Braid groups and Hodge theory
Curtis T. McMullen 25 April, 2009

Abstract This paper gives an account of the unitary representations of the braid group that arise via the Hodge theory of cyclic branched coverings of P1 , highlighting their connections with ergodic theory, complex reflection groups, moduli spaces of 1-forms and open problems in surface topology.

Contents
1 2 3 4 5 6 7 8 9 10 11 12 13 Introduction . . . . . . . . . . . . Topological preliminaries . . . . . From Hodge theory to topology . Action of the braid group . . . . Complex reflection groups . . . . The period map . . . . . . . . . . Arithmetic groups . . . . . . . . Factors of the Jacobian . . . . . Definite integrals . . . . . . . . . Complex hyperbolic geometry . . Lifting homological symmetries . The hyperelliptic case . . . . . . Polygons and Teichm¨ller curves u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 8 13 17 21 26 31 35 39 44 49 51 54

Research supported in part by the NSF.

1

Introduction

Every configuration of distinct points (b1 , . . . , bn ) in C determines a compact Riemann surface X by the equation y d = (x − b1 )(x − b2 ) · · · (x − bn ). (1.1)

This paper gives an account of the unitary representations of the braid group, and the geometric structures on moduli space, that arise via this branchedcovering construction. It also develops new connections with arithmetic groups, Teichm¨ller curves and ergodic theory, and highlights open problems u in surface topology and complex reflection groups. The general approach presented here pivots on the classification of certain arithmetic subgroups of U(r, s) which envelop the image of the braid group (§7). The case U(0, s) yields metrics of positive curvature on moduli space, finite representations of the braid group, algebraic definite integrals and rigid factors of the Jacobian of X. The case U(1, s) leads to the complex hyperbolic metrics on moduli space considered by Picard, Deligne–Mostow and Thurston. Other unitary groups U(r, s) provide moduli space with natural indefinite metrics. In particular, the hyperelliptic case (§12) yields an action of the braid group which, together with results of Ratner and Kapovich, illuminates the topology and ergodic theory of the period map u and its connection with the foliation of M∗ by Teichm¨ller geodesics. 0,n This section presents a overview of the discussion. Action of the braid group. We begin with topological considerations. Let ζd = e2πi/d . The braid group Bn is the fundamental group of the space C n of finite sets B ⊂ C with |B| = n. Each point configuration B determines, via equation (1.1), a compact Riemann surface X, whose cohomology group H 1 (X) forms the fiber of a flat, complex vector bundle over C n . The natural monodromy representation √ preserves the interesection form ( −1/2) α ∧ β, and commutes with the action of the deck group Z/d of X/C; thus it also preserves the eigenspace decomposition H 1 (X) =
q d =1

∼ ρ : Bn = π1 (C

n

, B) → Aut(H 1 (X))

H 1 (X)q = Ker(T ∗ − qI),

where T (x, y) = (x, ζd y). 1

As we will see in §3, the signature (r, s) of the Hermitian vector space H 1 (X)q can be determined by drawing a line through 1 and q in C; then r is the number of nth roots of unity strictly below the line, and s is the number strictly above. The signature changes from definite to hyperbolic to higher rank, and then back again, as q moves around the circle (Figure 1).
(3,2) (3,1) (4,0)

q 1 r

s
(2,2) (0,0)

(1,3)

(0,4)

Figure 1. The signature of H 1 (X)q for n = 6. Restricting attention to a single eigenspace, we obtain an irreducible unitary representation ρq : Bn → U(H 1 (X)q ) ∼ U(r, s). = In fact, Bn acts by a complex reflection group of type An−1 (q) (see §5); there n−1 is a spanning set (ei )1 for H 1 (X)q adapted to the standard generators τi of the braid group, such that   q−q 1−q 0 ... 0   √ q − 1 q − q 1 − q . . . 0  , ei , ej = −1    ...   0 ... 0 q−1 q−q τi∗ (x) = x − (q + 1) x, ei ei ei , ei

and

(for q = ±1). In particular, the generating twists τi of Bn act with finite order on H 1 (X)q . Although purely topological, these results are conveniently established using particular algebraic models such as y d = xn − 1 for X. We also use consistency of ρq under restriction to Bi × Bn−i ; geometrically, this amounts to letting X degenerate to a stable curve where a given Dehn twist becomes a holomorphic automorphism. (For a more algebraic approach to homological representations of Bn , see e.g. [KT].) 2

−k ∗ The period map. Now fix q = ζd with 1/n < k/d < 1, let T0,n denote r,s denote the space the Teichm¨ller space of n points in the plane, and let H u of positive lines in PH 1 (X)q ∼ PCr,s . The period map = ∗ fq : T0,n → Hr,s

sends a point (C, B) in Teichm¨ller space to the positive line spanned by u the holomorphic form [dx/y k ] ∈ H 1,0 (X) ⊂ H 1 (X). Thus it records a part of the Hodge structure on X. Using ideas familiar from polygonal billiards, in §6 we show that fq is a local homeomorphism when q n = 1. It is also equivariant with respect to the ∗ action of Bn via the mapping-class group on T0,n and via the representation r,s . Thus the period map gives the moduli space M∗ the structure of ρq on H 0,n a (G, X)-manifold, where G = U(r, s) and X = Hr,s . A similar construction applies to M0,n (the moduli space of n points in the sphere instead of in the −2 plane), when q = ζn . (For more on period mappings in general, see e.g. [CMP].) Arithmetic groups. It is a challenging problem to describe the image of the braid group under ρq . In particular, it is unknown when An−1 (q) ∼ = ρq (Bn ) is a lattice in U(r, s). To study the image of ρq in more detail, we begin by observing that the action of Bn on H 1 (X)q preserves the module Λn,q = H 1 (X, Z[q])q . Thus ρq (Bn ) is contained in the countable subgroup U(Λn,q ) ⊂ U(H 1 (X)) ∼ U(r, s) = of unitary automorphisms of this module. In §7 we determine all pairs (n, q) such that U(Λn,q ) is discrete (see Table 9). When discreteness holds, ρq (Bn ) is also discrete, and potentially arithmetic. There are infinitely many (n, q) such that U(Λn,q ) is discrete, but for n > 12 this only occurs when Z[q] is itself discrete (i.e. when d = 2, 3, 4 or 6). Special cases. In §8—§13 we develop the interaction between fq , ρq (Bn ) and U(Λq,n ) in more detail, in three special cases. 1. The finite case. Let q be a primitive dth root of unity. In §8 we show that ρq (Bn ) is finite iff U(Λn,q ) is finite. Referring to Table 9, we find there are exactly 9 pairs (n, d) such that this finiteness holds. In each case H 1 (X)q has signature (0, s) or (s, 0), and the Hodge structure on the corresponding part of H 1 (X) is rigid; equivalently, we have an isogeny Jac(X) ∼ J(X) × A 3

where A is independent of X. For example, when X is defined by y 10 = (x − b1 )(x − b2 )(x − b3 )

its Jacobian has a rigid factor isomorphic to (C2 /Z[ζ5 ])2 . In the other 7 cases, the rigid factor is a product of elliptic curves. Certain instances of this rigidity can be seen in terms of conformal mappings. For example, when all the (bi ) are real, the case (n, d) = (3, 4) is related to the fact that any (45,45,45,135)-degree quadrilateral can be mapped conformally onto the complement of a symmetric slit in a rightisosceles triangle (Figure 2). These polygons, when doubled, represent the sphere with the metrics |dx/y 3 | and |(x − a)dx/y 3 | respectively. Since the second polygon develops, under reflection through its sides, onto a periodic pattern in the plane, the form (x − a)dx/y 3 is pulled back from the square √ torus, giving Jac(X) a factor of C/Z[ −1].
b1 b1

b3 a

8

8

b3

b2

b2

Figure 2. Quadrilaterals and squares. By similar considerations, in §9 we determine 17 values of (n, µ) such that the definite integral
b2

I(b1 , b2 , . . . , bn ) =
b1

is an algebraic function of its parameters. The integral is evaluated explicitly for (n, µ) = (4, 3/4), as an example. 2. The complex hyperbolic case. In §10 we turn the case where H 1 (X)q has signature (1, s). In this case the period map
∗ fq : T0,n → H1,s ∼ CHs =

dx ((x − b1 )(x − b2 ) · · · (x − bn ))µ

is a holomorphic map to the complex hyperbolic ball. Using the Schwarz lemma, we find ρq (Bn ) ⊂ U(1, s) is a lattice whenever it is discrete. Examining Table 9, we obtain 24 cases where ρq (Bn ) is an arithmetic lattice. In 4

particular we find that for n = 4, 5, 6, 8 and 12, the period map fq : M0,n → CHn−3 /ρq (Bn )
−2 at q = ζn presents moduli space as the complement of a divisor in a finitevolume, arithmetic, complex-hyperbolic orbifold. These results are special cases of the work of Deligne–Mostow and Thurston, which we review in §10. We also develop parallels between non-arithmetic lattices in U(1, 2) and the known Teichm¨ller curves on Hilbert modular surfaces. u ∼ Next, consider the following purely topological problem. Let Sp(X)T = T denote the centralizer of T ∗ in the automorphism group of H 1 (X, Z), Sp2g (Z) and let Sp(X)T = Sp(X)T | Ker Φd (T ∗ ) d

where Φd (x) is the dth cyclotomic polynomial. We say a group homomorphism is almost onto if the image has finite index in the target. Problem: Is the natural map Bn → Sp(X)T almost onto? d In §11 we connect this problem to arithmeticity of lattices. We show that for n = 3, the answer is yes iff the (2, 3, p) triangle group is arithmetic, where p is the order of d−2 in Z/2d. (There are 16 such cases.) More generally, the answer is yes whenever (d, n) appear in Table 9 with signature (0, s) or (1, s). On the other hand, using a non-arithmetic lattice in U(1, 2) constructed by Deligne and Mostow, we show the answer is no when (n, d) = (4, 18). 3. The hyperelliptic case. In §12 we discuss the hyperelliptic case, where d = 2 and q = −1. In this case H 1 (X)q = H 1 (X), and ρq (Bn ) has finite ∼ index in U(Λn,q ) = Sp2g (Z) by [A’C]. For convenience, assume n = 2g + 1 is odd and g ≥ 2. Then there is a natural Weierstrass foliation of Teichm¨ller space by complex geodesics, u each of which is sent into a straight line under the period map
∗ fq : T0,n → Hg,g .

In fact there is an equivariant action of SL2 (R) on suitable circle bundles over the domain and range of fq . Using this action we find: 1. The period map fq is an infinite-to-one local homeomorphism, but not a covering map to its image;
∗ 2. It transports the discrete action of Bn on T0,n to an ergodic action on Hg,g ; and

5

∗ 3. The complement of the image fq (T0,n ) is contained in a countable union of lines.

The last assertion is deduced using Ratner’s theorem as in [Kap]. The Weierstrass foliation descends to a foliation W of M∗ . Although 0,n almost every leaf of W is dense, Veech showed the special leaf V passing i through [(bi = ζn )] is a properly embedded algebraic curve. In §13 we discuss this Teichm¨ller curve V from the perspective of braid groups and u flat metrics on the sphere. We first show that π1 (V ) is the (2, n, ∞) triangle group generated by the images of the braids α = τ1 τ3 . . . τ2g−1 and β = τ2 τ4 . . . τ2g . (Note that αβ is a lift of the Coxeter element for the An−1 diagram.) We then describe the abstract Euclidean polygons Q = (C, |ω|) associated to the two orbifold points of V : one is obtained from a regular n-gon by folding its vertices to a single point, while the other is the double of an immersed, rightangled disk whose edge lengths form an eigenvector for the An−1 adjacency matrix. These descriptions allow one to explicitly evaluate the period map at the orbifold points of V .

Figure 3. Line configuration associated to 5 points on a conic.

where Li (x, y) = y − 2bi x + b2 and Q(x, y) = y − x2 . This complex surface is i branched over the conic Q, its tangent lines Li at Pi = (bi , b2 ), and (possibly) i the line at infinity (Figure 3). After desingularizing X one obtains, for each eigenspace, a unitary action of Bn on H 2 (X)q . It would be interesting 6

Remark: Cyclic multiple planes. One can also associate to each point configuration (b1 , . . . , bn ) ∈ C n the cyclic covering X of P2 defined by the equation z d = L1 (x, y) · · · Ln (x, y)Q(x, y),

to study the associated period maps as Hodge-theoretic counterparts to the Lawrence–Krammer–Bigelow representations ξq of Bn [Law], [At], [Kr], [Bg2]. Notes and references. The hypergeometric functions of the form F = b2 n −µi dx have been studied since the time of Euler; the present 1 (x − bi ) b1 discussion is closely connected to works of Schwarz and Picard [Sch], [Pic]. For more on the classical theory of hypergeometric functions, see e.g. [Kl], [SG] and [Yo]. Deligne and Mostow developed a modern perspective on hypergeometric functions, and used their monodromy to exhibit non-arithmetic lattices in U(1, s) for certain s ≥ 2 [Mos1], [DM1], [DM2]; see also [Sau] and the surveys [Mos2] and [Par]. Thurston recast the work of Deligne and Mostow in terms of shapes of convex polyhedra, by observing that the integrand of F determines a flat metric on the sphere with cone-type singularities. Convexity of the polyhedron corresponds to signature (1, s) for the metric on moduli space [Th2]. For other perspectives, see [V3] and [Tr]. In this paper we have focused on the case where all µi assume a common rational value k/d. This case leads directly to algebraic curves and their Jacobians, and yields representations of the full braid group. At the same time it allows for non-convex polyhedra (as in Figure 2) and non-hyperbolic signatures (r, s), because |dx/y k | can be negatively curved at x = ∞. Analogous Hodge-theory constructions have been used to study, instead of finite sets in P1 , hypersurfaces of low degree in P1 × P1 or Pn , n ≤ 4; see e.g. [DK], [Ko], [ACT1], [ACT2], [Al2], [Lo2]. The action of the mappingclass group of a closed surface on the homology of its finite abelian covers is studied in [Lo1]. We note that ρq is essentially a specialization of the Burau representation (§5); as a representation of the Hecke algebra Hn (q), it corresponds to the partition n = 1 + (n − 1) [J], [KT]. The Lawrence–Krammer–Bigelow representations ξq correspond to the next partition, n = 2 + (n − 2). Our investigation began with the observation that ρq coincides with the quantum representations discussed in [AMU] when n = 3. It would be interesting to understand more fully the connection between classical and quantum representations. Acknowledgements. I would like to thank D. Allcock, J. Andersen, B. Gross, M. Kapovich, D. Margalit and J. Parker for useful conversations. Notation. We use x to denote the largest integer ≤ x, and x for the smallest integer ≥ x. We let U(V ) denote the unitary group of a Hermitian vector space V over C. The unitary group of the standard Hermitian form of signature (r, s) on Cr+s is denoted by U(r, s), and the corresponding 7

projective group by PU(r, s) ∼ U(r, s)/ U(1). The automorphism group of = a real symplectic vector space is denoted by Sp(V ) ∼ Sp2g (R). =

