Invariant functions with respect to the Whitehead-link

Transcription

1 Invariant functions with respect to the Whitehead-link Keiji Matsumoto (joint work with Haruko Nishi and Masaaki Yoshida) 1 Introduction The Gauss hypergeometric equation E(α, β, γ) x(1 x) d2 f df + {γ (α + β + 1)x} dx2 dx αβf = 0 for (α, β, γ) = (1/2, 1/2, 1) induces an isomorphism per : C {0, 1} H/M (taking the ratio of solutions), where H = {τ C Im(τ) > 0} and M is the monodromy group of E(1/2, 1/2, 1). Note that M is the level 2 principal congruence subgroup of SL 2 (Z), which is isomorphic to the fundamental group π 1 (C {0, 1}). We can regard per as the period map for the family of marked elliptic curves (double covers of P 1 branching at 4 points). As generalizations, we have some period maps and systems of hypergeometric equations; each of them induces an isomorphism between a certain moduli space of algebraic varieties and the quotient space of a certain symmetric domain by its monodromy group. 1

2 For examples, 1. Appell s F D with special parameters: from families of k- fold branched coverings of P 1 to complex balls studied by Terada and Deligne-Mostow, 2. the period map from the family of cubic surfaces to the 4-dimensional complex ball studied by Allcock-Carlson- Toledo, 3. period maps from some families of certain K3 surfaces to complex balls embedded in symmetric domains of type IV lectured by Dolgachev and Kondo, 4. E(3, 6; 1/2,..., 1/2): from the family of the double covers branching along 6-lines to the symmetric domain D of type I 22 studied by Sasaki-Yoshida-Matsumoto(the speaker). Interesting automorphic forms appear when we study the inverses of them! 2

3 In my talk, we construct automorphic functions on the real 3-dimensional upper half space H 3 = {(z, t) C R t > 0}, by observing the Whitehead link L = L 0 L in Figure 1. F 1 F 2 L 0 L F 3 Figure 1: Whitehead link The Whitehead-link-complement S 3 L is known to admit a hyperbolic structure: there is a discrete group W GL 2 (C) acting on on H 3, and a homeomorphism ϕ : H 3 /W = S 3 L. Note that the situation is quite similar to the inverse of per : per 1 : H/M C {0, 1}. 3

4 But one has never tried to make the homeomorphism ϕ explicit. We construct automorphic functions for W in terms of Θ ( a (τ) on the symmetric domain D of type I2,2 over the ring Z[i] appeared in Example 4, and express the homeomorphism ϕ in terms of these automorphic functions, which realize some branched coverings of real 3-dimensional orbifolds. Our automorphic functions derive some properties with respect to the Whitehead link: We can express the space S 3 L as a part of a real algebraic set (we need some inequalities). We can regard L 0 and L as the exceptional curves arising from the cusps. I expect that some link invariants can be obtained algebraically by our expression. We can realize symmetries of the Whitehead link as actions of (Z/2Z) 2 on these automorphic functions. Our automorphic functions give an arithmetical characterization of W. The group W is given by 2 generators (W π 1 (S 3 L) which is generated by 2 elements). By the definition of W, for a given g GL 2 (C) it is difficult to know if g belongs to W or not. 4

5 2 A hyperbolic structure on the complement of the Whitehead link Let H 3 be the upper half space model H 3 = {(z, t) C R t > 0} of the 3-dimensional real hyperbolic space. GL 2 (C) and an involution T act on H 3 as g (z, t) = ( ) g 11 ḡ 21 t 2 +(g 11 z+g 12 )(g 21 z+g 22 ) g 21 2 t 2 +(g 21 z+g 22 )(g 21 z+g 22 ), det(g) t g 21 2 t 2 +(g 21 z+g 22 )(g 21 z+g 22 ), T (z, t) = ( z, t), where Put ( ) g11 g g = 12 GL g 21 g 2 (C). 22 GL T 2 (C) := { GL 2 (C), T T g = ḡ T } for g GL 2 (C). 5

6 The Whitehead-link-complement S 3 L admits a hyperbolic structure. We have a homeomorphism where W := g 1, g 2, g 1 = ϕ : H 3 /W = S 3 L, ( 1 i 0 1 ), g 2 = ( i 1 We call W the Whitehead-link-complement group. ). A fundamental domain F D for W in H 3 is in Figure 2. #9 Im(z) 1 + i #5 i #7 #3 # #2 Re(z) #4 #8 #10 #6 i 1 i Figure 2: Fundamental domain F D of W in H 3 The group W has two cusps. (z, t) = (, + ), (0, 0) (±i, 0) (±1, 0) ( 1 ± i, 0). Remark 1 The monodromy groups of E(α, β, γ) for parameters satisfying are conjugate to W. cos(2πα) = 1+i 2, β = α, γ Z 6

