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

Explicit examples satisfying the homological criterion

We now give an infinite list of examples of pseudo-Anosov mapp ing classes using the homological criterion of Theorem 14.4. Let ([a1], [b1], [a2], [b2]) be the usual homology basis for S2 (cf. Figure 8.1). We consider the product

f = Ta1 Tb1 Ta1+a2 Tb2 Ta12−k ,

where, for instance, Ta1+a2 is the Dehn twists about any curve in the homology class [a1] + [a2]. Then Ψ(f ) is equal to:

Thus, the characteristic polynomial for Ψ(f ) is

P (t) = t4 − kt3 + t2 − kt + 1.

We now check that P (t) satisfies the hypotheses of Theorem 14.4 for each integer k > 1. First of all, it is obvious that P (t) is not a polynomial in tm for any m > 1. If P (t) is a nontrivial product of characteristic polynomials of symplectic matrices, then it factors into two integral quadratic polynomials, each of the form

Pi(t) = t2 − (λi + λ−i 1)t + 1 = (t − λi)(t − λ−i 1).

where λi + λ−i 1 is an integer. To check that this is not the case, we consider

the polynomial Q(x) obtained from P (t) by dividing by t2 and substituting x + x−1 for t:

Q(x) = x2 − kx − 1.

The polynomials Pi(t) have integral coefficients if and only if the roots of Q(x) are integers. It is easy to check that this is not the case when k 6= 0. It remains to check that P (t) is not a cyclotomic polynomial. But the

only degree 4 cyclotomic polynomials are t4 + t3 + t2 + t + 1, t4 + 1, t4 − t3 + t2 − t + 1, and t4 − t2 + 1. Thus, if |k| 6= 1, then P (t) is not

cyclotomic.

Let Mod(S, p) denote the mapping class group of a hyperbolic surface S with one marked point p. We have the Birman Exact Sequence:

Push

1→π1(S, p) → Mod(S, p) → Mod(S) → 1.

We say that an element γ of π1(S, p) fills S if every representative loop for γ intersects every essential simple closed curve in Sg .

Theorem 14.5 (Kra) Let S be a hyperbolic surface, and let γ π1(S). The mapping class Push(γ) is pseudo-Anosov if and only if γ fills S.

Because it is defined in terms of push maps, we like to think of T heorem 14.5 as a “fingerpainting” construction of pseudo-Anosovs mappp ing classes.

Note that Theorem 14.5 gives examples of pseudo-Anosov elements of the Torelli group I(Sg,1), and hence also provides counterexamples to the conjecture of Nielsen mentioned earlier in this section.

Kra's original proof is rooted in Teichm ¨uller theory; he directly shows that if γ fills, then the translation distance of the action of Push(γ) on Teichm ¨uller space is realized. We present an elementary approach, which is apparently new, but was inspired by the algebraic proof due to Kent– Leininger–Schleimer [99].

Proof. One direction of the theorem is obvious: if γ does not fill, then we can find an isotopy class of simple closed curves that is fixed b y Push(γ).

For the other direction, we assume that γ fills S. Since π1(S) is torsion free, the Nielsen–Thurston Classification tells us that we only ne ed to show that Push(γ) is not reducible, that is, if c is any simple closed curve in S − p, then Push(γ)(c) is not isotopic to c, where the isotopies are taken relative to p (equivalently, the curves are not isotopic in S − p).

So let c be a simple closed curve in S −p, and let c˜ denote the full preimage

collection of marked points (the lifts of p). The mapping class Push(γ) can

be lifted to ^; this homeomorphism can be realized by “pushing” each

(S, p)

of the marked points in H2 along a path lifting of γ; denote the resulting

relative homeomorphism of ^ by ^.

(S, p) Push

Since γ intersects c essentially, it follows that for any component c˜i of c˜, there is a path lifting of γ that connects marked points on different sides

of i. Therefore, there are marked points in ^ that lie between i and c˜ (S, p) c˜

^

Push(˜ci)

Any relative isotopy between Push(γ)(c) and c would lift to an equivariant

A CONSTRUCTION FOR BRAID GROUPS

We now give a construction for S0,n, the sphere with n punctures.