2

Topological preliminaries

A basic reference for the algebraic structure of the braid groups and mappingclass groups is [Bi]; see also [BZ], [KT] and [FM]. Moduli spaces and point configurations. Given n ≥ 2, let C n denote the space of unordered subsets B ⊂ C with |B| = n, and similarly for C n . Each of these spaces is naturally a connected complex manifold; for example, C n can be identified with the space of monic complex polynomials of degree n with nonvanishing discriminant. The quotient of C n by the group of affine automorphisms z → az + b yields the moduli space of n-tuples of points in the plane, M∗ = C n / Aut C. 0,n n / Aut C is the moduli space of n-tuples of points on Similarly M0,n = C the Riemann sphere C = C ∪ {∞}. Since some point configurations have extra symmetries, these moduli space are orbifolds. The natural maps C
n ∗ → M0,n and M∗ → M0,n 0,n

(2.1)

are both fibrations, with fibers Aut(C)/ Aut(C, B) and (C − B)/ Aut(C, B) respectively.

τ1

τ2

τ3

τ1

τ4

τ2

τ3

Figure 4. Right twists generating the braid group.

The braid group. The braid group Bn is the fundamental group of C n . The motion of B around any loop in C n can be extended to a motion of C with compact support (fixing a neighborhood of ∞); this gives an isomorphism Bn ∼ π1 (C n , B) ∼ Modc (C, B) = = between the braid group and the compactly-supported mapping-class group of the pair (C, B).

8

Standard generators τ1 , . . . , τn−1 for the braid group are given, for the configuration B = {1, 2, . . . , n} ⊂ C, by right Dehn half-twists around loops enclosing [i, i + 1] as in Figure 4. This means τi rotates the disk enclosing [i, i + 1] counter-clockwise by 180◦ . These n − 1 generators satisfy the braid relation τi τi+1 τi = τi+1 τi τi+1 (2.2) for i = 1, . . . , n − 2, as well as the commutation relation τi τj = τj τi for |i − j| > 1; and these generators and relations give a presentation for Bn . Alternatively, for the more symmetric configuration
2 n−1 B = {1, ζn , ζn , . . . , ζn },

ζn = exp(2πi/n)

shown at the right in Figure 4, we can take τi to be the right half-twist i+1 i interchanging ζn and ζn . Then a rotation of B by 2π/n is given by σ = τ1 τ2 · · · τn−1 , and we have τi+1 = στi σ −1 . In particular, we obtain an additional twist τn = στn−1 σ −1 . For n ≥ 3, the center of Bn is generated by σ n . Mapping-class groups. Like the braid group, the mapping-class groups of n-tuples of points in the plane and on the sphere can be defined by Mod∗ ∼ π1 (M∗ ) ∼ Mod(C, B) 0,n = 0,n = and Mod0,n ∼ π1 (M0,n ) ∼ Mod(C, B). = =

The groups at the right consist of the orientation-preserving diffeomorphisms f : (C, B) → (C, B) (resp. f : (C, B) → (C, B)), up to isotopy. From (2.1) we have natural surjective maps Bn → Mod∗ → Mod0,n . The kernel of 0,n the first is the center of Bn ; the kernel of the second is the smallest normal subgroup containing (τ2 · · · τn−1 )n−1 . The braid relation (2.2) can also be written as (τi τi+1 )3 = (τi τi+1 τi )2 ; in particular, B3 is a central extension of the (2, 3, ∞) triangle group
∗ a, b : a2 = b3 = 1 ∼ PSL2 (Z) ∼ Mod0,3 . = =

9

Cyclic covers. Fix an integer d > 0 and a point configuration B = {b1 , . . . , bn } ⊂ C. Let p(z) = n (z − bi ). There is a unique homomor1 phism w : π1 (C − B) → Z/d sending a loop to the sum of its winding numbers around the points of B; concretely, it is given by w(α) = 1 2πi p (z) dz mod d. p(z)

α

Let π : X → C denote the corresponding covering of C − B, completed to a branched covering of C. Let T : X → X be the generator of the deck group corresponding to 1 ∈ Z/d, and let ∞ and B denote the preimages of ∞ and B under π. It is easy to see that |B| = |B| and e = |∞| = gcd(d, n); in particular, X is unramified over infinity iff d divides n. By the RiemannHurwitz formula, the genus g of X is given by g= (n − 1)(d − 1) + 1 − e · 2

Eigenspaces. Recall that H 1 (X, C) = √ 1 (X) carries a natural intersecH tion form of signature (g, g), defined by ( −1/2) α ∧ β. Any orientationpreserving diffeomorphism of X preserves this form, so the map f → (f −1 )∗ |H 1 (X) gives a natural unitary representation of the mapping-class group Mod(X) → U(H 1 (X)) ∼ U(g, g). = Since [T ] ∈ Mod(X) has order d, we also have an eigenspace decomposition H 1 (X) = H 1 (X)q
q d =1

(2.3)

where H 1 (X)q = Ker(T ∗ − qI). Let (r, s) denote the signature of the intersection form on H 1 (X)q . The mapping-classes Mod(X)T commuting with T also preserve these eigenspaces, so we have a natural representation Mod(X)T → U(H 1 (X)q ) ∼ U(r, s) = for each q. 10

Clearly the invariant cohomology satisfies H 1 (X)1 ∼ H 1 (C) ∼ 0. For = = the sequel, we assume q d = 1 but q = 1. Lifting braids. To bring the braid group into the picture, observe that any compactly supported diffeomorphism φ : (C, B) → (C, B) has a unique lift φ : X → X which is the identity on a neighborhood of ∞. This lifting gives a natural map Bn ∼ Modc (C, B) → Mod(X)T . = Passing to cohomology, we obtain a natural representation ρq : Bn → U(H 1 (X)q ) ∼ U(r, s). = Our goal is to understand this representation. Lifting mapping classes. Note that any ψ ∈ Mod(C, B) has a lift ψ to X which is well-defined up to a power of T ; we pass to the braid group only to make the lift unique. Since T |H 1 (X)q = qI, the representation ρq descends to a map Mod(C, B) → PU(r, s). On the other hand, if X is unramified over infinity (i.e. if d|n), then ψ is uniquely determined by the condition that it fixes ∞ pointwise, so ρq factors through Mod(C, B). Similarly, if X is unramified over ∞ then any ψ ∈ Mod(C, B) has a lift to X which is well-defined up to the action of T . Thus we have a commutative diagram: Bn Mod∗ 0,n
 
ρq

(2.4)

// U(r, s) 99  // PU(r, s) 99

(2.5)

Mod0,n whose diagonal arrows exist when d|n. Invariant subsurfaces. To describe ρq , we first choose standard generators τ1 , . . . , τn for Bn ∼ Modc (C, B). Each τi is a half Dehn twist supported in = an open disk Di ⊂ C with Di ∩ B = {bi , bi+1 } (or {bn , b1 } when i = n). Let Xi = π −1 (Di ) ⊂ X. This is an open, connected, incompressible subsurface of X of genus (d − 1)/2 , with one or two boundary components (depending on the parity of d; see Figure 5). 11

bi

b i+1

Figure 5. The surface Xi is built from the Z/d-covering space of a pair of pants in C shown above, by capping off the inner and outer circles at the left with disks.

Let Yi be the closed subsurface X − Xi . Since τi is supported on Di and τi fixes ∞ ⊂ Yi , the lift τi is supported on Xi . In particular, both Xi and Yi are invariant under T and τi . Action on cohomology. For n ≥ 3, Yi is connected. In this case, by Mayer-Vietoris, we have an equivariant exact sequence
1 0 → Hc (Xi ) → H 1 (X) → H 1 (Yi ) → 0, 1 where Hc (Xi ) denotes cohomology with compact supports. We denote the corresponding exact sequence of eigenspaces of T by 1 0 → Hc (Xi )q → H 1 (X)q → H 1 (Yi )q → 0.

8

(2.6)

The same sequences are exact for n = 2, except when d is even. If d is even then Yi is disconnected (it is a pair of disjoint disks), and we have instead the exact sequence
1 H 1 (∂Xi ) → Hc (Xi ) → H 1 (X) → H 1 (Yi ) = 0.

(2.7)

In any case, since τi is supported in Xi , we have τi∗ |H 1 (Yi ) = I. (2.8)

1 In the next section we will show that τi∗ |Hc (Xi ) = −qI. Remark: stabilization. Note that if d divides d , then the corresponding branched covering X factors through X, and we obtain an equivariant isomorphism ι : H 1 (X)q ∼ H 1 (X )q . =

12

Thus the representation ρq of Bn depends only on q, not on d. The isomorphism ι is not, however, unitary; it multiplies the intersection form by d /d. To obtain a unitary isomorphism, one must rescale ι by d/d .

3

From Hodge theory to topology

In this section we analyze the action of the braid group further, by using holomorphic 1-forms to represent the cohomology of X. Hodge structure. A given point configuration B ⊂ C determines a complex structure on the genus g cyclic covering space π : X → C branched over B ∪ {∞}. The complex structure in turn determines the g-dimensional space of holomorphic 1-forms Ω(X) on X, and the Hodge decomposition H 1 (X) = H 1,0 (X) ⊕ H 0,1 (X) ∼ Ω(X) ⊕ Ω(X). = The intersection form α, β = √ −1 2 α∧β (3.1)

X

is positive-definite on H 1,0 (X) and negative-definite on H 0,1 (X). Symmetric curves. Now assume B coincides with the nth roots of unity. Then X is isomorphic to the smooth algebraic curve given by y d = xn − 1, with the projection to C given by π(x, y) = x and with the generating deck transformation of X/C given by T (x, y) = (x, ζd y). We then have an additional symmetry R : X → X given by R(x, y) = (ζn x, y). Eigenforms. Since the abelian group R, T preserves the intersection form and the Hodge structure, we have a decomposition H 1 (X) = H 1 (X)jk

of the cohomology of X into orthogonal eigenspaces
−k j H 1 (X)jk = Ker(R∗ − ζn I) ∩ Ker(T ∗ − ζd I),

13

H
5 4

1,0

k

3 2 1

H 0,1

1

2

3

4

5

6

7

8

j
Figure 6. Occupied eigenspaces H 1 (X)jk are shown in white. Those above the diagonal span H 1,0 (X), and those below span H 0,1 (X).

and a further refinement
1,0 0,1 H 1 (X)jk = Hjk (X) ⊕ Hjk (X).

We can assume 0 ≤ j/n, k/d < 1. To populate these eigenspaces, we consider the meromorphic forms defined by xj−1 dx . ωjk = yk The divisor (ωjk ) = (dx/x) + j(x) − k(y) is easily calculated using the fact that the fibers of X/C have multiplicity d/e over ∞, d over B, and 1 elsewhere, where e = gcd(d, n) = |∞|. Letting B, ∞ and 0 denote the preimages of B, 0 and ∞ on X, considered as divisors where each point has weight one, we obtain (x) = −(d/e)∞ + 0,

(dx/x) = −∞ − 0 + (d − 1)B, which yields

(y) = −(n/e)∞ + B, and

(ωjk ) = (kn/e − jd/e − 1)∞ + (j − 1)0 + (d − 1 − k)B. 14

(3.2)

Thus ωjk is holomorphic for all j, k such that 0 < j/n < k/d < 1, and hence in this range we have dim H 1,0 (X)jk ≥ 1. Similarly, by considering ω jk , we find dim H 0,1 (X)jk ≥ 1 for 0 < k/d < j/n < 1. But the total number of pairs (j, k) satisfying one condition or the other is exactly (n−1)(d−1)+(1− e) = 2g = dim H 1 (X). Thus ωjk and ω jk account for the full cohomology of X. In summary: Theorem 3.1 We have dim H 1 (X)jk = dim H 1,0 (X)jk = 1 if 0 < j/n < k/d < 1, and dim H 1 (X)jk = dim H 0,1 (X)jk = 1 if 0 < k/d < j/n < 1. The remaining eigenspaces are zero. The case of a degree d = 6 covering branched over n = 9 points (and infinity) is shown in Figure 6.
−k Corollary 3.2 When q = ζd , the signature of the intersection form on H 1 (X, C)q is given by (r, s) = ( n(k/d) − 1 , n(1 − k/d) − 1 ).

