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

We can distinguish among the parabolic isometries of Teich(S) those with τ = 0 and those with τ > 0. The latter correspond to mapping classes that have a pseudo-Anosov component, in the sense of Corollary 13.4.

It is natural to wonder if the axis for a pseudo-Anosov mapping class is unique. It turns out that it is unique, but this in no way follows from our proof of Theorem 13.2. For a proof, see [113, Theorem 9.2].

Bers' approach to proving the exclusivity statement for the classification is to show that a reducible mapping class gives a parabolic isometry of Teich(S)—this is the converse of Case 2 of the proof. The idea is that you can always shrink the reducing curves further to reduce the dilatation.

Chapter Fourteen

Pseudo-Anosov theory

The power of the Nielsen-Thurston Classification is that it g ives a criterion for an element f Mod(S) to be pseudo-Anosov, namely, f is neither finite order nor reducible. This criterion is only as useful a s our knowledge of pseudo-Anosovs. The purpose of this chapter is to study pseudo-Anosov homeomorphisms: their construction, their algebraic properties, and their dynamical properties.

2. A linear representative φ is called an Anosov diffeomorphism. Recall from Section 13.1 that φ comes with a kind of “Anosov package”, which we now recall.

The geometric picture of the action of φ on T 2 can be given quite explicitly. The map φ, thought of as an element in SL2(Z), has two distinct real eigenvalues, λ > 1 and λ−1. The diffeomorphism φ preserves two foliations Fu and Fs on T 2; these are the projections to T 2 of the foliations of R2 by lines parallel to the λ and λ−1 eigenspaces of the matrix φ. The diffeomorphism φ stretches each leaf of Fu by a factor of λ and contracts each leaf of Fs by a factor of λ−1. The eigenspaces are lines with irrational slope, from which it follows that each leaf of Fu and of Fs is dense in T 2. It is also an easy exercise to check that φ-periodic points are dense. Finally, from basic linear algebra one can see that, for the generic vector v R2, the vector φn(v) “converges to” the vector λnvu, where vu is a vector pointing in the direction of Fu; more precisely, the directions converge:

φn(v) → vu

|φn(v)| |vu|

and the magnitudes converge:

p

n |φn(v)| → λ.

One main goal of this chapter is to describe a “pseudo-Anosov package”,

which extends the above picture of Anosov diffeomorphisms from T 2 to higher genus surfaces.

14.1 FIVE CONSTRUCTIONS

The first basic question to address is: “Do pseudo-Anosovs ac tually exist?” The answer is, of course, “yes”. We now explain several const ructions of pseudo-Anosov mapping classes.

BRANCHED COVERS

One way to construct pseudo-Anosov mapping homeomorphisms is to lift Anosov maps of the torus via a covering map. Recall from Chapter 7 that an orbifold cover S′ → S is a map obtained by a finite group action on S′. We also call such a cover a A branched cover, since it is a covering off of a finite number of points, called branch points, and it is locally given by the complex map z 7→zki , with ki > 1, around each branch point xi S′. Recall from Section ?? that for g ≥ 2, there is a 2–fold regular branched cover Sg → T 2 with 2g − 2 branch points.

Let φ be an Anosov homeomorphism of the torus. Since all rational points of T 2 are periodic points of φ, we can change φ by isotopy and pass to a power of φ so that φ fixes pointwise the set P of branch points of p. Passing to further powers of φ, we may assume that φ lifts to a homeomorphism ψ of Sg (consider the action of φ on the finite set of index 2 subgroups of π1(T 2 − P )). As ψ has the same local properties as φ, we see that ψ is a pseudo-Anosov homeomorphism of Sg ; indeed, the stable and unstable foliations for ψ are the lifts of those for φ. Note that above each branch point in T 2, the foliations for ψ each have a singularity with 4 prongs.

By considering branched coverings over higher genus surfaces, we can use this construction to convert pseudo-Anosov mapping classes on any surface to pseudo-Anosov mapping classes on higher genus surfaces.

DEHN TWIST CONSTRUCTIONS

We now present an elementary construction of pseudo-Anosov mapping classes due to Thurston [166], and a related one due to Penner. Let S =

Sg,n. We say that a collection of curves fills S if any simple closed curve in S has nontrivial geometric intersection with some curve in the collection (see e.g. Figure 1.7). In other words, a collection of curves fills if the complementary regions in S are disks (each with at most one marked point in the interior).

Let A = {a1, . . . , an} and B = {b1, . . . , bm} be multicurves in S. Denote by TA and TB the products of the Tai and Tbi , respectively. We will see that if A B fills S, then TATB−1 is a pseudo-Anosov element of Mod(S). The construction builds on a construction of measured foliations given in Section ??.