Theorem 14.6 Let n be a prime number. If f is an infinite order element of Mod(S0,n) that permutes the punctures cyclically, then f is pseudo-Anosov.

Proof. Suppose that f is reducible. Since n is prime, the partition of the punctures induced by the reducing curves must have sets of different size. Note that f does not preserve this partition because it may permute the reducing curves. However, there is another nontrivial partition of the punctures, where we group together punctures that lie in subsets of the same size in the first partition. The mapping class f preserves this partition, and hence

In Section 7.1, we completely classified the finite order elem ents of Mod(S0,n). Since such elements are easy to avoid, it is not hard to write down explicit elements of Mod(S0,n) that satisfy the criteria of Theorem 14.6 .

The construction can be easily modified to work for braid grou ps: if an

and is not a root of a central element, then it is pseudo-Anosov.

14.2 DILATATIONS

In this section we present some basic properties of the set of dilatations of pseudo-Anosov mapping classes.

Orientation covers for foliations. In the proof of Theorem 14.7 below, we will need the following concept. Let (F, µ) be a measured foliation of a surface S, and let P be the set of singularities of F. Any based loop in S −P acts on the tangent space of F at the basepoint. If we fix some basepoint for π1(S − P ) and record this action, we get the orientation homomorphism

π1(S − P ) → Z/2Z.

An element γ gets mapped to the identity if and only if along some (any) representative loop, F can be consistently oriented. A loop that bounds a disk in S containing one singularity maps to the identity if and only if that singularity has an even number of prongs. Thus, the orientation homomorphism is nontrivial if F is nonorientable in the sense of Section 11.1.

If F is nonorientable, then by extending over the singularities, the orientation homomorphism gives rise to a two-fold branched cover

˜ →

S S.

What is more, there is an induced measured foliation e , which is ori-

(F, µ˜)

entable. The branch points are exactly the preimages of the singularities of F with an odd number of prongs.

Dilatations are algebraic integers. The following theorem was proven in Thurston's announcement of his proof of the Nielsen–Thurst on Classification [166].

Theorem 14.7 Let g ≥ 2. If λ is the dilatation of a pseudo-Anosov mapping class f of Sg , then λ is an algebraic integer whose degree is bounded above by 6g − 6.

Our proof of Theorem 14.7 really shows the stronger result that any eigenvalue of any pseudo-Anosov mapping class of Sg,n is an algebraic integer whose degree is bounded above by a constant depending only on Sg,n.

Proof. Let φ be a pseudo-Anosov representative of f , and let Fu be its unstable foliation. If Fu is orientable, then, as in Section ??, this means

algebraic integer of degree at most 2g. Since g ≥ 2, we have 2g < 6g − 6, and so we are done in this case.

In the case that Fu is nonorientable, we can give a similar argument. The only change is that we need to pass to the orientation double cover for Fu.

Say the singularities of Fu are s1, · · · , sk. By the Euler–Poincar´e formula (§11.1), each singularity contributes at least −1/2 to χ(S), so we have

given by τ ), and so the dimension of V− is at most 6g − 6. We have that ω is an element of V− (at a singular point, for example, τ rotates by π,

and this reverses all orientations). Also, since ˜ commutes with , we have

φ τ

that − is an integral subspace invariant under ˜ . Since is a root of the

V φ λ

characteristic polynomial for the action of f on V−, the theorem follows. 2

In his paper, Thurston states that the examples of Theorem 14.1 show that the bound of Theorem 14.7 is sharp. Also, Franks and Rykken showed that the dilatation of a pseudo-Anosov mapping class is an algebraic integer of degree 2 if and only if it is obtained by lifting through an n-fold branched cover over T 2 as in Section 14.1 [57].

As a corollary of the proof of Theorem 14.7 we have the following.

Corollary 14.8 The stable and unstable foliations for any pseudo-Anosov element of the Torelli group are not orientable.