Indeed, the signature (r, s) just counts the number of terms of type (1, 0) and (0, 1) in the expression H 1 (X)q = j H 1 (X)jk , or equivalently the number of terms above and below (but not on) the diagonal in the kth row of Figure 6. Corollary 3.3 The eigenspace H 1 (X)q has dimension n − 2 if q n = 1, and dimension n − 1 otherwise. Genus zero quotients. We note that the Riemann surfaces X/ R , X/ T and X/ RT all have genus zero. This provides another explanation for the vanishing of eigenspaces — any R, T or RT -invariant cohomology class descends to the sphere, and H 1 (C) = 0. Remark: Lefschetz formulas. The eigenspace decomposition of H 1 (X) shown in Figure 6 can be determined, alternatively, by applying the Lefschetz formulas Tr(g|H 1 (X)) = 2 − 1 and Tr(g|H 0,1 ) = 1 − (1 − g (z))−1

g(z)=z

g(z)=z

to compute the characters of G = R, T acting on H 1 (X) and H 0,1 (X). The Chevalley-Weil formula similarly gives the characters of G|H 1,0 (X) for general Galois coverings [ChW]; see also [Hu]. 15

Figure 7. A model for the branched covering X.

and extend it by linearity to Z[G]. Let R ⊂ Z[G] denote the radical of this form (the set of elements x such that [x, y] = 0 for all y). Theorem 3.4 The space H1 (X, Z) is isomorphic to Z[G]/R as a symplectic Z[G]-module.

Remark: the hexagon model. Here is an explicit model for H 1 (X, Z) as a symplectic lattice equipped with the action of G = Z/n × Z/d = R, T . The proof will also provide a global topological model for G acting on X. Let S = {(1, 0), (0, 1), (−1, −1)} ⊂ G. Define an alternating form on the standard basis of the group ring Z[G] ∼ Znd by =  1 if g − h ∈ S,  [g, h] = −1 if h − g ∈ S, and (3.3)   0 otherwise,

Sketch of the proof. Let A and B be a pair of 2-disks, colored white and black respectively. Construct a surface Y0 from (A B) × G by joining A × {g} to B × {g + s} with a band whenever s ∈ S (see Figure 7). The group G acts by deck transformations on Y0 , with quotient a surface Y0 /G of genus 1 with 1 boundary component. Now put a half-twist in each band of Y0 (as in Figure 5), to obtain a new surface Y1 . Then Y1 /G has genus 0 and 3 boundary components. By filling in the boundary components of Y1 and Y1 /G with disks, we obtain closed surfaces Y and Z ∼ S 2 and a degree nd branched covering = π : Y → S2 with deck group G. 16

Let C ⊂ Y be the core curve of one of the hexagons appearing in Figure 7. It is then straightforward to check that the intersection number (gC) · (hC) in H1 (Y, Z) is given by (3.3). By comparing the rank of this form with the first Betti number of Y , we find the inclusion Z[G]/R → H 1 (Y, Z) is an isomorphism. Now one can also regard Y /G as an orbifold with signature (n, d, nd/e), where e = gcd(d, n). The generators of G correspond loops around the points of orders n and d. Similarly, the map p : X → C given by p(x, y) = xn presents X/ R, T as a covering of the same orbifold, with singular points at 0, 1, ∞. The proof is completed by identifying p : X → C with π : Y → S 2 .

4

Action of the braid group

In this section we determine the representation ρq : Bn → U(H 1 (X)q ). Theorem 4.1 There is a spanning set (ei )n for H 1 (X)q such that the in1 tersection form is given by ei , ei ei , ei+1 ei , ej = 2 Im q, √ = −1(1 − q),

and

= 0 if |i − j| > 1;

and the action of the braid group is given by √ −1 ∗ x, ei ei τi (x) = x − 2 if q = −1, and otherwise by τi∗ (x) = x − (q + 1) x, ei ei . ei , ei

(4.1)

(4.2)

Derivatives at infinity. Recall that e = |∞| = gcd(d, n). Near infinity we have y d ∼ xn , and thus z = y d/e /xn/e maps ∞ to the eth roots of unity. It follows that the pointwise stabilizer of ∞ in R, T ∼ Z/n × Z/d is the = cyclic group generated by RT . Similarly, a local coordinate near a point of ∞ is provided by t = 1/(xa y b ), 17

(Here the indices i, j ∈ Z/n and |i − j| > 1 means i − j = −1, 0 or 1 mod n.)

where ad + bn = e. Computing in this coordinate, we find
−ad−bn −e −a −b (RT ) (∞) = ζn ζd = ζnd = ζnd .

(4.3)

Action of rotations. Now recall that σ = τ1 · · · τn−1 ∈ Modc (C, B) represents a right 1/n Dehn twist around a loop enclosing B. The actions of R and T on H 1 (X) are given transparently by the eigenspace description just presented. To bring the braid group into play, we show: Proposition 4.2 We have [σ] = [RT ] in Mod(X). Proof. The mapping-class σ is represented by a homeomorphism which agrees with the rigid rotation R0 (x) = ζn x except on a small neighborhood of ∞. Thus σ agrees with some lift RT i of R0 outside a small neighborhood of ∞. Now near infinity, σ twists by angle −2π/n back to the identity. Since d/e sheets of X come together at each point of ∞, upon lifting this becomes of twist by angle (−2πe)/nd. It follows that
−e (RT i ) (∞) = ζnd = (RT ) (∞)

by (4.3), and thus i = 1. Corollary 4.3 The kernel of the lifting map Bn → Mod(X)T is the cyclic central subgroup generated by σ nd/e . Proof. The lifted map φ ∈ Mod(X)T determines [φ] ∈ Mod(C, B), so the kernel of lifting must be contained in the kernel of the map Bn → Mod(C, B) which is generated by σ n ; now apply the preceding Proposition. Action of twists. We can now determine the action of twists on H 1 (X). As in §2 we assume τi is supported in a disk with Di ∩ B = {bi , bi+1 }, and let Xi = π −1 (Di ). Theorem 4.4 For each i and q = 1, we can choose ei = 0 such that
1 Hc (Xi )q = Cei , τi∗ (ei ) = −qei , and

ei , ei = − Im q.

Proof. Since the statement concerns only the covering Xi → Di , which is branched over 2 points, it suffices to treat the case where n = 2, i = 1 and X is defined by y d = x2 −1. In this case R∗ = −I on H 1 (X); indeed, y : X → C 18

presents X as a hyperelliptic Riemann surface, with R(x, y) = (−x, y) its hyperelliptic involution. Moreover τ1 = σ, so we have
∗ τ1 = σ ∗ = (RT )∗ = −T ∗ on H 1 (X)

(4.4)

by Proposition 4.2. Now recall from (2.7) that for each q we have an exact sequence
1 H 1 (∂Xi )q → Hc (Xi )q → H 1 (X)q → 0.

If q = −1, then the first term is zero and we get an equivariant isomorphism 1 Hc (Xi )q ∼ H 1 (X)q . By Theorem 3.1, the latter space has dimension one = −k and satisfies H 1 (X)q = H 1,0 (X)q when Im q = Im ζd > 0 (i.e. 1/2 < k/d < 1 (X) = H 0,1 (X) when Im q < 0. Thus H 1 (X ) admits a basis 1), and H q q i q vector with ei , ei = Im q; and we have
∗ τ1 (ei ) = −T ∗ (ei ) = −qei

by (4.4). On the other hand, when q = −1 we have H 1 (X)q = 0, and by equation (2.6) we have an equivariant isomorphism H 1 (∂Xi )−1 ∼ Hc (Xi )−1 . = 1 In this case d is even and ∂Xi has two components, which are preserved by τ1 but interchanged by T ; thus H 1 (∂Xi )−1 is a one-dimensional La1 grangian subspace of Hc (Xi ), spanned by a vector ei fixed by τi∗ and satisfying ei , ei = Im q = 0.

1 Corollary 4.5 The lifted twist satisfies τi∗ = −T ∗ on Hc (Xi ) ⊂ H 1 (X).

Example of genus 1. Consider the case (n, d, q) = (4, 2, −1). Then the elliptic curve y 2 = x4 − 1 can be identified with the Riemann surface √ √ X = C/Z[ −1] in such a way that R(z) = −1z, X1 is a vertical annulus, and the lift of τ1 is the full right twist τ1 (z) = z + Im z. The other lifted twists are given (up to isotopy) by τi+1 = Ri τ1 R−i , i = 2, 3, 4. Noting that √ −1 ∗ τ1 (dz) = dz + d Im z = dz − (dz − dz), 2

19

we find the action of the braid group on H 1 (X)q = H 1 (X) = C[dz] ⊕ C[dz] is given by τi∗ (x) =x− −1 x, ei ei , 2 where e1 = dz − dz and ei+1 = (R−i )∗ e1 . Since Cei is the unique eigenspace 1 for τi∗ |H 1 (X), it must coincide with Hc (Xi ). With these normalizations, we have ei , ei = 0 and √ √ e1 , e2 = ei , ei+1 = −1 dz − dz, dz + dz = 2 −1 for all i. Proof of Theorem 4.1. We can assume X is defined by y d = xn − 1 and the disks Di are chosen symmetrically with respect to rotation, as in Figure 4; then R(Xi ) = Xi+1 . Using Theorem 4.4, we can choose ei ∈ H 1 (X)q such that 1 Hc (Xi )q = Cei , ei , ei = 2 Im q and R∗ (ei+1 ) = ei . Then ei , ej = 0 if |i − j| > 1, since Xi and Xj are disjoint up to isotopy. If q = −1, then ei , ei = 0 as well, and ei can be normalized so that √ ei , ei+1 = 2 −1 and τi∗ acts by (4.1), by comparison with the case y 2 = x4 − 1 above. If q = 1 then H 1 (X)q = 0 and the theorem also holds. Now suppose q = −1 (and q = 1, as usual). Since the value of A = ei , ei+1 = e1 , e2 is independent of n, to compute it we may assume n = d. Then by Theorem 3.1, 1 and q −1 do not occur as eigenvalues of R|H 1 (X)q , and thus n ei = n q −i ei = 0. Pairing these vectors with e1 , we find 1 1 A + 2 Im q + A = qA + 2 Im q + qA = 0, √ which gives A = −1(1 − q). Once the inner product is known, we can also compute that
n −ji ζd ei , e1 i=1 j whenever ζd = 1 or q −1 ; thus the eigenspaces of R|H 1 (X)q that do occur are all contained in the span of (ei )n . It follows that (ei )n spans H 1 (X)q . 1 1 1 The intersection form on Hc (Xi ) is nondegenerate, so we have an orthogonal splitting 1 1 1 H 1 (X)q = Hc (Xi )q ⊕ Hc (Xi )⊥ ∼ Hc (Xi )q ⊕ H 1 (Yi )q q = j −j = ζd A + 2 Im q + ζd A = 0

√

20

preserved by τi∗ . Since τi∗ acts by −q on the first factor by the identity on the second (cf. (2.8)), its action on H 1 (X)q is given by the complex reflection formula (4.2). Remark: The cases q = ±1. When q = 1 we have H 1 (X)q = 0, and the formulas in Theorem 4.1 give ei , ej = 0. On the other hand, if we rescale by (Im q)−1 as q → 1 in S 1 , then the inner product converges to the Coxeter matrix   2 −1 −1    −1 2 −1     ei , ej =   −1 2 −1    ··· −1   −1 −1 2

for the affine diagram An , and the action of Bn on H 1 (X)q converges to the action of Sn as a reflection group. 1 √ As q → −1 in S , no rescaling is necessary: we have (q + 1)/ ei , ei → −1/2, and thus the transvection (4.1) arises as a limit of the complex reflections (4.2). In the case q = −1 we can also express the action of Bn by τi∗ (x) = x + [x, ei ]ei , where [ei , ei+1 ] = 1 and [ei , ej ] = 0 for |i − j| > 1. This alternating form is a positive multiple of the usual intersection pairing on H 1 (X, R).

5

Complex reflection groups

In this section we put the action of the braid group on H 1 (X)q in context by relating it to Artin systems, complex reflection groups and the Burau representation. Artin systems. Let Γ be a finite graph with vertex set S. The associated Artin group A(Γ) is the group freely generated by S, modulo the following relations: 1. If s and t are joined by an edge of Γ, then sts = tst (we say s and t braid); 2. Otherwise, s and t commute.

21

A n−1
*

~ An

Figure 8. The braid groups of n points in C and C∗ correspond to the Coxeter diagrams An−1 and An .

Complex reflections. Let V be a finite-dimensional complex vector space, and suppose q ∈ S 1 , q = −1. A linear map T : V → V is a q-reflection if it acts by v → −qv on a 1-dimensional subspace of V , and by the identity on a complementary subspace. Every Artin group admits a finite-dimensional representation ρq : A(Γ) → GL(CS ) such that each s ∈ S acts by a q-reflection through es . One such representation is given by the action s(x) = x − (q + 1) x, es es es , es (5.1)

(For weighted graphs one allows longer relations of the form stst · · · s = tsts · · · t, but we will only need the simply-laced systems above.)

with respect to the (possibly degenerate) real-valued inner product satisfying  2 if s = t,  es , et = 0 if s and t commute, and   − sec(θ/2) if s and t braid.

Here θ = arg q ∈ (−π, π). The value − sec(θ/2) is determined (up to sign) by the condition that the braid relation sts = tst holds in the subspace C{s,t} .

