µu) assigning a mea-

sure to each arc transverse to Fs (resp. µu), obtained by realizing the foliations by straight lines in some flat metric on T 2, and declaring the measure of a transverse arc to be the total variation in the direction perpendicular to the foliation.

In this case we say that f is Anosov. We thus have the following theorem.

THEOREM 13.1 Each nontrivial element of Mod(T 2) is of exactly one of the following types: finite order, reducible, Anosov.

We can be even more specific in the first two cases. A nontrivial finite order element of Mod(T 2) has order 2, 3, 4, or 6. Also, a nontrivial reducible element of Mod(T 2) is either a power of a Dehn twist or the product of a power of a Dehn twist with the hyperelliptic involution.

374 CHAPTER 13

Using just the isomorphism Mod(T 2) ≈ SL(2, Z), and without appealing to hyperbolic geometry, we can give a more algebraic approach to the classification for Mod(T 2). Let A SL(2, Z), and let f Mod(T 2) denote the corresponding mapping class. The characteristic polynomial for A is x2 −trace(A)x + 1. It follows that the eigenvalues of A are inverses of each other. call them λ and λ−1. There are then three cases to consider:

1.|trace(A)| {0, 1}

2.|trace(A)| = 2

3.|trace(A)| > 2

The three cases are equivalent to the cases: λ and λ−1 are complex, λ = λ−1 = ±1, and λ and λ−1 are distinct reals. In the first case it follows from the Cayley–Hamilton theorem that A, hence f , has finite order. In the second case A has a rational eigenvector, and from this it follows that f is reducible. In the third case we see that A has two real eigenvalues, and so f is Anosov.

We summarize the results of this section in the following table.

13.2 THREE TYPES OF MAPPING CLASSES

We now describe three kinds of elements of the mapping class group. Each is an analogue of one of the three types of elements of Mod(T 2) in the statement of Theorem 13.1. The Nielsen–Thurston classifica tion theorem (Section 13.3) states that every mapping class falls into (at least) one of these three categories.

PERIODIC MAPPING CLASSES AND HYPERBOLIC ISOMETRIES

We have already studied periodic, or, finite order, mapping classes. A basic example is shown in Figure 13.1; see also Figures 2.1, 2.1, and 2.3. Also, see Chapter 7 for a general discussion of finite order mapping classes.

Figure 13.1 A finite order element of the mapping class group.

Theorem 7.1 states that every periodic mapping class has a representative diffeomorphism that is finite order ( a priori, we only know that there is a representative with a power isotopic to the identity). We deferred the proof of this theorem because we wanted to give a proof that uses the basic properties of Teichm ¨uller space. While this is a natural point in the text to give the proof, we emphasize that the proof is classical and elementary, relying only on definitions and the fact that Teich(S) is contractible.

We will actually prove something stronger than Theorem 7.1: we will show that each periodic element of Mod(S) can be realized as an isometry of S with respect to some hyperbolic metric. The idea is to show that a periodic mapping class, thought of as a homeomorphism of Teichm ¨uller space, has a fixed point. Nielsen's gave a direct proof of Theorem 7.1 in 1 942 [137]. The proof we present here was first suggested by Fenchel (1948 ) [55] [56] and Macbeath [111].

Proof of Theorem 7.1. Recall from Section 12.1 that an element of Mod(S) fixes a point of Teich(S) if and only if it has a representative homeomorphism that is an isometry of S with respect to some hyperbolic metric. Thus, to prove the theorem, it suffices to show that each perio dic element of

Say f Mod(S) has finite order n. Since Teich(S) is contractible (Theorem 10.5), the group hf i cannot act freely on Teich(S), for otherwise the quotient would be a finite dimensional K(Z/pZ, 1). Thus, f k · X = X for some 1 ≤ k ≤ n and some X Teich(S).

In the case that n is prime (or even if gcd(n, k) = 1), we have that f is a power of f k. Thus f fixes X Teich(S) and we are done.

Teich(S) is realized by an isometry φ′ of a hyperbolic surface X that is homeomorphically identified with S. The quotient X/hφ′i is a hyperbolic orbifold. Denote the fixed set of f ′ in Teich(S) by Fix(f ′).

Claim 1: Fix(f ′) ≈ Teich(X/hφ′i).

Claim 2: Teich(X/hφ′i) ≈ Teich(X/hφ′i − cone points).

Indeed, complex structures on the punctured surface correspond to complex structures on X minus the branch points. Also, any complex structure on X minus the branch points can be extended to a complex structure on all of X. As a result, we see that the fixed set is contractible.

