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

The corresponding bijection is not true if we replace c g by g and

M(S ) M(S )

by Mod(Sg ): one can have a bundle with finite monodromy corresponding to the trivial homotopy class. The problem is precisely with the torsion in Mod(Sg ), and so it disappears upon passing to .

12.4 MUMFORD'S COMPACTNESS CRITERION

Moduli space is not compact. For instance, in the case of the torus, using the identification of Teich(T 2) with the upper half-plane, we see that the sequence of points of moduli space [i/n] leaves every compact subset of M(S1). For a general surface S, allowing a Fenchel–Nielsen length parameter to tend towards zero will produce a sequence of points leaving every compact set in moduli space. This is because the “length-of- the-shortest- curve” function is a continuous function on moduli space and hence must attain a minimum on any compact set.

Mumford's compactness criterion states that moduli space does however have a natural exhaustion by compact sets. Before stating the theorem, we need to introduce a particular function on moduli space. For X M(S), let ℓ(X) denote the length of the shortest essential closed curve in X.

When S is closed the positive real number ℓ(X)/2 is the same as the injectivity radius of X, which we now define. The injectivity radius of a Riemannian manifold X at a point x is defined to be the radius of the largest embedded disc in X centered at x. Then, the injectivity radius of X is the infimum of these injectivity radii over all points of X.

The ǫ–thick part of M(S) is the set

Mǫ(S) = {X M(S) : ℓ(X) ≥ ǫ}.

Clearly, {Mǫ(S) : ǫ > 0} is an exhaustion of M(S).

THEOREM 12.8 (Mumford's compactness criterion) Let g ≥ 2. For all

ǫ > 0, the set Mǫ(Sg ) is compact.

In other words, Theorem 12.8 states that, in order for a sequence of points in M(S) to leave every compact set, the injectivity radii must tend to 0.

MAHLER'S COMPACTNESS CRITERION

The arithmetic version of Mumford's compactness criterion is the following theorem of Mahler. The injectivity radius of a lattice is the length of the shortest element; in other words, in the corresponding quotient of Euclidean space, this is the length of the shortest essential curve.

THEOREM 12.9 (Mahler's compactness criterion) The space of unit volume lattices in Rn with injectivity radius bounded below by any fixed ǫ > 0 is compact.

We now give the proof of Mahler's compactness criterion for n = 2, which is exactly Mumford's compactness criterion for the torus. The proof contains all the ideas needed for the general case, but is much simpler notationally.

Proof. Suppose ≈ Z2 is any lattice in R2 with injectivity radius bounded below by ǫ. Let v be the shortest nontrivial vector of , and let w be the vector with shortest nonzero distance to the (real) subspace spanned by v. By choice of v and w, there are no points of in the interior of the parallelogram spanned by v and w, and so v and w generate .

We will show that the norms of v and w are bounded above by a function of ǫ (independent of ) and that w2, the projection of w to v , is bounded from below by a function of ǫ. The first property will ensure that any infinite sequence of lattices has a convergent subsequence, and the second property will ensure that the limiting lattice is nondegenerate.

Let w1 be the projection of w to the real subspace spanned by v. We have

1

|v| ≤ |w| ≤ |w1| + |w2| ≤ 2 |v| + |w2|,

and so |w2| ≥ |v|/2 ≥ ǫ. Since |v||w2| = 1, we have |v| = 1/|w2| ≤ 1/ǫ and |w2| = 1/|v| ≤ 1/2ǫ. Without loss of generality, w is shortest among {w + kv : k Z}, and so we may assume |w1| ≤ |v|/2 ≤ 1/2ǫ. This

BERS' CONSTANT

To prove Mumford's compactness criterion, we will need the following theorem of Bers [13].

THEOREM 12.10 (Bers' constant) Let S be a compact surface with χ(S) < 0. There is a constant L = L(S) such that for any hyperbolic surface X homeomorphic to S (with totally geodesic boundary), there is a pants decomposition {γi} of X with ℓX (γi) ≤ L for all i.

Bers' constant is the smallest L that satisfies the conclusion of the theorem. Buser shows that Bers' constant is at most 21(g − 1) for a closed surface of genus g (although he suggests the actual bound is on the order of √g) [38, §5.2.5]. Our proof gives a bound that grows faster than exponentially in g, but this suffices for our purposes.

Proof. Suppose that S has genus g and b boundary components. Recall that a pants decomposition for S has 3g − 3 + b simple closed curves. We will prove the following statement by induction on k for 0 ≤ k ≤ 3g − 3 + b: there is a constant Lk (S) so that for every hyperbolic surface X with totally geodesic boundary that is homeomorphic to S, there is a set of k distinct, disjoint, essential closed geodesics, each of length at most Lk. This inductive statement is true for k = 0 since we may take Lk = 0.