Phase shift. More generally, we can modify this representation of A(Γ) by specifying a phase shift φ(s, t) ∈ S 1 whenever s and t braid (cf. [Mos1]), to obtain a new inner product es , et = φ(s, t) sec(θ/2). 22

We assume φ(s, t) = φ(t, s), so this inner product is Hermitian. Then the action defined by (5.1) for the new inner product still respects the braiding and commutation relations. The shifted representation of A(Γ) depends only on q and the cohomology class of φ in H 1 (D, S 1 ) (represented by the cochain sending the edge [s, t] to φ(s, t) ∈ S 1 .) In particular, if Γ is a tree, then the q-reflection representation is unique up to conjugacy, and we refer to its image as a complex reflection group of type Γ(q). Reduction. The quotient of CS by the subspace N = {x : x, y = 0 ∀y} yields the reduced complex reflection representation ρq : A(Γ) → U(CS /N ). Proposition 5.1 If Γ is connected, then ρq is irreducible for all q = −1. Proof. Let V ⊃ N be an invariant subspace of CS with V = N . Then v, es = 0 for some v ∈ V and s ∈ S. Therefore v −sv = (q +1) v, es es ∈ V and hence es ∈ V . Now if s and t braid, then es , et = 0 and thus et ∈ V as well. Connectivity of Γ then implies V = CS . Coxeter groups and Shephard groups. The Shephard group A(Γ, d) is the quotient of A(Γ) by the relations sd = 1 for all s ∈ S (cf. [KM]). Note that ρq factors through A(Γ, d) whenever (−q)d = 1. The quotient A(Γ, 2) is the usual Coxeter group associated to Γ, and ρ1 is its usual reflection representation. Braid groups. Recall that the Coxeter diagrams An and An are simply an interval and a circle, each with n vertices. The braid group Bn , with its standard generators, can be identified with the Artin group of type An−1 . Other spherical and affine Coxeter diagrams arise from braid groups for other orbifolds [Al1]. In particular, the Artin group of the type An can be identified with the braid group Bn of n points in C∗ moving with total winding number (about z = 0) equal to zero (see Figure 8). Theorem 5.2 The braid group acts on H 1 (X)q as a reduced complex reflection group of type An−1 (q) with generators si = τi∗ . Proof. The inner product on H 1 (X)q given in Theorem 4.1 satisfies |1 − q| 1 1 θ | ei , ei+1 | = = = sec , | ei , ei | 2| Im q| |1 + q| 2 2 23

so it is proportional to the inner product for An−1 (q) up to phase factor; but this phase factor determines an equivalent representation, since An−1 is a tree. Corollary 5.3 The action of Bn on H 1 (X)q is irreducible. The lifted representation. If we declare that the vectors (ei )n are linearly 1 independent, the formulas in Theorem 4.1 give a lifting of τi∗ to a matrix Ti ∈ GLn (C) for i = 1, . . . , n. It is given explicitly by   1 0 0   1 −q q  ⊕ In−3 T2 =   0 0 1

and Ti = P i−2 T2 P 2−i , where P is a cyclic permutation of the coordinates (e1 , . . . , en ). The cyclically ordered braid relations are still satisfied, so we obtain a lifted representation ρq : Bn → GLn (C); and the same inner product considerations show: Theorem 5.4 The lifted representation ρq gives a complex reflection group √ of type An (q), with phase shift φ(si , si+1 ) = −1(1 − q)/|1 − q|. Note however that ρq does not in general factor through Bn , because the relation (τ1 · · · τn−1 )τn = τ1 (τ1 · · · τn−1 ) does not lift to a relation between the matrices Ti . The Burau representation. The classical Burau representation β : Bn → GLn (Z[t, t−1 ]) is given by β(τi ) = Ii−1 ⊕ 1−t t 1 ⊕ In−i−1 .

0

These matrices preserve the vector (1, 1, 1, . . . , 1); taking the quotient yields the reduced Burau representation β. Theorem 5.5 For q n = 1, the representation ρq : Bn → U(H 1 (X)q ) is dual to the reduced Burau representation specialized to t = q. 24

(When q n = 1, the reduced Burau representation has an extra trivial summand of dimension one.) Proof. Let t = q. Then β(τ1 ) is a q-reflection through v1 = (1, −q, 0, . . . , 0), and similarly for β(τi ). When q n = 1 the vectors (v1 , . . . , vn−1 ) are linearly independent modulo (1, 1, 1, . . . , 1), so the reduced Burau representation, like ρq , gives a complex reflection group of type An−1 (q). Note, however, that ρq (τi ) = (τi∗ )−1 , because the braid group is acting on cohomology (see equation (2.3)). Thus ρq (τi ) is actually a q-reflection, and hence the representations ρq and β are dual. One can also see this duality by noting that the reduced Burau representation gives the action of Modc (C, B) on the first homology of the associated infinite cyclic covering of C − B.

Examples: n = 3. For n = 3 and p > 6, the image of An−1 (−ζp ) in PU(1, 1) is simply the (2, 3, p)-triangle group acting on the hyperbolic plane. It has the presentation Γ(2, 3, p) = {a, b : a2 = b3 = (a−1 b)p = e}, where a is the image of τ1 τ2 τ1 and b is the image of τ1 τ2 . (Note that a2 = b3 is the image of the central element (τ1 τ2 )3 ∈ B3 , so it is trivial by irre−k ducibility.) This group arises as ρq (B3 ) when q = ζd and k/d = 1/2 ± 1/p. For p = 3, 4, 5, we obtain the Platonic symmetry groups A4 , S4 and A5 in PU(2); while for p = 6, the reduction of An−1 (−ζp ) lies in PU(1), so it is trivial. Question 5.6 For what values of n and q is the An (q) reflection group a lattice in U(r, s)? This question is open even for An (ζ3 ) with n ≥ 7. For related questions, see §11. Notes. A Hermitian form preserved by the Burau representation is given in [Sq]; see also [Pe] and [KT, Ch.3]. It is known that the Burau representation is unfaithful for n ≥ 5 [Bg1]. The reflection groups An (q) are particular representations of the Hecke algebras Hn (q). For more on complex reflection groups, Coxeter diagrams and braids, see e.g. [Sh], [Cox], [Mos1], [Mos2], [Al1], [J] and [BMR].

25

6

The period map
∗ fq : T0,n → Hr,s

In this section we introduce the period map

from the Teichm¨ller space of n points in the plane to the space Hr,s of u positive lines in Cr,s . This map records, for each configuration B ⊂ C and −k for q = ζd , the line in H 1 (X)q ∼ Cr,s spanned by the holomorphic 1-form = k ]. It is defined so long as 1/n < k/d < 1. [dx/y We then show: Theorem 6.1 The period map is a holomorphic local diffeomorphism when q n = 1. Otherwise it is a submersion, with 1-dimensional fibers. Corollary 6.2 Provided q n = 1, the period map gives M∗ the structure 0,n of a (G, X)-manifold, where G = U(r, s) and X = Hr,s. Note: When q = −1, the structure group can be reduced from U(g, g) to Sp2g (R) (see §12). a Corollary 6.3 The moduli space M∗ is endowed with a natural K¨hler 0,n metric of constant negative curvature for each k/d ∈ (1/n, 2/n), and constant positive curvature for each k/d ∈ (1−1/n, 1). These metrics are modeled on the symmetric spaces CPn−2 and CHn−2 respectively. (For more on the complex hyperbolic case, see §10 and [Th2].)
−2 Theorem 6.4 When q = ζn , the period map descends to a holomorphic local diffeomorphism fq : T0,n → CHn−3 . −2 Corollary 6.5 The period map for q = ζn gives M0,n the structure of a complex hyperbolic orbifold.

Positive lines. Given (r, s) with r > 0, we let Hr,s ∼ U(r, s)/ U(r−1, s) × U(1) = denote the space of positive lines in the Hermitian vector space Cr,s . Thus Hr,0 ∼ CPr−1 and H1,s ∼ CHs . In general, Hr,s carries an invariant Her= = mitian metric of signature (r − 1, s). (It should not be confused with the 26

and the absolute periods by its image in H 1 (X).

Holomorphic 1-forms. Let Ω(X) denote the g-dimensional vector space of holomorphic 1-forms ω on X ∈ Mg . Assume ω = 0, and let Z ⊂ X denote its zero set. The absolute periods of ω are given by C ω, where C ranges all closed loops on X. The relative periods include, more generally, integrals along paths joining the zeros of ω. The relative periods are recorded by the cohomology class [ω] ∈ H 1 (X, Z) = Hom(H1 (X, Z), C),

bounded symmetric domain U(r, s)/(U(r) × U(s)), whose natural metric is always Riemannian [Sat, Appendix, §3].)

Moduli spaces, Teichm¨ ller spaces and periods. We will regard the u moduli spaces
∗ M∗ = T0,n / Mod∗ 0,n 0,n

and Mg = Tg / Modg

as quotients of the associated Teichm¨ller spaces of marked Riemann suru faces. Any two different markings of the same surface are related by the action of the mapping-class group. The set of pairs (X, ω) with ω = 0 forms the moduli space of 1-forms ΩMg → Mg . This moduli space is a union of strata ΩMg (p1 , . . . , ps ) labeled by partitions of 2g − 2; a form (X, ω) lies in a given stratum iff the zero set of ω consists of s distinct points with multiplicities (p1 , . . . , ps ). The bundle ΩMg → Mg pulls back to a bundle ΩTg → Tg which is stratified in the same way. The cohomology groups H 1 (X) form a trivial bundle over Tg ; thus if we fix a basepoint (X0 , ω0 ) ∈ ΩTg , we have a natural absolute period map α : ΩTg → H 1 (X0 )

given by α(X, ω) = [ω] ∈ H 1 (X) ∼ H 1 (X0 ). = Similarly, the cohomology groups H 1 (X, Z) form a locally trivial bundle over any stratum, so we have a locally defined relative period map π : ΩTg (pi ) → H 1 (X0 , Z0 ). By [V2] and [MS, Lemma 1.1], these relative period maps are holomorphic local homeomorphisms.

27

Point configurations. Now fix an integer n ≥ 2, a degree d > 1 and a −k level k with 1/n < k/d < 1. Set q = ζd and e = gcd(d, n). As usual, we associate to a point configuration B = (b1 , . . . , bn ) ∈ C n the branched covering space X : y d = (x − b1 ) · · · (x − bn ) with deck transformation T . The level k picks out a holomorphic eigenform ω = dx/y k ∈ H 1,0 (X)q , and F (B) = (X, ω) gives a holomorphic map F :C
n

→ ΩMg (p1 , . . . , ps ).

Here the partition (pi ) depends only on (n, k, d); it can be read off from the formula (ω) = (kn/e − d/e − 1)∞ + (d − 1 − k)B, which is a special case of equation (3.2). Note that X and the divisor (ω) depend only the pair (C, B) up to isomorphism, and (ω) determines ω up to scale. Thus F descends to a map M∗ → PΩMg . If d|n then X is unramified over infinity, so it only depends 0,n on the pair (C, B); and if k/d = 2/n as well, then the same is true of the divisor (ω) = (n − 3)B. Summing up, we have a commutative diagram: C

n F

// ΩMg  // PΩMg ::  // Mg

(6.1)

M∗ 0,n M0,n


whose bottom arrow exists when d|n, and whose diagonal arrow exists when we also have k/d = 2/n. The period map. Lifting to the universal cover of the domain, we obtain a map F : C n → ΩTg whose image we denote by (ΩTg )q . By construction, the absolute periods of any form in the image of F determine a positive vector [ω] ∈ H 1 (X)q ∼ H 1 (X0 )q ∼ Cr,s , = =

28

because the trivialization of H 1 (X) over (ΩTg )q respects the action of T . The line [Cω] depends only the location of B in Teichm¨ller space, so α ◦ F u descends to a function ∗ fq : T0,n → Hr,s which we call the period map. Since Hr,s is constructed from the flat bundle H 1 (X)q over C n with holonomy ρq : Bn → U(r, s), the map fq is equivariant with respect to the actions coming from Bn → Mod∗ and Bn → PU(r, s) (see equation (2.5)). 0,n Lemma 6.6 The map F covers a bijection
∗ T0,n → (PΩTg )q ,

except when d|n and k/d = 2/n, in which case it factors through a bijection on T0,n . Proof. For each (X, ω) ∈ (ΩTg )q there is a T ∈ Aut(X) which is in the same isotopy class as the deck transformation T0 at the basepoint X0 , and which satisfies T ∗ (ω) = qω. The data (X, ω, T ) generally allows us to reconstruct ∗ the original configuration (C, B, ∞) ∈ T0,n , by setting B ∪ ∞ = Z(ω) ∪ Fix(T e ) (6.2)

and passing to the quotient X/T ∼ C. This reconstruction succeeds unless = d|n and k/d = 2/n. In this special case, X is unramified over ∞ and (ω) = (d − 3)B, so (6.2) does not hold. But then (X, ω) depends only on the location of [B] in T0,n , and the same procedure recovers (C, B). Lemma 6.7 The relative period map from (ΩTg )q to H 1 (X0 , Z0 )q is a local homeomorphism. Proof. Recall that the relative period map gives a local homeomorphism ΩTg ⊃ U → H 1 (X0 , Z0 ). The points in H 1 (X0 , Z0 )q correspond to eigenforms (X, ω, T ) with the same combinatorics as (X0 , ω0 , T0 ), so they come from branched coverings of the sphere.

29

Corollary 6.8 The image of M∗ in PΩMg is locally a union of linear 0,n submanifolds with respect to period coordinates. Lemma 6.9 The map H 1 (X0 , Z0 )q → H 1 (X0 )q is surjective. Proof. The map H 1 (X0 , Z0 ) → H 1 (X0 ) is surjective, and T has finite order so its action is semisimple. Proof of Theorems 6.1 and 6.4. The three lemmas above show fq is a composition of submersions. By Corollary 3.3 we have dim Hr,s = ∗ n − 2 if q n = 1 and n − 3 otherwise. Since dim T0,n = n − 2, fq is a local homeomorphism in the first case and fq has one-dimensional fibers in the second. When d|n and k/d = n/2 the map fq factors through T0,n by Lemma 6.6, and (r, s) = (1, n − 3) by Corollary 3.2 so its target is CHn−3 . Proof of Corollaries 6.5 and 6.3. By Corollary 3.2 we have (r, s) = (n − 3, 0) or (1, n − 2) in these two cases, so fq gives an equivariant map to CPn−2 or CHn−2 . Pulling back the U (r, s)-invariant metric on the target gives the desired Mod∗ -invariant metric on the domain. 0,n Example: conformal mapping. For a concrete instance of the period −k map, suppose n = 3 and q = ζd with 1/3 < k/d ≤ 1. We can identify the moduli space M∗ of ordered triples of points in C with C − {0, 1}. Then 0,3 any t = 0, 1 determines a Riemann surface of the form y d = x(x − 1)(x − t), and the (multivalued) period map fq : M∗ → P1 is given by 0,3 fq (t) = [
1 k 0 dx/y ∞ k 1 dx/y ].