7 3 Discrete subgroups of GL 2 (C), especially Λ We define some discrete subgroups of GL 2 (C) : Γ = GL 2 (Z[i]), SΓ 0 (1+i) = {g = (g jk ) Γ det(g) = ±1, g 21 (1+i)Z[i]}, SΓ(1+i) = {g SΓ 0 (1+i) g 12 (1+i)Z[i]}, Γ(2) = {g Γ g 11 1, g 12, g 21, g Z[i]}, W = T W T = {ḡ g W }, Ŵ = W W, W = W, W. Convention: We regard these groups as subgroups of the projectified group P GL 2 (C). For G in Γ, we denote G T = G, T in GL T 2 (C). 7

8 Γ T (2) is a Coxeter group generated by the eight reflections, of which mirrors form an octahedron in H 3, see Figure 3. Im(z) 1 + i i 1 0 Re(z). Figure 3: Weyl chamber of Γ T (2) We put Λ := Γ T (2), W. Λ = SΓ T 0 (1+i) SΓ T (1+i) SΓ 0 (1+i) W = W, W Γ T (2) W W Γ(2) Ŵ = W W 8

9 Lemma 1 1. Γ T (2) is normal in Λ; Λ/Γ T (2) the dihedral group D 8 of order [Λ, W ] = 8, W is not normal in Λ: T W T = W. 3. The domain bounded by the four walls and by the hemisphere a : Im(z) = 0, b : Re(z) = 0, c : Im(z) = 1 2, d : Re(z) = 1 2, #9 : z 1 + i = is a fundamental domain of Λ, see Figure 4. 1+i 2 Im(z) i Re(z). Figure 4: Fundamental domain of Λ 4. Λ = SΓ T 0 (1+i) and [SΓ 0 (1+i), W ] = 4. (We will see SΓ 0 (1+i)/W = (Z/2Z) 2.) 9

10 H 3 /Γ T (2) H 3 /W \ Z/(2Z) H 3 / W, W \ D8 Z/(2Z) H 3 /SΓ 0 (1+i) Our strategy is following. \ / Z/(2Z) H 3 /Λ At first, we realize the quotient space H 3 /Γ T (2) by using theta functions Θ ( a (τ) on D. Next we construct D8 -invariant functions which realize H 3 /Λ. This step corresponds to the construction of the j-function from the λ-function. Finally, we construct the 3 double covers in the right line step by step. We must know the branch locus of each of double covers. We investigate the symmetry of the Whitehead link. 10

11 4 Symmetry of the Whitehead link Orientation preserving homeomorphisms of S 3 keeping L fixed form a group (Z/2Z) 2. The group consists of π-rotations with axes F 1, F 2 and F 3, and the identity. F 1 F 2 L 0 L F 3 Figure 5: The Whitehead link with its symmetry axes There is also a reflection of S 3 keeping a mirror (containing L) pointwise fixed. These rotations and the reflection can be represent as elements of Λ. We give the axes and the mirror in the fundamental domain of H 3 /W. 11

12 Proposition 1 The three π-rotations with axes F 1, F 2 and F 3, and the reflection can be represented by the transformations ( ) 1 1 γ 1 :, γ : respectively, of H 3 modulo W. ( ) 1 1, γ : ( ) 1 0, T, 0 1 The fixed loci in F D, as well as in H 3 /W, of the rotations γ 1, γ 2 and γ 3 are also called the axes F 1, F 2 and F 3 ; they are depicted in F D as in Figure 6. A bullet stands for a vertical line: the inverse image of the point under π. Figure 6: The fixed loci of γ 1, γ 2, γ 3 12

13 5 Orbit spaces under W, SΓ 0 (1 + i) and Λ A fundamental domain for W and the orbifold H 3 / W 2 L L 0 F 2 F 3 A fundamental domain for SΓ 0 (1 + i) and the orbifold F 1 H 3 /SΓ 0 (1 + i) 2 L L 0 2 A fundamental domain for Λ and the boundary of H 3 /Λ F 1 F 2 F 3 2 c 0 d b a a d b c 13