Combining the two claims, plus the fact that Teich(X/hφ′i−cone points) is contractible, we deduce that Fix(f ′) is contractible. Since f commutes with f ′, it follows that hf i acts on Fix(f ′). As hf ′i is contained in the kernel of this action, the action of hf i acts on Fix(f ′) factors through an action of hf i/hf ′i ≈ Z/psZ on Fix(f ′). As before, the latter action must have a fixed point, and so since ps is prime we are again able to deduce that f has

Our argument for Theorem 7.1 really demonstrates something even stronger: every finite solvable subgroup of Mod(S) is realized as a subgroup of the isometry group of S for some hyperbolic metric on S.

REDUCIBLE MAPPING CLASSES

We say that an element f of Mod(S) is reducible if there is a nonempty collection {c1, . . . , cn} of distinct isotopy classes of essential simple closed curves in S so that i(ci, cj ) = 0 for all i and j and so that {f (ci)} = {ci}.

The collection is called reduction system for f . In this case, we can further understand f via the following procedure:

1.Choose representatives {γi} of the {ci} with γi ∩ γj = for i 6= j

2.Choose a representative φ of f with {φ(γi)} = {γi}

3.Consider the homeomorphism of the noncompact surface S − γi induced by φ

(Note that the second step is an application of the Alexander Method plus Proposition 1.11.) As the connected components of S− γi are each simpler than S itself (as measured by, say, Euler characteristic), we can hope to understand f by induction. In particular, we can decompose f into irreducible pieces.

Examples. A typical reducible mapping class is obtained as follows. Let γ be a separating simple closed curve in S2, and let S′ and S′′ be the two subsurfaces of S2 bounded by γ. Choose φ′ and φ′′ to be homeomorphisms of S′ and S′′ that fix γ pointwise. Even better, choose φ and φ′ so that neither fixes the isotopy class of any essential simple close d curve in S′ or S′′ (in this case it is enough to check that the actions on H1(S′; Z) and H1(S′′; Z) do not have any fixed vectors). Let σ be a homeomorphism of S2 that switches the two sides of γ. The homeomorphism

σ ◦ Tγ ◦ φ′ ◦ φ′′

represents a reducible element of Mod(S2). The isotopy class of γ is the unique reduction system in this case.

A simple example of a reducible mapping class is a Dehn twist Ta: any isotopy class of curves b satisfying i(a, b) = 0 is fixed.

Another example is the one given in Figure 13.1 (which curves, or collections of curves, are fixed?). Thus, we see that there is overla p between the sets of periodic and reducible mapping classes.

Canonical reduction systems. In each of the last two examples, there are many choices for the reduction system, as there are many collections of curves fixed by either mapping class. Say that a reduction s ystem for a mapping class f is maximal if it is maximal with respect to inclusion of reduction systems for f . We can then consider the intersection of maximal reduction systems for f . This intersection is clearly canonical, and we call it the canonical reduction system for f .

Canonical reduction systems were introduced by Birman, Lubotzky, and McCarthy [24]. In their approach, the isotopy class c of an essential simple closed curve in S is in the canonical reduction system for f exactly when it satisfies the following two criteria: f (c) = c and f (b) 6= b whenever i(b, c) 6= 0. The advantage of their definition is that it gives qualitati ve information about the isotopy classes in an essential reduction system. It is a simple exercise, though, to show that their definition is eq uivalent to ours.

Periodic versus reducible. Dehn twists are examples of mapping classes that are reducible but not periodic. The example of Figure 13.1 is reducible and periodic. One element of Mod(Sg ) that is periodic but not reducible is the example that realizes the upper bound of Theorem 7.5, that is, the periodic element of maximal order in Mod(Sg ). Recall that this mapping class is realized by representing Sg as a (4g + 2)–gon and rotating the polygon by one “click,” that is, by 2π/(4g + 2). The quotient surface is a sphere with three cone points: one corresponding to the center of the polygon, one corresponding to the vertices of the polygon (all of which get identified in the quotient), and one of which corresponds to the midpoints of the edges of the polygon (again, all of these get identified in the quotien t). In the complement of the cone points, there are no essential curves on the sphere, and it follows that the mapping class is not reducible.

The question remains: what can we say about mapping classes that are neither periodic nor reducible?

PSEUDO-ANOSOV MAPPING CLASSES

We say that f Mod(S) is pseudo-Anosov if there is a pair of transverse measured foliations (Fu, µu) and (Fs, µs) on S, a number λ > 1, and a representative homeomorphism φ so that

φ · (Fu, µu) = (Fu, λµu) and φ · (Fs, µs) = (Fs, λ−1µs).