:

This Schwarz triangle function is univalent on the upper halfplane, and T = fq (H) is an equilateral circular triangle with internal angles 2π|k/d − 1/2|; compare [SG, §14, §16.6]. The triangle T is Euclidean for k/d = 2/3, spherical for k/d > 2/3, and hyperbolic for k/d < 2/3. Filtrations. We remark that differentials p(x) dx/y k with deg p < i determine a filtration 0 ⊂ F1 ⊂ F2 ⊂ · · · Fr ⊂ H 1 (X)q , whose variation as a function of B could also be studied. We have focused our attention on the first term so that the period map becomes a local homeomorphism, providing a geometric structure on moduli space. 30

Notes. An elegant characterization of those classes C ∈ H 1 (X) which lie in H 1,0 (X) for some complex structure on X is given in [Kap] (see also §12 below). Another approach to Theorem 6.1 appears in [DM1, Prop. 3.9].

7

Arithmetic groups

We now turn to arithmetic constraints on the image of the braid group. Consider the Z[q]-module Λn,q = H 1 (X, Z[q])q ⊂ H 1 (X)q . The unitary automorphisms preserving this module form a countable subgroup U(Λn,q ) ⊂ U(H 1 (X)q ) ∼ U(r, s). = Since our representation of the braid group preserves H 1 (X, Z[q]), it also preserves Λn,q , and hence it factors as: ρq : Bn → U(Λn,q ) ⊂ U(r, s). In this section we will show: Theorem 7.1 The values of n ≥ 3 and q = 1 such that U(Λn,q ) is a discrete subgroup of U(H 1 (X)q ) ∼ U(r, s) are those given in table Table 9 and their = complex conjugates. In these cases, either: 1. U(Λn,q ) is an arithmetic lattice in U(r, s), or 2. We have q = −1, and U(Λn,q ) ∼ Sp2g (Z) is a lattice in Sp2g (R) ⊂ = U(g, g). Corollary 7.2 The image of the braid group ρq (Bn ) is discrete in the cases appearing in Table 9. Table of arithmetic groups. In Table 9, p gives the order of ρq (τi ) in U(Λn,q ). Note that the values k/d = 1/6, 1/4, 1/3 and 1/2 occur for every n ≥ 3. These are exactly the cases where Z[q] is a discrete subring of C, and they account for all cases of discreteness when n > 12. For the entries with n = 3, U(Λn,q ) is an arithmetic triangle group of signature (2, 3, p) (or its finite image in U(1), when p = 6). The complete list of arithmetic triangle groups can be found in [Tak]. Note that the (2, 3, 5)

31

n 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4

k/d 1/6 1/4 1/10 3/10 1/3 5/14 3/8 7/18 2/5 9/22 5/12 3/7 7/16 4/9 11/24 7/15 1/2 1/6 1/4 3/10 1/3 5/14 3/8 2/5 5/12 4/9 1/2

(r, s) (0,2) (0,2) (0,2) (0,2) (0,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (0,3) (0,2) (1,2) (1,2) (1,2) (1,2) (1,2) (1,2) (1,2) (1,1)

p 3 4 5 5 6 7 8 9 10 11 12 14 16 18 24 30 ∞ 3 4 5 6 7 8 10 12 18 ∞

n 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 7 7 7 7 7 7 7 8 8 8 8 8 8 8

k/d 1/6 1/4 3/10 1/3 3/8 2/5 5/12 1/2 1/6 1/4 3/10 1/3 3/8 5/12 1/2 1/6 1/4 3/10 1/3 3/8 5/12 1/2 1/6 1/4 3/10 1/3 3/8 5/12 1/2

(r, s) (0,4) (1,3) (1,3) (1,3) (1,3) (1,2) (2,2) (2,2) (0,4) (1,4) (1,4) (1,3) (2,3) (2,3) (2,2) (1,5) (1,5) (2,4) (2,4) (2,4) (2,4) (3,3) (1,6) (1,5) (2,5) (2,5) (2,4) (3,4) (3,3) (r, s)

p 3 4 5 6 8 10 12 ∞ 3 4 5 6 8 12 ∞ 3 4 5 6 8 12 ∞ 3 4 5 6 8 12 ∞

n 9 9 9 9 9 9 10 10 10 10 10 10 11 11 11 11 11 12 12 12 12 12

k/d 1/6 1/4 3/10 1/3 5/12 1/2 1/6 1/4 3/10 1/3 5/12 1/2 1/6 1/4 1/3 5/12 1/2 1/6 1/4 1/3 5/12 1/2

(r, s) (1,7) (2,6) (2,6) (2,5) (3,5) (4,4) (1,8) (2,7) (2,6) (3,6) (4,5) (4,4) (1,9) (2,8) (3,7) (4,6) (5,5) (1,9) (2,8) (3,7) (4,6) (5,5)

p 3 4 5 6 12 ∞ 3 4 5 6 12 ∞ 3 4 6 12 ∞ 3 4 6 12 ∞

n n > 12

k/d 1/6 1/4 1/3 1/2

p 3 4 6 ∞

( n/6 − 1 , 5n/6 − 1 ) ( n/4 − 1 , 3n/4 − 1 ) ( n/3 − 1 , 2n/3 − 1 ) ( n/2 − 1 , n/2 − 1 )

−k Table 9. For q = ζd above, the braid group maps into an arithmetic lattice U(Λn,q ) ⊂ U(r, s).

32

triangle group occurs twice in our table, once for each complex place of Q(ζ5 ). In all other cases (including those with n > 3), we have 1/p + k/d = 1/2. Arithmetic perspectives. To establish Theorem 7.1, we will relate the group U(Λn,q ) to the full Q-algebraic group G ∼ Sp2g (R)T of automorphisms = of H 1 (X) commuting with T and respecting the symplectic structure. 1. We first note that the real points of the centralizer factor as follows: G(R) ∼ Sp2g (R)T = ∼ Sp(H 1 (X, R)−1 ) × =

U(H1 (X)q ).
Im q>0

There are (d − 1)/2 terms in the second factor, one for each d-th root of unity q in the upper halfplane. This factorization of G(R) is obtained from the splitting of H 1 (X, R) into eigenspaces Vq+q of T + T −1 |H 1 (X, R). We have also used the fact that for q = ±1, T determines a complex structure on Vq+q which yields an isomorphism Sp(Vq+q )T ∼ U(H1 (X)q ). = 2. A fundamental result of Borel and Harish-Chandra then implies that G(Z) ∼ Sp2g (Z)T is a lattice in G(R) ∼ Sp2g (R)T . = = Indeed, the real characters of the symplectic and unitary factors of G(R) satisfy XR (Sp2h (R)) = XR (U(r, s)) = 0, and hence XQ (G) ⊂ XR (G) = 0 as well. By [BH, Thm. 9.4], this implies vol(G(R)/G(Z)) < ∞. 3. Let Φe (x) denote the cyclotomic polynomial for the primitive e-th roots of unity, and let Ge = G| Ker(Φe (T ∗ )) We then have a similar factorization of Q-algebraic groups, namely G = e|d Ge . Note that G1 is trivial, G2 is the symplectic group of Ker(T ∗ + I), and for e ≥ 3 we have U(H1 (X)q ). Ge (R) ∼ =
Φe (q)=0,Im q>0

Let Ge (Z) ⊂ Ge (R) denote the stabilizer of the lattice Le = Ker(Φe (T ∗ ))∩ H 1 (X, Z). Then G(Z) is commensurable to e|d Ge (Z), and as above we find: Ge (Z) is a lattice in Ge (R). 33

4. Now suppose q is a primitive d-th root of unity. Then: The projection of Gd (Z) to U(H1 (X)q ) is commensurable to U(Λn,q ) . This comes from the fact that the Z[T ]-module Le projects orthogonally to a Z[q]-module commensurable to Λn,q in H 1 (X)q . Clear G(Z) preserves Λn,q , so we also find: The group G(Z) projects to a subgroup of finite index in U(Λn,q ). 5. The lattice property of Gd (Z) then implies: U(Λn,q ) is a discrete, arithmetic subgroup of U(H1 (X)q ) iff the other factors of Gd (R) are compact.
k 6. Suppose q = ζd with 0 < k < d/2 and gcd(k, d) = 1. Using Corollary 3.2 to compute the signature (r , s ) of H 1 (X)q for the other primitive d-th roots of unity, we deduce:

U(Λn,q ) ⊂ U(H1 (X)q ) is a lattice ⇐⇒ Every k = k in the range 0 < k < d/2 with gcd(k , d) = 1 satisfies k ≤ d/n. Table 9 is now readily verified by applying the criterion above, completing the proof of Theorem 7.1. Notes and references. More generally, one can show that if T ∈ Sp2g (Q) is semisimple, then Sp2g (Z)T is a lattice in Sp2g (R)T iff the irreducible factors of det(xI−T ) are reciprocal polynomials (their roots are invariant under z → 1/z). (In particular, Sp2g (Z)T is a lattice in Sp2g (R)T whenever det(xI − T ) is irreducible.) The pseudo-Anosov mapping in genus 3 studied in [AY] and [Ar] gives an interesting example where the centralizer Sp6 (Z)T is not a lattice in Sp6 (R)T . In this example, the reciprocal sextic polynomial det(xI − T ) factors as a product (x3 + x2 + x − 1)(x3 − x2 − x − 1) of two non-reciprocal cubics, and ∼ Sp6 (Z)T is commensurable to T Z in Sp6 (R)T = R∗ × C∗ . For background on arithmetic and algebraic groups, see e.g. [BH], [Bor2, §23], [Bor1], [Mg, Ch. I]. More on centralizers and conjugacy classes can be found in [SS], [Re] and [Gr].

34

8

Factors of the Jacobian

In this section we elaborate some of the algebraic geometry underlying the cases where ρq (Bn ) is finite. In particular, we describe a basis of elliptic differentials for the representative examples (n, d) = (3, 4) and (3, 6). We begin by showing: Theorem 8.1 The representation of the braid group ρq (Bn ) at a primitive dth root of unity is finite iff n = 3 and d = 3, 4, 6 or 10; or n = 4 and d = 4 or 6; or n = 5, 6 and d = 6. Proof. In the 8 cases given we find, consulting Table 9, that H 1 (X)q is definite for all primitive dth roots of unity q; hence U (Λn,q ), and its subgroup ρq (Bn ), are both finite. To see the list is complete for n = 3, note that if −q is a primitive pth root of unity and p > 6, then ρq (Bn ) maps onto the (2, 3, p) triangle group (see §5) and hence it is infinite. In particular, for q = −1 we have ρq (Bn ) ∼ SL2 (Z). = To see the list is complete for n > 3, let
−1 −1 ±1 g = τ1 τ2 τ3 τ4 · · · τn

and let A = ρq (g). Observe that ρq (Bn ) contains a copy of ρq (Bn−1 ), so the list of d for which ρq (Bn ) is finite only gets shorter as n increases; and if ρq (Bn ) is finite, then the spectral radius r(A) must be 1. For n = 4 we need only test d = 3, 4, 6 and 10. We have seen that d = 4 and 6 give finite groups, and we can rule out d = 3 and 10 by checking that 3 r(A) > 1 for q = ζ3 and q = ζ10 . Similarly, for n = 5 we can rule out d = 4 by checking that r(A) > 1 at q = ζ4 ; and for n = 7 we can rule out d = 6 by checking that r(A) > 1 at q = ζ6 . (Here we have used the fact that the representations ρq at different primitive dth roots of unity are all Galois conjugate.) Corollary 8.2 The image of the braid group ρq (Bn ) is finite iff U(Λn,q ) is finite.