It is an interesting exercise to construct a pair of filling cu rves for an any given surface. On the other hand, it is quite easy to find a pair of multicurves that fill a given surface.

Let N be the intersection matrix associated to A and B, namely:

Nj,k = i(aj , bk ).

In the case that a and b are single curves, the matrix N is 1×1, and the entire discussion leading up to the theorem degenerates. Since this case is useful in its own right, we recommend that, on a first pass, the reader concentrate on this case.

The matrix N is well-known in graph theory. Given N , let G be the abstract bipartite graph with m red vertices and n blue vertices, and Nj,k edges be-

tween the jth red vertex and the kth blue vertex. Then the (j, k) entry of the dth power (N N t)d is equal to the number of paths in G of length 2d

between the jth and kth red vertices.

We say that a matrix (or vector) is positive, or nonnegative, if each of its entries has the given property. The above description of N N t clearly implies that it is nonnegative.

Recall that a nonnegative matrix is irreducible if it has a power that is a positive matrix. If we assume that A B fills S, then it follows from our characterization of (N N t)d that N N t is irreducible.

Now, any irreducible matrix has a unique (up to scaling) nonnegative eigenvector, called the Perron–Frobenius eigenvector . Even better, the Perron– Frobenius eigenvector is a positive vector, and has a positive eigenvalue, called the Perron–Frobenius eigenvalue (see [60, §XIII.2]).

Denote the Perron–Frobenius eigenvalue and eigenvector fo r N N t by µ and V , respectively:

N N tV = µV.

Interchanging the roles of A and B, the Perron–Frobenius eigenvalue for N tN is still µ:

N tN V ′ = µV ′.

Here, V ′ is chosen to be µ−1/2N tV . In this case, we have the formula

V = µ−1/2N V ′.

Theorem 14.1 (Thurston) Suppose A and B are multicurves that together fill a surface S. The subgroup hTA, TB i of Mod(S) has a representation ρ into PSL2(R) given by

where µ is determined as above. The image of ρ is a discrete group, and an element of hTa, Tbi is periodic, reducible, or pseudo-Anosov according to whether its image under ρ is elliptic, parabolic, or hyperbolic. When ρ(f ) is hyperbolic, the pseudo-Anosov f has dilatation equal to the larger eigenvalue of ρ(f ).

We remark that if µ1/2 ≥ 2, then ρ(hTA, TB i) is isomorphic to a free group,

it contains no elliptics, and any parabolic element it contains is conjugate to a power of TA, TB or (when µ1/2 = 2) to TATB .

We now describe explicitly the representation ρ of Theorem 14.1. First, we choose representatives of the curves of A and B that are in minimal position. As A B gives a cell division of S, there is a dual cell division C. Note that the 2-cells of C are squares; each square corresponds to a point of intersection of an element of A with an element of B. (If a 2-cell of the first cell decomposition has a marked point in its interior, then that marked point is taken to be a vertex of the dual decomposition.)

Any choice of side lengths for the (combinatorial) squares of C gives rise to a singular Euclidean structure for S. Our side lengths will come from the entries of the vectors V and V ′: we assign the length Vi to each edge transverse to (the representative for) each ai and Vj′ to each edge transverse

to, but not homotopic to, α.

One victory of Thurston's construction (Theorem 14.1), which accompanied the announcement of his proof of the classification [166], is that it makes it easy to see that Nielsen's conjecture is false—just take all curves in the construction to be separating.

Corollary 14.2 Let S be a closed surface of genus g ≥ 2. The Torelli subgroup of Mod(S) contains pseudo-Anosov elements.

It even possible to use the Thurston construction to find a pse udo-Anosov element of I(S) − K(S); that is, an pseudo-Anosov element of the Torelli group that is not the product of Dehn twists about separating curves. This is tricky because, in order to use a bounding pair map, the two curves of the bounding pair must belong to different multicurves of the construction.

Consider the mapping class Ta−21Ta−11Tb2 Tb1 , where the curves are as shown in Figure 14.1. This mapping class is pseudo-Anosov by Theorem 14.1, it is in I(S) since it is the product of a bounding pair map with a pair of Dehn twists about separating curves, and it is not in K(S) since it is not in the kernel of the Johnson homomorphism (see Section 8.4).

a1

Figure 14.1 A multicurve that yields a pseudo-Anosov element of I(S) − K(S) via the Thurston construction.

Penner's construction

Penner gives the following construction.

Theorem 14.3 (Penner [142]) Let A = {a1, . . . , an} and B = {b1, . . . , bm} be multicurves that fill a surface S. Any product of positive powers of the Tai and negative powers of the Tbi , where each ai and each bi appears at least once, is pseudo-Anosov.