14 F 1 c d F 3 a b 0 F 2 Figure 7: A better picture of the fundamental domain for SΓ 0 (1 + i) Proposition 2 The branch locus of the double cover H 3 /SΓ 0 (1+i) of H 3 /Λ is the union of the walls a, b, c, d. That of the double cover H 3 / W of H 3 /SΓ 0 (1+i) is the union of the axes F 2 and F 3 (the axes F 2 and F 3 are equivalent in the space H 3 / W ). That of the double cover H 3 /W of H 3 / W is the axis F 1. 6 Theta functions on D The symmetric domain D of type I 2,2 is defined as D = {τ M 2,2 (C) τ τ } is positive definite. 2i The group U 2,2 (C) = { h GL 4 (C) gjg = J = 14 ( )} O I2 O I 2

15 and an involution T act on D as h τ = (h 11 τ + h 12 )(h 21 τ + h 22 ) 1, T τ = t τ, ( ) h11 h where h = 12 U h 21 h 2,2 (C), and h jk are 2 2 matrices. 22 We define some discrete subgroups of U 2,2 (C): U 2,2 (Z[i]) = U 2,2 (C) GL 4 (Z[i]), U 2,2 (1+i) = {h U 2,2 (Z[i]) h I 4 mod (1+i)}. Theta functions Θ ( a (τ) on D are defined as ( ) a Θ (τ) = e[(n + a)τ(n + a) + 2Re(nb )], b n Z[i] 2 where τ D, a, b Q[i] 2 and e[x] = exp[πix]. By definition, we have the following fundamental properties. 15

16 Fact 1 1. If b 1 1+i Z[i]2, then Θ ( a i (τ) = Θ ( a (τ). If b 1 2 Z[i]2, then Θ ( a (τ) = Θ ( a (τ). 2. For k Z and m, n Z[i] 2, we have ( i k ) ( ) a a Θ (τ) = Θ (τ), i k b b ( ) ( ) a + m a Θ (τ) = e[ 2Re(mb )]Θ (τ). b + n b 3. If (1+i)ab / Z[i] for a, b 1 1+i Z[i]2, then Θ ( a (τ) = 0. It is known that any action of U 2,2 (Z[i]) on τ D can be decomposed into the following transformations: (1) τ τ + s, where s = (s jk ) is a 2 2 hermitian matrix over Z[i]; (2) τ gτg, where g GL 2 (Z[i]); (3) τ τ 1. Fact 2 By T and these actions, Θ ( a (τ) is changed into ( ) (ā ) a Θ (T τ) = Θ (τ), b b ( ) ( ) a Θ (τ + s) = e[asa a ]Θ b b + as + 1+i 2 (s (τ), 11, s 22 ) ( ) ( ) a ag Θ (gτg ) = Θ (τ) for g GL b b(g ) 1 2 (Z[i]), ( ) ( ) a b Θ ( τ 1 ) = det(τ)e[2re(ab )]Θ (τ). b a 16

17 In order to get the last equality, use the multi-variable version of the Poisson summation formula. equality. ( ) a Θ (gτg ) b = We show the 3rd e[(n + a)(gτg )(n + a) + 2Re(n(gg 1 )b )] n Z[i] 2 = e[(ng + ag)τ(ng + ag) + 2Re(ng(b(g ) 1 ) )] n Z[i] ( 2 ) ag = Θ (τ), b(g ) 1 since m = ng rnus over Z[i] 2 for any g GL 2 (Z[i]). Proposition 3 If a, b 1 1+i Z[i]2 then Θ 2( a (τ) is a modular from of weight 2 with character det for U 2,2 (1+i), i.e., ) (T τ) = Θ 2 ( a b ( a Θ 2 b for any h = (h jk ) U 2,2 (1+i). ) (τ), ( a Θ 2 b ) ( ) a (h τ) = det(h) det(h 21 τ + h 22 ) 2 Θ 2 (τ), b By following the proof of Jacobi identity for lattices L 1 = Z[i] 2, L 2 = Z[i] 2 A, L = L 1, L 2, where A = 1 + i ( ) 1 1, AA = I , A 2 = ii 2, we have quadratic relations among theta functions Θ ( a (τ). 17