35

Jacobians. When ρq (Bn ) is finite, the Hodge structure on the corresponding part of the H 1 (X) is rigid; equivalently, we have an isogeny Jac(X) → J(X) × A where dim A > 0 and A is independent of X. We refer to A as a rigid factor of the Jacobian of X. Theorem 8.3 The Jacobian of the curve X defined by y d = (x − b1 ) · · · (x − bn ) (for n distinct points) has a rigid factor in the 8 cases shown in Table 10. n 3 3 3 3 d 3 4 6 10 g(X) 1 3 4 9 Factor of Jac(X) C/Z[ζ3 ] (C/Z[ζ4 ])2 (C/Z[ζ3 ])2 (C2 /Z[ζ5 ])2 n 4 4 5 6 d 4 6 6 6 g(X) 3 7 10 10 Factor of Jac(X) (C/Z[ζ4 ])2 (C/Z[ζ3 ])3 (C/Z[ζ3 ])4 (C/Z[ζ3 ])4

Table 10. Rigid factors of the Jacobian. In Table 10, g(X) indicates the genus of X, and the rigid factor listed comes from H 1 (X)q over the primitive dth roots of unity. The factor C2 /Z[ζ5 ], which occurs for (n, d) = (3, 10), denotes quotient of C2 by the image of Z[ζ5 ] under x → (x, x ), where x → x is the Galois 2 automorphism satisfying ζ5 = ζ5 . The other factors listed are elliptic curves with automorphisms of orders 4 or 6. Note that the cases (n, d) = (3, 4) and (4, 4) yield the same set of curves X, as do the cases (5, 6) and (6, 6). Note also that curves of type (n, d) = (3, 6) cover curves of type (3, 3), so they also have a second rigid factor A ∼ C/Z[ζ3 ]. = Proof of Theorem 8.3. Let S = H 1 (X)q , where the sum is taken over the primitive dth roots of unity. This subspace is defined over Q, so it meets H 1 (X, Z) in a lattice SZ of the same rank as dimC S. For the 8 cases listed, the induced Hodge structure on S is given by     S = S 1,0 ⊕ S 0,1 = 
Im q>0

H 1 (X)q  ⊕  36

Im q<0

H 1 (X)q  ,

where again the sum is over the primitive dth roots of unity (see Corollary 3.2). That is, the Hodge structure is refined by the eigenspace decomposition of T |S, which is independent of the complex structure on X; and hence the complex torus A = S 1,0 /SZ is a rigid factor of Jac(X). The dimension of A is determined by Corollary 3.3, and the CM-type of the endomorphism T |A (see [Shi, §5.5A]) is determined by the condition Im q > 0. This information determines A itself, up to isogeny, yielding Table 10. Genus 3 curves branched over 4 points. For added perspective we describe some details in one of the simplest cases, namely genus 3 curves X of the form y 4 = (x − b1 )(x − b2 )(x − b3 ). 1. √ Eigenspaces. The most interesting eigenspace here is H 1 (X)q for −3 q = −1 = ζ4 . This eigenspace satisfies H 1 (X)q ∼ H 1,0 (X)q ; its signature = is (2, 0), and it is spanned by the holomorphic 1-forms dx/y 3 and x dx/y 3 . 2. Factors of Jac(X). By Theorem 8.3 we have an isogeny Jac(X) → E × A × A, (8.1)

3. Elliptic differentials. These maps can be seen directly as follows. First, observe that X is branched over B = B ∪{∞}, and that Aut(C, B ) ∼ = o Z/2 × Z/2. Applying a M¨bius transformation to B , we can normalize so that its symmetry group is generated by x → −x and x → 1/x; then the defining equation for X takes the form X : y 4 = x4 + bx2 + 1. (8.2)

where A ∼ C/Z[ζ4 ] is the square torus and E is an elliptic curve that depends = on X. Composing with the inclusion X → Jac(X), we obtain a dominant map p0 : X → E and a pair of dominant maps pi : X → A, i = 1, 2.

Now consider the elliptic curves and holomorphic 1-forms defined by E : z 2 = x4 + bx2 + 1, ωE = dx/z, A : y 4 = u2 + bu + 1, ωA = du/y 3 . and

√ Since A has an order 4 automorphism (u, y) → (u, −1y), it is evidently a square torus. Setting z = y 2 and u = x2 , we obtain degree two maps p0 : X → E and p1 : X → A. Composing with an automorphism of X, we define p2 (x, y) = p1 (1/x, y/x). 37

We claim the triple of maps (p0 , p1 , p2 ) induces the desired isogeny (8.1). To see this, just note that the pullbacks of the forms (ωE , ωA , ωA ) under (p0 , p1 , p2 ) respectively yield a basis = (dx/y 2 , dx/y 3 , −x dx/y 3 ) for Ω(X) consisting of elliptic differentials. 4. Action of the braid group. The braid group acts on PH 1 (X)q as the octahedral group Γ ∼ S4 ⊂ PU(2), preserving the spherical metric. = The generating twists τ1 , τ2 act by rotations R1 , R2 ∈ Γ of order 4 about orthogonal axes, corresponding to a pair of elliptic differentials as above. 5. The period map. The period map determines an embedding ∼ fq : M∗ → (H2,0 = P1 )/S4 0,3 whose image omits just one point. The domain is the (2, 3, ∞) orbifold and the target is the (2, 3, 4) orbifold; and fq sends the cusp to the cone point of order 4. These facts can be verified by showing the metric completion of M0,4 in the induced spherical metric is an orbifold; or by explicitly computing the period map, using the elliptic forms above. Remark: Shimura–Teichm¨ ller curves. Equation (8.2) determines a u remarkable curve V ⊂ M3 (parameterized by b). M¨ller shows V is the only o subvariety of any moduli space Mg , g ≥ 2 which is both a Shimura curve and a Teichm¨ller curve [Mo] u Irregular covers of elliptic curves. The preceding example was particularly simple because the maps pi : X → A were Galois coverings. To give the flavor of a more typical example, we describe a map p : X → A in the case (n, d) = (3, 6). In this case X is a genus 4 curve, presented as a covering of C branched over 4 points. We can change the x-coordinate by a M¨bius transformation o so the defining equation for X becomes X : y 6 = (x3 − 3x − b)(x − 2)3 for some b ∈ C (because the cross-ratio of the roots of the polynomial on the right takes on all possible values). Now consider the elliptic curve defined by A : z 6 = (b − 2)4 w(w − 1)3 . Because of its order 6 symmetry S(w, z) = (w, ζ6 z), we have A ∼ C/Z[ζ3 ]; = (w, z) = x3 − 3x − b , y(x + 1) 2−b 38 (8.3)

and the formula

defines an irregular, degree 3 map p : X → A. This map shows A occurs as one of the rigid factors of Jac(X). This calculation is motivated by the observation that the desired map p : X → A induces a map p : X/ T → A/ S between a pair of orbifolds with signatures (2, 6, 6, 6) and (2, 3, 6) respectively. Since both orbifolds have genus zero, we can regard p as a rational map of the form w = p(x). The signatures of these orbifolds suggest there exists such a map with deg(p) = 3, sending the 3 points of order 6 in the domain to the unique point of order 6 in the range. Any such map would be totally ramified over the singular point of order 3, so in suitable coordinates it would be a cubic polynomial; and we are led to formula (8.3). Remark. In the case (n, d) = (3, 10), the projection to the rigid factor Jac(X) → A ∼ C2 /Z[ζ5 ] is not induced by a map p : X → Y with = Jac(Y ) ∼ A. This can be established by showing there is no map from the = (10, 10, 10, 10) orbifold to the (2, 5, 10) orbifold which lifts to the required Z/10 covering spaces. (It is, however, induced by a correspondence.)

9

Definite integrals

In this section we present another consequence of finiteness of the representation of the braid group. Theorem 9.1 Suppose n ≥ 3 and 0 < µ < 1. Then
b2

I(b1 , b2 , . . . , bn ) =
b1

dx ((x − b1 )(x − b2 ) · · · (x − bn ))µ

is an algebraic function of (b1 , . . . , bn ) iff ρq (Bn ) is finite for q = exp(−2πiµ), and µ = 1/n. Here the integral is initially defined for distinct points bi ∈ R, and then extended to C by analytic continuation. Using Theorem 8.1 to check the finiteness of ρq (Bn ), we obtain: Corollary 9.2 The definite integral I(b1 , . . . , bn ) is algebraic exactly for the 17 values of (n, µ) given in Table 11.

39

n µ n µ 3/4 2/3 1/4 4 1/6 5/6 1/4 3/4 1/6

3 5/6 5 3/4 1/6 5/6 1/10 3/10 7/10 6 5/6 9/10

Table 11. Algebraic definite integrals.

D p Cp

φ (D)

Figure 12. The cycle D is modified by a loop around p .

The integral for the case (n, µ) = (4, 3/4) will be evaluated explicitly at the end of this section. Meromorphic differentials. To set the stage for the proof, let ω be a meromorphic 1-form on a compact Riemann surface X with a finite set of poles P ⊂ X. Then ω determines a cohomology class [ω] ∈ H 1 (X − P ). If Res(ω, p) = 0 for every p ∈ P , then ω is traditionally called a differential of the second kind; it gives a well-defined cohomology class [ω] ∈ H 1 (X) ⊂ H 1 (X − P ). Now let φ ∈ Mod(X, P ) be a mapping-class stabilizing P , and assume φ is isotopic to the identity on X (it is in the kernel of the natural map Mod(X, P ) → Mod(X)). Then in the course of the isotopy, each p ∈ P traces out a relative cycle Cp ∈ H 1 (X, P ) joining p to p = φ(p). It is then easy to compute the action of φ on H1 (X − P ): it satisfies (with suitable orientations) [φ(D)] = [D] + (Cp · D)[Up ],
p∈P

where Up is a small loop around p (see Figure 12).

‘
p

40

This computation facilitates the study of periods of general meromorphic 1-forms (which may have residues); in particular, it shows that φ∗ (ω) =
D D

ω+
P

(Cp · D) Res(ω, p ).

(9.1)

Proof of Theorem 9.1. The singularities of I(b) = I(b1 , . . . , bn ) along the loci bi = bj in Cn are algebraic iff µ is rational, so we may restrict attention to this case. Consider a generic configuration of distinct points B = {b1 , . . . , bn }. −k Let µ = k/d in lowest terms, let q = ζd , let X be the curve defined by y d = (x − b1 ) · · · (x − bn ) as usual, and let ω = dx/y k . Then T ∗ (ω) = qω. The form ω potentially has poles on the set P = ∞. Let L ⊂ X − P denote a lift of the segment [b1 , b2 ] to an arc connecting the corresponding points of B, chosen so that ω|L agrees with the chosen branch of the integrand I(b). Observe that D = L − T (L) is a cycle on X − P , and that I(b) =
L

ω = (1 − q)−1

L

ω − T ∗ (ω) = (1 − q)−1

ω
D

= (1 − q)−1 D, ω . Thus I(b) simply measures one of the periods of ω. Now suppose b is allowed to vary, keeping its coordinates distinct. Then I(b) can be analytically continued, along many different paths, from b to any point in Sn · b (where the coordinates of b are permuted). The set of possible values for I(σ · b) so obtained are given, up to a factor of (1 − q), by J(b) = { φ(D), ω : φ ∈ Mod(C, B) ∼ Bn }. = (9.2)

Since I(b) has algebraic singularities, it is a (multivalued) algebraic function of b iff J(b) is finite. To evaluate |J(b)|, first assume k/d = 1/n. Then ω is a differential of the second kind. To see this, let e = gcd(n, d). Then T e fixes every p ∈ P , and satisfies Res(ω, p) = Res((T e )∗ ω, p) = q e Res(ω, p); so the residues of ω vanish if e < d. On the other hand, if e = d then k/d ≥ 2/n, which implies ω is actually a holomorphic 1-form on the whole of X.

41

Clearly J(b) is finite if ρq (Bn ) is finite. We also note that 0 = [D] ∈ H 1 (X)∗ , q

Since ω is of the second kind, it defines a cohomology class in H 1 (X). Thus the pairing in (9.2) depends only on the class of φ(D) ∈ H1 (X), and therefore J(b) = D, ρq (Bn ) · ω . (9.3)

because there exist configurations b such that I(b) = 0 (e.g. when all bi ∈ R). So conversely, if J(b) is finite, then one of the ‘matrix entries’ of ρq : Bn → U(H 1 (X)q ) assumes only finitely many values; but ρq is irreducible (Corollary 5.3), so this implies ρq (Bn ) is finite as well. It remains to show that J(b) is infinite when k/d = 1/n. In this case ω has simple poles along P and |P | = d. Write P = {p0 , . . . , pd−1 } with pi = T (p0 ). Since ω is an eigenform, it satisfies Res(ω, pi ) = q i Res(ω, p0 ) = 0. The idea is to show that I(b) can be made to take on infinitely many different values by pushing the path of integration repeatedly through infinity, so it picks up a residue of ω. Note that (9.3) continues to hold, e.g. by considering one of the Galois conjugates of q. Choose a cycle C ∈ H1 (X, Z) such that
d

0=E=
i=1

q i T i (C) ∈ H1 (X)q ∼ H 1 (X)∗ . = q

(9.4)

(Most cycles have this property). Represent C by a closed loop on X that begins and ends at p0 , and otherwise avoids P ∪ B. Then the projection of C to C gives a loop γ ∈ π1 (C − B, ∞). Let φ ∈ Modc (C, B) be a mapping class that pushes ∞ once around γ, and maps to the identity in Mod(C, B). Then its canonical lift φ ∈ Mod(X − P ) pushes pi once around the cycle Ci , where C0 = C and Ci = T i (C0 ); and φ maps to the identity in Mod(X). In particular, by (9.1) we have
d

φ(D), ω

= =

D, ω +
1

(Ci · D) Res(ω, pi )

D, ω + Res(ω, p0 )(E · D).

Similarly, we have φk (D), ω = D, ω + k Res(ω, p0 )(E · D). 42

Thus J(b) is infinite provided E · D = 0, or more generally if E · ψ(D) = 0 for some ψ ∈ Modc (C, B). But if these intersection numbers are all zero, then E = 0 or D = 0, since Bn acts irreducibly on H1 (X)q . This contradicts equations (9.3) and (9.4). Genus 3 revisited. Here is an explicit formula for one of the algebraic integrals provided by Theorem 9.1. We have normalized so that (b1 , b2 , b3 , b4 ) = (0, 1, a, b). Theorem 9.3 For 1 < a < b < ∞, the definite integral
1

I(a, b) =
0

dx (x(x − 1)(x − a)(x − b))3/4

satisfies I(a, b)4 = c(1 + c − 2ac)4 16π 2 Γ(1/4)4 · 2 , Γ(3/4)4 a (a − 1)2 (a − b)3 (c2 − 1)4

where c > 1 is the largest root of the quadratic equation (a − b)c2 + (4ab − 2a − 2b)c + (a − b) = 0. Eliminating c, we obtain the expression: I(a, b)4 = ((2a − 1) b(b − 1) + (2b − 1) a(a − 1))4 π 2 Γ(1/4)4 · , Γ(3/4)4 a2 b2 (a − 1)2 (b − 1)2 (a + b − 2ab − 2 ab(a − 1)(b − 1))3

which exhibits the solution’s symmetry in a and b. Proof of Theorem 9.3. As in §8, we use the fact that the configuration B = {0, 1, a, b} ⊂ C has an order-two M¨bius symmetry g(x) satisfying o g(0) = 1 and g(a) = b. (The point c has a natural meaning: it is the unique fixed point of g between a and b.) Thus by changing coordinates so the fixed points of g become x = 0 and x = ∞, we can transform I(a, b) into an integral of the form:
1 −1

((x2

(Ax + C) dx · − 1)(x2 − d2 ))3/4

The integrand is now a sum of elliptic differentials for the square torus, as discussed in §8. The term Ax can be ignored because it is odd; evaluating the remaining integral in terms of Γ-functions, we obtain the formula given in Theorem 9.3.