The spectrum of dilatations. Theorem 14.7 constrains the structure of the set D(S) of all dilatations of pseudo-Anosov homeomorphisms on a given surface S. However, it does not keep track of which (or how many) conjugacy classes have a given dilatation. The following theorem of Arnoux– Yoccoz and Ivanov gives such an account.

THEOREM 14.9 For any C ≥ 1 there exists only finitely many conjugacy classes of pseudo-Anosov homeomorphisms in Mod(S) with dilatation at most C.

We give the proof of Theorem 14.9 not only for its own interest, but because it applies a number of other results we have proven up to this point. This proof follows Ivanov [85].

Proof. Let {fn} be a set of conjugacy classes of pseudo-Anosov homeomorphisms of S = Sg,n, each having dilatation at most C. Let A(fn) denote the axis of fn in Teich(S). Choose ǫ < δ/C3g−3+n, where δ is the constant from Corollary 13.8. In particular any two distinct isotopy classes of simple closed curves of length < ǫ on any closed hyperbolic surface must be disjoint. Let K be a compact subset of Teich(S) with the property that K surjects onto the ǫ-thick part Mǫ(S) of moduli space M(S) (cf. Section 12.4). Let V be the set of fn's with the property that A(fn) maps entirely into Mǫ(S). Each conjugacy class fi V has a representative whose axis intersects K. Since K is compact, and since each fi has a Teichm ¨uller translation length that is bounded above by log(C), it follows from the proper discontinuity of the action of Mod(S) on Teich(S) (Theorem 12.2) that V is a finite set.

If f {fn} but f 6 V , then the projection of A(f ) has some point X lying in M(S)\Mǫ (S). In other words there is a marked hyperbolic surface X

A(f ) whose shortest simple closed curve α has length ℓX (α) < ǫ. Consider the simple closed curves α, f (α), f 2(α), . . . , f 3g−3+n(α). By Wolpert's

Lemma (Lemma 12.6), each of these curves has length at most

ℓX (f i(α)) ≤ C3g−3+n · ℓX (α) < δ for i = 1, . . . , m.

Since there are at most 3g − 3 + n distinct disjoint isotopy classes of simple closed curves on S, it must be that two of the curves on the list are isotopic,

which implies that f j (α) is isotopic to α, contradicting the fact that f is pseudo-Anosov. Since such an f cannot exist, we have {fn} = V , which is

By definition, dilatations of pseudo-Anosov maps are greate r than 1. By the comments about D(S) preceding Theorem 14.9, if we fix g, there is a smallest number λg that appears as the dilatation of a pseudo-Anosov element of Mod(Sg ). Penner proved the following beautiful theorem about λg [141] (see also [122]). In the statement, we write f (x) g(x) for real-valued functions f and g with f (x)/g(x) [1/C, C] for some C > 1.

THEOREM 14.10 The function λg : N → R satisfies

log λg 1/g.

14.3 PROPERTIES OF THE FOLIATIONS Fs AND Fu

In this section we will explore special properties of measured foliations that are invariant by a pseudo-Anosov homeomorphism. Much of our treatment follows that of [54].

We will be forced to consider measured foliations on compact surfaces with boundary. Refer to Section 11.8 for the definition.

Since a pseudo-Anosov mapping class contracts the leaves of its stable foliation, we immediately obtain the following.

Lemma 14.11 No leaf of a stable or unstable foliation of a pseudo-Anosov mapping class is closed, and no leaf connects two singularities.

We therefore deduce the following.

Corollary 14.12 Any simple closed curve has nonzero measure with respect to the stable (or unstable) foliation of any given pseudo-Anosov homeomorphism.

POINCARE´ RECURRENCE FOR FOLIATIONS

Say that a subset of S is a polygon with respect to a foliation F if it is a closed simply connected region, containing at most one singularity of F, that is bounded by finitely many arcs that alternate between b eing subarcs of leaves F and being arcs transverse to F. Call the transverse arcs the faces of the polygon, and call the other arcs the sides. Say that a polygon is standard if it contains at most one singularity of F, in its interior, and there is one face for each prong of the singularity (two faces total if there are no singularities); see Figure 14.3.