In the statement of the theorem, the twists can appear in any order, for in-

stance Ta1 Tb−3

1

Penner has conjectured that every pseudo-Anosov element of the mapping class group has a power that is given by this construction. This is a difficult conjecture to disprove. For instance, one can use the Thurston construction to find pseudo-Anosov mapping classes that are not a priori given by Theorem 14.3. However, how can one tell if there is or is not another way to write the same element as a product of Dehn twists so that, in that form, it is given by the Penner construction?

The idea of Penner's proof is that one can explicitly find the t rain track (see Chapter 15) associated to the square of any such element. The train track is obtained by “smoothing” out the subset A B of S; see [142].

HOMOLOGICAL CRITERION

We now show how to detect pseudo-Anosov maps via the symplectic representation of Mod(S) of Section 8.4; here S is Sg or Sg,1. The original version of this criterion is due to Casson–Bleiler [40].

We say that an polynomial is symplectically irreducible over Z if it cannot be written as a product of two polynomials, each of which is the characteristic polynomial of a matrix in Sp(2g, Z). In particular irreducible polynomials are symplectically irreducible.

As noted in Section 8.2, the roots of the characteristic polynomial of a symplectic matrix come in pairs λ, λ−1. Since the coefficients of a polynomial are symmetric functions of its roots, and since the roots are paired, an easy argument gives that characteristic polynomials of integral symplectic matrices are monic and palindromic, i.e., the coefficients are the same read forwards or backwards. Thus it is much easier to be symplectically irreducible than to be irreducible.

Theorem 14.4 Let f Mod(S) and let Ψ(f ) be its image in Sp(2g, Z). The mapping class f is pseudo-Anosov if the characteristic polynomial of

the matrix Ψ(f ) is symplectically irreducible over Z, is not a cyclotomic polynomial, and is not a polynomial in tk for k > 1.

Note that, if f satisfies the criterion of the theorem, then every element of the coset f I(S) is pseudo-Anosov. Of course, in consideration of Corollary 14.2, there is no hope for any kind of converse to Theorem 14.4.

Proof. We show that if f is not pseudo-Anosov, then the characteristic polynomial of Ψ(f ) fails to satisfy one of the given conditions. By the Nielsen– Thurston Classification, if f is not pseudo-Anosov, then there are two overlapping cases: f is periodic, or f is reducible. We deal with each in turn.

If f is periodic of order n, then Ψ(f )n = 1, so each root of the characteristic polynomial of Ψ(f ) are nth roots of unity. It is a fact that an integral polynomial that has a root of unity as a zero is either symplectically reducible over Z or is a cyclotomic polynomial, and so we are done in this case.

When f is reducible, we split into two (again, overlapping) subcases: where some power of f fixes a nonseparating reducing curve, and where f fixes a collection of disjoint separating curves.

For the first subcase, say f n(c) = c for some nonseparating curve c. Since c is nontrivial in H1(S, Z), it follows that Ψ(f )n has an eigenvalue of 1, and hence Ψ(f ) has an eigenvalue that is an nth root of unity. As in the periodic case, this implies that the characteristic polynomial is either symplectically reducible or is cyclotomic, and this completes the proof in this subcase.

For the second subcase, let c be one curve in the collection of separating curves that are fixed, and assume that the other curves in the c ollection all lie on one side of c, i.e., choose an “innermost” curve in the collection. Let R be a closed subsurface of S that has c as its boundary and that does not contain any other curves of the collection. It follows that the subsurfaces {f i(R)} are mutually disjoint (there may be only one of them); see Figure 14.2. Suppose that f n(c) = c and let T be the complement of the f i(R) (really, we are using representatives of f and c here). We have

H1(S, Z) = H1(R, Z) H1(f (R), Z)· · ·H1(f n−1(R), Z) H1(T, Z).

This decomposition gives rise to a choice of basis for H1(S, Z); namely, the first set of basis elements are a basis for H1(R, Z), then the images of these elements under f , f 2, etc., and finally the last set of basis elements are a basis for H1(T, Z). Under such a basis, Ψ(f ) is of the form

f 2(R)

f 2(c)

f (c)

f 3(R)

T

f (R)

f 3(c)

c = f 4(c)

R = f 4(R)

Figure 14.2 A mapping class f that fixes a union of disjoint separating curves.

where C is the induced action of f on H1(T, Z). In this case, the characteristic polynomial of Ψ(f ) is |Ψ(f ) − tI| = |B − tnI||C − tI|. If T has genus 0, then C is a 0 × 0 matrix (meaning, it is not really there), and the characteristic polynomial is a polynomial in tn with n > 1. If T has positive genus, then C is an m × m matrix with m ≥ 1, and so the polynomial is