Now, assume the inductive hypothesis for some fixed k ≥ 0. Let X be a hyperbolic surface with totally geodesic boundary that is homeomorphic to S. Choose a collection of k geodesics in X as in the inductive hypothesis, and cut X along these curves. Let Y be any component of the cut surface that is not homeomorphic to a pair of pants, and let y be a point on Y that is furthest from the boundary. We must find an essential geodesi c in Y whose length is bounded above by a function of S.

Let D(y, ρ) be the disk of radius ρ in Y centered at y. More precisely, D(y, ρ) is the image under the exponential map of the ball of radius ρ in the tangent space Ty (Y ). For small ρ this is an embedded disk isometric to a disk of radius ρ in H2. Therefore, its area is given by the formula

Z Z

2π ρ

sinh(r) dr dθ = 2π(cosh(ρ) − 1).

00

The key point for us is that the area of D(y, ρ) is a proper function of ρ.

We can increase ρ until either D(y, ρ) bumps into itself or the boundary. Let ρy denote the supremum over ρ so that D(y, ρ) is an embedded disk disjoint from ∂Y . Since the area of Y is less than or equal to the area of X (which is −2πχ(S)), we have that ρy is finite and is bounded above by a function of S. The disk D(y, ρy ) is either not embedded in Y or it intersects ∂Y .

In the first case, there are two radii of ∂D(y, ρy ) that meet at both endpoints. The union of these two arcs is a closed geodesic of length 2ρy , which, as discussed, is bounded above by a function of S. The geodesic is necessarily essential by the uniqueness of geodesics in a hyperbolic surface.

In the second case, we note that D(y, ρy ) must intersect ∂Y in at least two points, for otherwise, we could find a point in Y that is further from ∂Y . Thus, we have two arcs from y to ∂Y , which we think of as an arc γ between components δ1 and δ2 of ∂Y (possibly δ1 = δ2). Let N be a regular metric neighborhood of γ δ1 δ2. By making N arbitrarily small, the length of the simple closed curve α = ∂N is arbitrarily close to 2ℓY (γ) + ℓY (δ1) + ℓY (δ2), and so the geodesic in the class of α is strictly less than this. Since Y is not a pair of pants (and is not an annulus), α is essential in Y , hence in

THE PROOF OF MUMFORD'S COMPACTNESS CRITERION

We can now prove Mumford's compactness criterion, that the ǫ–thick part of moduli space is compact.

Proof of Theorem 12.8. Since M(Sg ) inherits the Teichm ¨uller metric from Teich(Sg ), it suffices to show that Mǫ(Sg ) is sequentially compact. Let ǫ > 0, and let {Xi} be a sequence in Mǫ(Sg ), and let Xi Teich(Sg ) be a lift of Xi for each i. To prove that some subsequence of {Xi} converges in Mǫ(Sg ), we will show that for a fixed choice of Fenchel–Nielsen coor-

dinates, the Xi lie in a compact rectangular region of the Euclidean space

(R+)3g−3 × R3g−3.

By Theorem 12.10, for each Xi there is a pants decomposition Pi of S with ℓXi (γ) [ǫ, L] for each γ Pi (L is Bers' constant). Since there are only finitely many topological types of pants decomposition s of Sg , we can choose a subsequence, also denoted {Xi}, and a sequence fi Mod(Sg ) so that fi(Pi) = P1.

Now, in Fenchel–Nielsen coordinates adapted to P1 (with arbitrary seams), the Yi = fi · Xi have length parameters in [ǫ, L].

Since Dehn twists about the curves of P1 change the twist parameters by 2π, there is for each i a product hi of Dehn twists about the curves of P1 so

Again, the proof of Mumford's compactness criterion generalizes easily to non-closed surfaces.

12.5 THE TOPOLOGY OF MODULI SPACE AT INFINITY

Now that we know that M(S) is not compact, we can ask some finer questions. In this section we give a kind of quantification of how m any ways there are of “going to infinity” in M(S). One way of saying what we will be doing is to say that we will be computing the groups π0 and π1 of M(S) “relative to infinity.”

The main technical result in this subsection is the following, from which we will deduce connectedness properties of the cusp of M(S).

Proposition 12.11 Let g ≥ 2. Let X, Y Teich(Sg ) and suppose that their images X, Y M(Sg ) lie in M(Sg ) − Mǫ(Sg ). Then there is a path from X to Y whose projection to M(Sg ) lies in M(Sg ) − Mǫ(Sg ).