Following [54], a good atlas for a foliation F of S consists of two collections of standard polygons, {Ui} and {Vi}, with the following properties.

1.S is the union of the interiors of the Ui.

2.For each i, Ui is contained in Vi, and the faces of Ui are contained in the faces of Vi.

3.For each i, the measure of any transverse arc that connects a side of Ui to a side of Vi is at least ǫ0, where ǫ0 is some fixed number.

4.Each singular point belongs to exactly one Ui.

5.Whenever i 6= j, the intersection Ui ∩ Uj is a rectangle; see Figure 14.4.

We leave it as an exercise to show that any measured foliation has a good atlas. If we also allow foliations to be transverse to the boundary, we can still find a good atlas (adjusting the definition appropriate ly). While such

foliations do not come up in the discussion of quadratic differentials, Teichm ¨uller maps, or pseudo-Anosov homeomorphisms, they do come up in inductive arguments about foliations, where we cut along a transversal and consider the induced foliation on the cut surface.

Figure 14.4 The intersections of two charts of a good atlas.

Lemma 14.13 (Poincare´ Recurrence for foliations) Let (F, µ) be a measured foliation on Sg . Let L be a leaf of F that avoids the singularities of F and is not closed. If α is an arc transverse to F that intersects L, then α intersects L infinitely many times.

Lemma 14.13 has an analogue for non-closed surfaces; we simply need to add the assumption that L does not run into a marked point or the boundary, then α ∩ L is infinite.

Proof. It is enough to show that α ∩ L can never be a single endpoint of α. Assume, for contradiction, that this is the case. Now, cut S along α. The result is a surface S′ with one boundary component, equipped with an induced measured foliation that has two singular points on the boundary, corresponding to the endpoints of α; see Figure 14.5. Note that ∂S′ is transverse to F′.

cut

Figure 14.5 Cutting along a transverse arc.

Denote by L′ the leaf of F′ corresponding to L. The leaf L′ starts from one of these singular points of F′ on the boundary of S′, say s. We see that L′

does not return to s, for that would mean that L is closed. Our assumption on α ∩ L translates now to the statement that L′ does not return to ∂S′.

Choose a good atlas for F′, with constant ǫ0. Let β be an arc of ∂S′ that has s as one of its endpoints, and whose µ′-length is ǫ < ǫ0. Further assume that any leaf of F′ starting from β avoids the singularities of F′; this is possible because each leaf starting from a singularity can hit ∂S′ at most once.

In order to obtain a contradiction, we will show that we can push β along the foliation for infinite time, that is, we can create a strip of (F′, µ′) of width ǫ and of infinite height.

To make this formal, we define a map

P : β × R≥0 → S′

by the rule that P (x, R≥0) is the entire leaf of F′ starting from x.

Let Ls be the leaf of F′ emanating from the singularity s on ∂S′. Given any particular point of Ls, there is a chart Ui of the good atlas containing that point, and so, by the properties of a good atlas, we obtain an entire strip of width ǫ in the interior of the corresponding Vi. It follows that P is an immersion. Since P never hits a singularity (by assumption on β) and since P −1(β) = β × {0}, we see that in fact P is injective.

Any two of the strips of the image of P are disjoint. Since F′ has no closed

Lemma 14.13 may be counterintuitive. It may seem all of a sudden that we have greatly restricted the kinds of foliations we are looking at. But that is in fact the case: our definition of a measured foliatio n precludes the possibility that a leaf spirals towards a singularity, or spirals towards a curve, or that a foliation has a “Reeb component” (a closed annulus f oliated by its two boundary curves and infinitely many leaves homeomorphic to R). The existence of the good atlas is indeed a very strong condition.

A COMBINATORIAL DESCRIPTION OF THE STABLE AND UNSTABLE FOLIA-

TIONS

Let F be the stable or unstable foliation for a pseudo-Anosov homeomorphism. In this section, we will explain how to take the foliation F and