43

Remark. If we change the exponent µ = 3/4 to 1/2, we essentially obtain the complete elliptic integral of the first kind,
1

K(k) =
0

√

1−

x2

dx √ . 1 − k2 x2

This is a transcendental function of k, defined classically by the relation sn(K, k) = 1 [WW, §22.3]. Notes and references. The values (α, β, γ) such that
1

I(b) =
0

xα (x − 1)β (x − b)γ dx

is an algebraic function of b were determined by Schwarz in 1873 [Sch, §VI]; see [CW2] for a generalization to multivariable hypergeometric integrals, which includes Theorem 9.1 as a special case. The failure of algebraicity when ω has nonzero residues (e.g. when µ = 1/n in Theorem 9.1) is discussed by Klein in [Kl, §49], which corrects a statement of Riemann.

10

Complex hyperbolic geometry

In this section we discuss the cases where moduli space acquires a complex hyperbolic metric of finite volume. For example, we will see: Theorem 10.1 For n = 4, 5, 6, 8 and 12, the period map fq : M0,n → CHn−3 /ρq (Bn )
−2 at q = ζn presents moduli space as the complement of a divisor in a finitevolume, arithmetic, complex-hyperbolic orbifold.

(This divisor corresponds to certain Dehn twists whose images under ρq have finite order.) Since M2 ∼ M0,6 we also have: = Corollary 10.2 The moduli space M2 can be completed to a complex hyperbolic orbifold of finite volume. Note that the complex hyperbolic structure on M2 comes from Hodge structures on surfaces of genus four (obtained as triple covers of C branched over six points). These results are special cases of the work of Deligne–Mostow and Thurston, which we recall below. We will also give a self-contained proof of: 44

Theorem 10.3 If ρq (Bn ) ⊂ U(1, s) is discrete, then it is a lattice. Corollary 10.4 Whenever U(Λn,q ) ⊂ U(1, s) is a lattice, so is ρq (Bn ). Table 9 furnishes 24 values of (n, q) with n ≥ 4 where this Corollary applies, including the 5 cases covered by Theorem 10.1 above. Hypergeometric functions and shapes of polyhedra. Let M0,n denote the moduli space of ordered points on C (an Sn -covering space of M0,n ). Let Pn denote the pure braid group of n points on the sphere. There is a natural surjective map Pn → π1 (M0,n ). Following [DM1] and [DM2], let µ = (µ1 , . . . , µn ) be a sequence of rational weights satisfying 0 < µi < 1 and Given b = (b1 , . . . , bn ) ∈ C
n

µi = 2.

, consider the multivalued algebraic 1-form (x − b1 )µ1 dx · · · · (x − bn )µn µ i bi

ωb = Note that

K = (ωb ) = −

ratios become well-defined on the universal cover of M0,n . These period coordinates give M0,n a complex hyperbolic metric gµ , whose holonomy determines a representation ξµ : Pn → Isom(CHn−3 ) ∼ PU(1, n). = Alternatively, following [Th2], one can consider |ωb | as a flat metric on C with cone angles 2π(1 − µi ) at each point bi . The space (C, |ωb |) is then isometric to a convex polyhedron in R3 of total area A = C |ωb |2 . By considering A as a quadratic form, Thurston obtains the same complex hyperbolic metric gµ on M0,n as Deligne and Mostow.1 The orbifold condition. The weights µ satisfy the orbifold condition if whenever s = µi + µj < 1, either
The work [Th2] grew out of a discussion of problem 3 of the 27th Mathematical Olympiad (July 9, 1986).
1

can be regarded as a generalized canonical divisor on C, since deg(K) = b −2. The periods bij ωb are multivariable hypergeometric functions, whose

45

• (1 − s)−1 ∈ Z, or • µi = µj and 2(1 − s)−1 ∈ Z. A list of the 94 different weights µ satisfying this condition, with n ≥ 4, is given in the Appendix to [Th2]. This condition allows us to formulate following two important results. Theorem 10.5 (Deligne–Mostow) If the weights (µi ) satisfy the orbifold condition, then the image ξµ (Pn ) of the pure braid group is a lattice in PU(1, n − 3). Theorem 10.6 (Thurston) The metric completion of (M0,n , gµ ) is a complex hyperbolic cone manifold of finite volume. It is an orbifold iff the weights (µi ) satisfy the orbifold condition. See [DM1], [Mos1], and [Th2, Thm. 0.2], as well as [Par, Lemma 3.5], which clarifies the relationship between the two results. Equal weights. It is straightforward to see that when µ1 = · · · = µn = 2/n, the constructions of Deligne–Mostow and Thurston give the same complex hyperbolic structure on M0,n ∼ M0,n /Sn = as that furnished by the period mapping fq : T0,n → H1,n−3
−2 of §6 at q = ζn (see Corollary 6.5). Similarly, if µ1 = · · · = µn−1 = k/d = µn , then k/d ∈ (1/(n − 1), 2/(n − 1)) and the complex hyperbolic metric on

∼ M∗ 0,n−1 = M0,n /Sn−1
−k agrees with that furnished by the period mapping at q = ζd (see Corollary 6.3). In particular, the images of the holonomy maps ξµ (Pn ) and ρq (Bn ) are commensurable, and hence Theorem 10.5 provides a criterion for ρq (Bn ) to be a lattice. Arithmetic lattices. To complement the results above, we present a short proof that ρq (Bn ) ⊂ U(1, s) is a lattice whenever it is discrete.

Proof of Theorem 10.3. Assume ρq (Bn ) ⊂ U(1, s) is discrete. Let −2 M = M∗ or M0,n depending on whether q = ζd or not, and let Z = 0,n s /ρ (B ). As we have seen in §6, the period map defines an analytic CH q n local homeomorphism of orbifolds fq : M → Z. By the Schwarz lemma, 46

fq is distance-decreasing from the Teichm¨ller metric on M to (a suitable u multiple of) the complex hyperbolic metric on Z. Let M be the Deligne-Mumford compactification of M by stable curves with marked points, and let ∂M = M − M . There is a unique maximal analytic continuation of fq to an open set M 0 with M ⊂ M 0 ⊂ M . A local model (U, ∂U ) for (M, ∂M ) near any x ∈ M is given by a finite quotient of (∆a × (∆∗ )b , 0 × ∆b near (0, 0). Since the puncture of ∆∗ is a cusp, every element of π1 (U ) is represented by a short loop in M . Thus every element of H = f∗ (π1 (U )) ⊂ π1 (Z) is represented by a short loop in Z. If H is finite, then fq lifts to a map fq from a finite cover of U into the bounded domain CHs , and hence fq itself extends analytically to a neighborhood of x. That is, x ∈ M 0 . Otherwise, there is an h ∈ H which has no fixed point in CHs but whose minimal translation distance is zero. Then by the Schwarz lemma, fq (U ) is contained in the thin part of Z, and hence fq tends to infinity at x. It follows that fq : M 0 → Z is proper. In particular, fq (M 0 ) = Z and vol(Z) = vol(fq (M )) since dim ∂M < dim M = dim Z. Since fq is a contraction, the volume of fq (M ) ⊂ Z is no more than a constant multiple of the volume of M in the Teichm¨ller metric, which is finite (see e.g. [Mc1][Thm. u 8.1]). Thus vol(Z) < ∞, and hence π1 (Z) = ρq (Bn ) is a lattice. Note that Corollary 10.4 to Theorem 10.3 also follows from Theorem 10.5 and the following observation regarding the 36 entries in Table 9 with hyperbolic signature: Proposition 10.7 Whenever U(Λn,q ) ⊂ U(1, s) is a complex hyperbolic lattice, the corresponding weights µ satisfy the orbifold condition. Proof of Theorem 10.1. For these values of n, the entry k/d = 2/n appears in Table 9, so we may apply Theorem 10.6. Arithmetic hyperbolic structures for n ≤ 12. An examination of Table 9 also shows there is at least one arithmetic complex hyperbolic structure on M∗ for every value of n with 3 ≤ n ≤ 11. (For n > 12, hyperbolic sig0,n nature no longer occurs.) When n = 3, M∗ can be completed to give the hyperbolic orbifold 0,3 H/Γ(2, 3, p) for any p > 6; this completion is arithmetic for 12 values of ∗ p. There are 6 arithmetic hyperbolic structures on M0,4 besides the one −1 induced from the covering map M∗ → M0,5 ; the case n = 4, q = ζ3 0,4 47

was considered by Picard in 1883 [Pic]. There are also several arithmetic hyperbolic structures on M∗ for n = 5, 6, 7, but only one for each n = 0,n 8, 9, 10, 11, and none for M∗ . 12 Non-arithmetic examples. Non-arithmetic lattices in U(1, s), s > 1, were first discovered in [Mos1]. For the braid group, Theorem 10.6 shows there is a unique case where ρq (Bn ) is nonarithmetic but (M∗ , gµ ) is still 0,n a complex hyperbolic orbifold: namely n = 4 and µ = (7, 7, 7, 7, 8)/18. In −7 this example q = ζ18 and ρq (Bn ) ⊂ U(1, 2). Other values of µ giving non-arithmetic lattices are enumerated in [Th2, Appendix]. Only a finite number of non-arithmetic lattices in U(1, s) are presently known (up to commensurability). Modular embeddings. Let us consider the non-arithmetic lattice coming from n = 4 and d = 18 in more detail. In this case the equation y 18 = x(x − 1)(x − a)(x − b) determines a 2-dimensional family of curves X of genus g = 25. As in §7, we can regard ∆ = Sp2g (Z)T | Ker Φd (T ∗ ) as an irreducible, arithmetic lattice in the product of unitary groups U (H 1 (X)q ) ∼ U (1, 2)2 × U (2)2 . =

Φd (q)=0,Im q<0

Because of the U(2) factor, ∆ has no unipotents, so it determines a compact, 4-dimensional Shimura variety S = (CH2 × CH2 )/∆.
−7 Now let Γ = ρq (Bn ) ⊂ U(1, 2) where q = ζ18 . Then Γ is a non-arithmetic −5 lattice. Let Z = CH2 /Γ and let r = ζ18 (so H 1 (X)q and H 1 (X)r both have signature (1, 2)). Using the period mappings at these two roots of unity, we obtain a holomorphic map F : Z → S making the diagram

OOO OOO q ,fr ) (f OOO OOO  '' F // (CH2 × CH2 )/∆ CH2 /Γ

M∗ 0,4 
_

commute. When suitably normalized on the universal covers of domain and range, F determines a modular embedding F : CH2 → CH2 × CH2 of the form F (z) = (z, F2 (z)). The map F2 : CH2 → CH2 is a transcendental function which intertwines the action of Γ with its Galois conjugate Γ = 48

ρr (Bn ). The locus F (Z) ⊂ S is a Kobayashi geodesic subvariety, reminiscent of the Teichm¨ller curves on Hilbert modular surfaces studied in [Mc2] and u [Mc3]. For more details and further examples, see [CW3]. A similar construction can be carried out for nonarithmetic triangle groups [CW1].

11

Lifting homological symmetries

For brevity of notation, let Sp(X) = Sp(H 1 (X, Z)) ∼ Sp2g (Z), let Sp(X)T = denote the centralizer of T ∗ , and let Sp(X)T = Sp(X)T | Ker Φd (T ∗ ), d where Φd (x) is the dth cyclotomic polynomial. As in §7, we have natural maps Bn → Mod(X)T → Sp(X)T → Sp(X)T . d

We say a group homomorphism φ : G1 → G2 almost onto if [G2 : φ(G1 )] is finite. In this section we study the following purely topological question: For example, by [A’C] we have: Question 11.1 When is the natural map Bn → Sp(X)T almost onto? d

Theorem 11.2 When d = 2, the natural map Bn → Sp(X)T ∼ Sp(X) is d = almost onto for all n ≥ 3. We will show that additional cases of Question 11.1 can be resolved using arithmeticity of lattices. Proposition 11.3 The following are equivalent: 1. The map Bn → Sp(X)T is almost onto. d 2. The map Mod(X)T → Sp(X)T is almost onto. d 3. There is a primitive dth root of unity q such that ρq (Bn ) has finite index in U(Λn,q ). Proof. Up to isotopy, any mapping-class [φ] commuting with T can be represented by a homeomorphism commuting with T (e.g. we may assume T is conformal and take φ to be a Teichm¨ller mapping). This easily implies u that Bn → Mod(X)T is almost onto and hence (1) and (2) are equivalent. We have also seen, in §7, that the projection of Sp(X)T to U(Λn,q ) is almost d onto, so (1) is equivalent to (3).

49

Theorem 11.4 Suppose q is a primitive dth root of unity, and Γ = ρq (Bn ) ⊂ U(1, s) is a lattice. Then Bn → Sp(X)T is almost onto if and only if Γ is d arithmetic. Proof. If Γ = U(Λn,q ) is discrete then the hypotheses imply both it and Γ are arithmetic lattices, by Theorem 10.3, and hence [Γ : Γ] < ∞. Otherwise Γ is indiscrete and [Γ : Γ] = ∞. Corollary 11.5 The map B3 → Sp(X)T is almost onto if and only if the d Fuchsian (2, 3, p) triangle group is finite or arithmetic, where p is the order of (d − 2) in Z/2d. (Explicitly, using Table 9, we find B3 → Sp(X)T is almost onto iff d = d 2, 3, 4, 5, 6, 7, 9, 10, 14, 15, 16, 18, 22 or 24.) Proof. We may assume d = 2 or d ≥ 7, since otherwise Sp(X)T is finite. d Let q be the primitive dth root of unity closest to −1 with Im q ≤ 0. Then ρq (B3 ) ⊂ U(1, 1) is a lattice, namely the discrete (2, 3, p) triangle group (with p = ∞ when d = 2), and we may apply the Theorem above. Corollary 11.6 The map Bn → Sp(X)T is almost onto whenever d and n d appear in Table 9 with associated signature (r, s) = (0, s) or (1, s). Proof. In these cases either Sp(X)T is finite or ρq (Bn ) ⊂ U(1, s) is arithd metic by Corollary 10.4.