Proof. By the assumptions on X and Y, there are nontrivial simple closed curves α and β in Sg with ℓX(α) < ǫ and ℓY(β) < ǫ. By Theorem 5.2, there is a sequence of essential simple closed curves α = γ1, . . . , γn = β such that γi ∩γi+1 = for all i. Taking γ1 and γ2 to be part of a Fenchel–Nielsen coordinate system of curves, it is clear that there is a path in Teich(Sg ), starting from X, where γ1 has length less than ǫ at each point in the path, and where the path ends at a point X2 in which γ2 has length less than ǫ. Repeating this procedure from γ2 to γ3, etc., we obtain a path in Teich(Sg ) from X to some Y′, where each point on the path projects to M(Sg ) − Mǫ(Sg ), and in particular where the length of γn = β in Y′ is less than ǫ. We can then vary the last set of Fenchel–Nielsen coordinates to obtain a path from Y′ to Y where the length of β remains less than ǫ. The concatenation

THE END OF MODULI SPACE

The theory of ends of spaces is a way to encode the number of “no ncompact directions” of a space. We will only need the notion of “o ne end.” A connected, locally compact topological space X has one end if for every

compact set B X the space X \ B has only one component whose closure is noncompact. For example, compact spaces do not have one end; neither does the real line, as the complement of a closed interval has two unbounded components.

Suppose that X is a connected, locally compact metric space, and that Xi is an exhaustion of X by compact sets with X \ Xi path connected. Then X has one end. This holds for example for X = Rd with d ≥ 2, where one can choose Xi to be the ball of radius i about any fixed point.

Proposition 12.11 allows us to deduce the following.

Corollary 12.12 Let g ≥ 1. The moduli space M(Sg ) has one end.

Proof. If g = 1 then M(S1) is the modular surface. The fact that M(S1) has one end follows directly from the explicit description of M(S1) above.

If g ≥ 2, then M(Sg ) − Mǫ(Sg ) is connected for any ǫ > 0 by Proposition 12.11. Since the Mǫ(Sg ) form an exhaustion of M(Sg ), we conclude

The key fact needed for Proposition 12.11 is that the complex of curves is connected. Thus, Proposition 12.11 and Corollary 12.12 both hold for Sg,n with 3g − 3 + n ≥ 2.

LOOPS IN MODULI SPACE

Taking the fundamental group of the topological space underlying M(S) misses its salient features. Indeed, we have the following fact:

M(Sg ) is simply connected for all g ≥ 1.

For g = 1 this follows from the fact that M(S1) is the (0; 2, 3, ∞) hyperbolic orbifold. The simple connectivity of M(Sg ) is due to Maclachlan [112], and follows from the following three facts: Mod(Sg ) is generated by finite order elements (Theorem 7.16), the action of each fin ite order element on Teich(Sg ) has a fixed point (see Section 12.1), and the cover Teich(Sg ) → M(Sg ) enjoys the path-lifting property (the path-lifting propery holds any time we take the quotient of a simply connected space by

a properly discontinuous action [35, 4]). To get simple connectivity from this, take any loop in M(Sg ) based at the image in M(Sg ) of a fixed point of one of the generators of Mod(Sg ); the lift of this loop is a closed loop in Teich(Sg ), and any null-homotopy in Teich(Sg ) descends to a nullhomotopy in M(Sg ).

In light of this, we are led to consider the orbifold fundamental group of moduli space. The orbifold fundamental group of the quotient X/ of a simply connected space X by a group acting properly discontinuously (but not necessarily freely) is defined to be

π1orb(X/ ) ≈ .

Since M(S) = Teich(S)/ Mod(S) and since Teich(S) is simply connected, we have

π1orb(M(S)) ≈ Mod(S).

Two loops in X/ are considered to be homotopic if they have lifts to X that are –equivariantly homotopic. For example, a loop around the co ne point

of order two in M(S1) is trivial in the topological category, but nontrivial in the orbifold category; it has order two in π1orb(M(S)).

With the above comments in hand, we now consider loops in M(S) considered as an orbifold. The orbifold M(S1) has a unique homotopy class of loops that can be freely homotoped (in the orbifold sense) outside every compact subset of M(S1). This contrasts with the behavior of M(Sg ) when g ≥ 2.

Corollary 12.13 Let g ≥ 2. Any loop in M(Sg ) can be freely homotoped (even in the orbifold sense) outside every compact set in M(Sg ).

Proof. It suffices to consider loops that are essential and compact s ets that are of the form Mǫ(Sg ). Let α be any essential loop in M(Sg ), and X any point in M(Sg ) − Mǫ(Sg ). Since M(Sg ) is path connected, α can be freely homotoped to a loop β based at X. As above β can be lifted to β in Teich(Sg ). Proposition 12.11 gives a path γ between the endpoints

Another way to state Corollary 12.13 is that, for g ≥ 2, the (orbifold) fundamental group of M(Sg ) “relative to infinity” is trivial. More formally,

the inclusion map M(Sg ) − Mǫ(Sg ) ֒→ M(Sg ) induces an isomorphism of orbifold fundamental groups.

We also remark that both Corollary 12.12 and Corollary 12.13 are true with M(Sg ) replaced by the manifold Teich(Sg )/ , where is a finite-index torsion-free subgroup of Mod(Sg ).

PART 3

The classification and pseudo-Anosov

theory