The measured foliations (Fu, µu) and (Fs, µs) are called the unstable foliation and stable foliation, respectively, and the number λ is called the dilatation. We call φ a pseudo-Anosov homeomorphism.

Of course, the representative φ is not unique; we can change the stable and unstable foliations by an isotopy, and then conjugate φ by the homeomorphism at the “end” of the isotopy. It is a theorem, however, th at this is the only nonuniqueness; in other words, any two homotopic pseudo-Anosov homeomorphisms are conjugate by a homeomorphism that is isotopic to the identity [54, Expos´e 12, Thm III].

The map φ is a diffeomorphism away from the singularities of the stable and unstable foliations, and, because both the stable and unstable foliations span the tangent space at the singularities, φ is not smooth at the singularities. One should compare the definition of a pseudo-Anosov homeomo rphism with the statement of Teichm ¨uller's Existence Theorem (Theorem 11.6).

Does φ stretch or shrink the leaves of, say, the stable foliation? It is certainly not obvious. To check, let α be an arc of the stable foliation. We want to compare µu(φ(α)) with µu(α). By definition (Section ??), the former is equal to φ−1 · µu(α) = λ−1µu(α), where λ > 1. Therefore, the correct statement is that φ shrinks the leaves of the stable foliations and stretches the leaves of the unstable foliation. Perhaps one way to remember this is that, if we take a point p on a stable leaf that emanates from a singularity s of Fs, then φn(p) approaches s as n goes to infinity; we think of this as a stability condition.

Punctures and boundary. The definition of a pseudo-Anosov homeomorphism carries over for surfaces with punctures and/or boundary. See the end of Chapter 11 for the definition of a foliation on such a surfac e. Again, the definition is not so natural for surfaces with boundary, and, so, if we prefer, we can define a pseudo-Anosov homeomorphism for a surface wit h boundary as a homeomorphism that restricts to a pseudo-Anosov homeomorphism on the punctured surface obtained by removing the boundary. Note, for example, that, given any pseudo-Anosov homeomorphism on any surface, we may remove any finite orbit to get a pseudo-Anosov homeomorph ism on a punctured surface.

13.3 STATEMENT OF THE CLASSIFICATION

Theorem 13.2 (Nielsen–Thurston Classification) Let g, n ≥ 0. Each f

Mod(Sg,n) is either

1.periodic,

2.reducible, or

3.pseudo-Anosov.

Further, pseudo-Anosov mapping classes are neither periodic nor reducible.

The main content of the theorem is that every irreducible (i.e., not reducible), infinite order mapping class has a representative that has a v ery special form, namely, that of a pseudo-Anosov homeomorphism.

We will give two consequences of the Nielsen–Thurston Class ification, both of which may be thought of as restatements of the theorem. The idea is that, if a mapping class is reducible, then we can cut along the reducing curves and analyze the pieces by re-applying the theorem.

Corollary 13.3 (Nielsen–Thurston Classification restated , I) For each f

Mod(S), there is a representative φ, and a (possibly empty) collection of pairwise disjoint simple closed curves {ci} in S so that φ({ci}) = {ci} for all i, and, if {Sj } are the connected components of S − ci, and ki is any integer so that φkj (Sj ) = Sj , then φkj |Sj represents either a finite order element or a pseudo-Anosov element of Mod(Sj ).

In the corollary, it is important that we are removing the ci and not cutting along the ci. In other words, each ci becomes a pair of punctures on S − ci. Therefore, all twists along the ci disappear when we consider φkj |Sj (cf. Equation (??)).

Corollary 13.3 is analogous to the Jordan Canonical Form of a matrix. The φki |Si are the analogues of irreducible Jordan blocks.

By taking powers, we can assume that a mapping class fixes each of its reducing curves, fixes each of the complementary subsurface s, and is either the identity or is pseudo-Anosov on each of these subsurfaces. The resulting statement is as follows.

Corollary 13.4 (Nielsen–Thurston Classification restated , II) For each f

Mod(S), there is a representative φ, an integer k, and a (possibly empty) collection of pairwise disjoint simple closed curves {ci} in S so that, if {Sj } denote the closures in S of the connected components of S − ci, and ηj : Mod(Sj ) → Mod(S) denotes the homomorphism induced by the inclusion Sj → S, we have

1.φk (ci) = ci.

2.φk (Sj ) = Sj .

identity or is pseudo-Anosov.