Corollary 11.7 The map Bn → Sp(X)T is almost onto for d = 3 and n = 3, 4, 5, 6; d = 5 and n = 3, 4, 5; and d = 7, n = 3. Proof. These cases appear in Table 9 as required, and d is prime, so Sp(X)T = Sp(X)T . d On the other hand, since ρq (B4 ) ⊂ U(1, 2) is a non-arithmetic lattice for −7 q = ζ18 (see §10), we find: Corollary 11.8 The image of B4 has infinite index in Sp(X)T . 18

50

Notes and references. It is well-known that for any closed orientable surface Σg of genus g, the natural map Mod(Σg ) → Sp(Σg ) is surjective [MKS, Thm. N13] (see also [Bu, §7], [FM, Thm. 7.3]). One can then ask, which subgroups H ⊂ Sp(Σg ) lift to Mod(Σg )? This is a homological form of the Nielsen realization problem; cf. [Ker]. Question 11.1 is related to certain special cases of this question, where H is an abelian group of the form S, T . A version of question 11.1 for finite abelian coverings of closed surfaces is addressed in [Lo1].

12

The hyperelliptic case
y 2 = (x − b1 ) · · · (x − bn )

In this section we discuss the hyperelliptic case, where X takes the form (12.1)

and ω = dx/y. A special feature of this case is that the symmetry T ∗ ω = −ω is preserved by the Teichm¨ller geodesic flow. Consequently, the period map u
∗ fq : T0,n → Hg,g

is compatible with a natural action of SL2 (R). We will use this compatibility to analyze fq and its image, and (in the next section) to describe fq explicitly for the Teichm¨ller curves coming from regular polygons [V1]. u The period domain. In the hyperelliptic case we have d = 2, q = −1, and H 1 (X)q = H 1 (X). Thus the period domain Hg,g ⊂ PH 1 (X) ∼ PCg,g = coincides with the space of all positive lines in the cohomology of X (§6). The real points Rg ∼ RP2g−1 = of its ambient projective space PH 1 (X) correspond to lines of the form [Cv], v ∈ H 1 (X, R). Since v, v = 0 for v ∈ H 1 (X, R), we have Rg ⊂ ∂Hg,g . By restricting our attention to automorphisms that preserve this real structure, we may regard the period domain as the homogeneous space Hg,g = Sp2g (R)/(SO(2, R) × Sp2g−2 (R)). 51 (12.2)

Secant lines. We say a line L ⊂ PH 1 (X) is a secant if it joins a pair of real points p, q ∈ Rg . Every secant has the form L = PV , where V ⊂ H 1 (X) is invariant under complex-conjugation and has signature (1, 1). The intersections S = L ∩ Hg,g form the leaves of the secant foliation S of Hg,g . Each leaf is a totally geodesic copy of CH1 (up to the sign of its metric). The secant foliation can also be defined by observing that SL2 (R) acts C ∼ R2 , and hence on H 1 (X) = Hom(π1 (X), C); the orbits of this action = project to the leaves of S. Weierstrass forms. Now assume n = 2g + 1 is odd. Let Hg denote the moduli space of hyperelliptic Riemann surfaces of genus g. Using (12.1), each point configuration [B] = [{b1 , . . . , bn }] in M∗ determines an X ∈ Hg 0,n and a line Cω ⊂ Ω(X), where ω = dx/y. We refer to ω as a Weierstrass form, since its divisor (ω) = (2g − 2)∞ is supported on a single Weierstrass point of X. The map [B] → [(X, ω)] gives a bijection ∼ ∼ M∗ 0,2g+1 = PΩHg (2g − 2) = Hg , (12.3)

where Hg is the degree (2g+2) covering space of Hg parameterizing Riemann surfaces with a chosen Weierstrass point. Each Weierstrass form determines a quadratic differential ω 2 on C − B hence a complex Teichm¨ller geodesic W ∼ H through ([C, B]). These u = ∗ geodesics form the leaves of the Weierstrass foliation W of T0,2g+1 . Under the isomorphism (12.3), the leaves of W correspond to the orbits of SL2 (R) on ΩHg (2g − 2) (see e.g. [KZ]). Since the actions of SL2 (R) on ΩHg and H 1 (X) are compatible, we find:
∗ Theorem 12.1 The period map fq : T0,2g+1 → Hg,g sends the Weierstrass foliation W to the secant foliation S. Its restriction to each leaf is an isometry.

Ergodicity. For g = 1, fq simply gives the standard homeomorphism ∗ between T0,3 and H1,1 ∼ H. = For g > 1 and n = 2g + 1, the period map is still an equivariant local homeomorphism (§6). But the braid group now acts very differently on its domain and its range: while the orbits of Bn in Teichm¨ller space are u discrete, a typical orbit in the period domain is dense. In fact: Proposition 12.2 The action of ρq (Bn ) on Hg,g is ergodic for g > 1. Proof. Equation (12.2) gives Hg,g = G/F , where F is a closed subgroup of G = Sp2g (R); and Γ = ρq (Bn ) is a lattice in G by Theorem 11.2. But for 52

g > 1, F is noncompact, so Γ acts ergodically on G/F by the Howe–Moore theorem [HM]. Thus the image of fq is an open set of full measure, and fq is not a covering map to its image (else ρq (Bn ) would have discrete orbits).
∗ Proposition 12.3 The group Ker ρq has infinite orbits on T0,2g+1 , and hence the period map fq is infinite-to-one for g > 1.

Proof. The square of a Dehn twist about a loop enclosing an odd number of points of B lifts to a Dehn twist in Mod(X) which acts trivially on homology. For more on ‘isoperiodic forms’, see [Mc6, §9]. The image of the period map. Although the period map has a complicated structure, its image is reasonably tame. Indeed, the results of [Kap] easily imply: Theorem 12.4 For g > 1, the complement of the image of the period map
∗ fq : T0,2g+1 → Hg,g

is a nonempty, countable union of secant lines. Proof. Let G = Sp2g (R), H = Sp2 (R) × Sp2g−2 (R), Γ = ρq (B2g+1 ). By Theorem 11.2, Γ is a lattice in G. The image of the period map, modulo the secant foliation, is a nonempty, open, Γ-invariant subset U ⊂ Hg,g /S ∼ = G/H. By Ratner’s theorem, for any x ∈ G there is a connected Lie group J with H ⊂ J such that ΓxH = ΓxJ and Γ ∩ Jx is a lattice in Jx = xJx−1 [Rat]. We will show G/H − U is countable. Suppose [xH] ∈ U . Then ΓxH is not dense in G, and hence J = G. As shown in [Kap, §3], this implies J = H. Thus Γx = Γ ∩ Hx is a lattice Hx . There are only countably many possibilities for Γx , since it is a finitely-generated subgroup of Γ; and Hx is the Zariski closure of Γx , so there are only countably many possibilities for Hx . But NG (H)/H is finite (in fact trivial for g > 2), so Hx determines [xH] up to finitely many choices. Thus the complement of the image of fq is a countable union of leaves of the secant foliation. The complement is nonempty by [Kap, §1]: a cohomology class p∗ : π1 (X) → Z[i] determined by a degree one map p : X → C/Z[i] can never be realized by a holomorphic 1-form.

53

The case n = 2g + 2: relative periods and quadratic differentials. ∗ Similar results hold for the period map fq : T0,2g+2 → Hg,g . In this case ω = dx/y has a pair of distinct zeros at the points {P, Q} lying over ∞ in X, and fq is a submersion with 1-dimensional fibers. The fibers form ∗ the leaves of the absolute period foliation A of T0,2g+2 . Along any fiber A, one can normalize ω so its absolute periods are constant; then the (locally well-defined) relative period function
Q

z=
P

ω

determines a (globally well-defined) holomorphic quadratic differential q = ∗ u dz 2 on A. The natural foliation of T0,2g+2 by complex Teichm¨ller geodesics (analogous to W) is transverse to A, so it gives rise to a homeomorphism map φ : A → A whenever fq (A) and fq (A ) are joined by a secant; and in u fact φ is a Teichm¨ller mapping, with dilatation proportional to q/|q|. See [Mc6, §8] for a more detailed discussion, in the case of genus two.

13

Polygons and Teichm¨ ller curves u

Finally we describe certain totally geodesic Teichm¨ller curves in moduli u space from the perspective of braid groups and polyhedra. Regular polygons. We continue in the hyperelliptic setting, with g > 0 ∗ and n = 2g + 1. The Weierstrass foliation of T0,2g+1 descends to a foliation ∗ of M0,2g+1 , which we also denote by W. Let α = τ1 τ3 . . . τ2g−1 and β = τ2 τ4 . . . τ2g (13.1) be the elements of Bn obtained by grouping its even and odd generators together. The generators in each product commute, so their ordering is immaterial. Note that αβ is a natural lifting of the Coxeter element of Sn = W (An−1 ) to the corresponding Artin group Bn . Thus the subgroup α, β ⊂ Bn is a sort of halo of the Coxeter element. Now let B ⊂ C be the vertices of a regular polygon with n sides, and let V ⊂ M∗ be the leaf of the Weierstrass foliation through [(C, B)]. Although 0,n most leaves of W are dense, Veech showed the special leaf V is closed [V1]. More precisely, using [Lei] or [Mc4] for the statement on π1 , we have: Theorem 13.1 The leaf V is a rational, totally geodesic algebraic curve in moduli space; and the image of the inclusion π1 (V ) → π1 (M∗ ) 0,n 54

is the (2, n, ∞) triangle group generated by the mapping-classes [α] and [β]. We remark that the triangle group relations follow from the identity (αβ)n = ((αβ)g α)2 = σ n in the braid group (recall that the mapping class [σ n ] is trivial). As shown in [Loch] (see also [Mc5, §5]), the point configurations arising in V are given explicitly by B(t) = {ζ + tζ −1 : ζ n = 1}, where t ∈ C and tn = 1. (Note that B(t), B(1/t) and B(ζn t) all represent the same point in moduli space.)

Figure 13. The sphere with the flat metric |dx|/|xn − 1|1/2 , n = 3, 5, 7. Polyhedra. Following [Th2], one can associate to each point configuration B ⊂ C the abstract Euclidean polyhedron Q = (C, |ω|) determined by the flat metric |ω| = |dx/y| = |p(x)|−1/2 |dx|, where p(x) = B (x − b). This polyhedron has a cone angle of π at the points x ∈ B, and a cone angle of (n − 2)π at x = ∞.

Folded polygons and immersed disks. As B(t) moves along V , one obtains a family of polyhedra related by Teichm¨ller mappings. These polyu hedra take on particularly simple shapes at the orbifold points of V , which come from t = 0 and t = −1. At t = 0 we have p(x) = xn − 1, and Q can be obtained from a regular n-gon by folding its edges together at their midpoints, so that all n vertices are identified (see Figure 13). In the case g = 1 the result is a regular tetrahedron, but for g > 1 there is negative curvature at the Weierstrass point at infinity, so the polyhedron is no longer convex.

55

√ For t = −1, it is convenient to set B = ( −1/2)B(t); then
g

p(x) =
B

(x − b) =

k=−g

x − sin

2πk n

has real zeros x1 < x2 < · · · < xn . The Schwarz-Christoffel formula
z

f (z) =
0

dt

p(t)

yields a conformal immersion f : H → C, which bends the real axis by 90 degrees at each point xi , and |ω| is nothing more than the pullback of the Euclidean metric on C under f .

Figure 14. Immersed disks for 1 ≤ g ≤ 8. Thus Q can be visualized as the double of the immersed disk D = f (H) across its boundary. The disk D is a square for g = 1, an L-shaped region for g = 2, and an immersed right-angled polygon for g ≥ 3 (see Figure 14.) The immersion f for g = 4 is shown in more detail in Figure 15, where it has been factored as the composition f (z) = h(z)2 of an embedding h : H → C with the squaring map. Periods and eigenvectors. The lengths of the edges of D are given explicitly by the absolute values of periods of ω, namely
xi+1

Li =
xi

|p(x)|−1/2 dx.

These lengths can also be determined algebraically. 56

−→

z2

Figure 15. The immersion f , factored through z 2 , for g = 4.

Theorem 13.2 The lengths (L1 , . . . , Ln−1 ) form a positive eigenvector for the adjacency matrix (Cij ) of the An−1 Coxeter diagram. Here Cij = 1 if |i − j| = 1 and 0 otherwise. Proof. Let (hi ) be a positive eigenvector for Cij , and let X be the hyper∗ elliptic surface defined by y 2 = p(x). The generators [τi ] of Mod0,n lift to Dehn twists about simple closed curves on X satisfying Si · Sj = Cij . By a construction of Thurston, this implies there is a holomorphic quadratic differential q on X such that the loops (Si ) are represented by a cylinders of heights (hi ) on (X, |q|), and the lifts of α and β are realized by affine automorphisms [Th1]; cf. [Mc4, §4]. In the case at hand, it is easy to see that q is a multiple of ω 2 , and hence (hi ) is proportional to (Li ). Remark. The fact that the ratios Li /Lj are algebraic reflects the fact that the Jacobian of the curve y 2 = p(x) admits complex multiplication by −1 Q(i, ζn + ζn ).

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