18 Theorem 1 ( ) a 4Θ (τ) 2 b = e[2re((1+i)be )] e,f 1+i 2 Z[i]2 /Z[i] 2 ( ) ( ) e + (1+i)a e Θ (τ)θ (τ). f + (1+i)b f For a, b ( Z[i] 1+i /Z[i])2, there are 10 non-vanishing Θ ( a (τ). Corollary 1 The ten Θ ( a (τ) 2 satisfy the same linear relations as the Plücker relations for the (3, 6)-Grassmann manifold, which is the linear relations among the 10 products D ijk (X)D lmn (X) of the Plücker coordinates, where x x 16 x 1i x 1j x 1k X = x x 26, D ijk (X) = det x 2i x 2j x 2k x x 36 x 3i x 3j x 3k and {i, j, k, l, m, n} = {1,..., 6}. There are 5 linearly independent Θ 2( a (τ). Remark 2 τ can be regarded as periods of the K3-surface coming from the double cover of P 2 branching along 6 lines given by the 6 columns of X. 18

19 7 Embedding of H 3 into D We embed H 3 into D by ı : H 3 (z, t) i t ( t 2 + z 2 ) z z 1 D; we define a homomorphism ( ) g/ det(g) O j : GL 2 (C) g O (g / det(g) ) 1 They satisfy U 2,2 (C). ı(g (z, t)) = j(g) ı(z, t) for any g GL 2 (C), ı(t (z, t)) = ( T ( ı(z, t), 0 1 (ı(z, t)) 1 = j 1 0 ) ) T ı(z, t). We denote the pull back of Θ ( a (τ) under the embedding ı : H 3 D by Θ ( a (z, t). By definition, we have the following. Fact 3 1. For a, b 1 2 Z[i]2, each Θ ( a (z, t) is real valued. If 2Re(ab ) + 2Im(ab ) / Z then Θ ( a (z, t) For a, b 1 1+i Z[i]2, each Θ ( a (z, t) is invariant under the action of Γ T (2). 3. The function Θ = Θ ( 00) (z, t) is positive and invariant under the action of Γ T. 19

20 8 Automorphic functions for Γ T (2) and an embedding of H 3 /Γ T (2) Set and Θ p = Θ p q q (z, t) = Θ( p 2 q 2 x 0 = Θ 0, 0, x 1 = Θ 0, 0 x 2 = Θ 1+i, 0, x 3 = Θ 0, 1+i ) (z, t), p, q Z[i] 2 1+i, 1+i, 1+i, 1+i 0, 1+i. 1+i, 0 Theorem 2 The map H 3 (z, t) 1 (x 1, x 2, x 3 ) R 3 x 0 induces an isomorphism between H 3 /Γ T (2) and the octahedron Oct = {(t 1, t 2, t 3 ) R 3 t 1 + t 2 + t 3 1} minus the six vertices (±1, 0, 0), (0, ±1, 0), (0, 0, ±1). 20

21 9 Automorphic functions for Λ and an embedding of H 3 /Λ Proposition 4 g 1, g 2 induce transformations of x 1, x 2, x 3 : x 1 1 x 1 x 2 g 1 = 1 x 2, x 3 1 x 3 x 1 1 x 1 x 2 g 2 = 1 x 2. x 3 1 x 3 This a representation of the dihedral group D 8 of order 8. Theorem 3 x x 2 2, x 2 1x 2 2, x 2 3, x 1 x 2 x 3 are Λ-invariant. The map λ : H 3 (z, t) where ξ j = x j /x 0, (λ 1, λ 2, λ 3, λ 4 ) = (ξ ξ 2 2, ξ 2 1ξ 2 2, ξ 2 3, ξ 1 ξ 2 ξ 3 ) R4, induces an embedding of H 3 /Λ into the subdomain of the variety λ 2 λ 3 = λ

22 10 Automorphic functions for W Set y 1 = Θ 0, 1, y 2 = Θ 1+i, 1, 1+i, 0 1+i, 0 z 1 = Θ 0, 1, z 2 = Θ 1+i, 1. 1, 0 1, 1+i We define functions as Φ 1 = x 3 z 1 z 2, Φ 2 = (x 2 x 1 )y 1 + (x 2 + x 1 )y 2, Φ 3 = (x 2 1 x 2 2)y 1 y 2. Theorem 4 Φ 1,( Φ 2 and) Φ 3 are W -invariant. By the actions g = I Γ(2) and T, p q r s Φ 1 g = e[re((1+i)p + (1 i)s)]φ 1, Φ 2 g = e[re(r(1 i))]φ 2, Φ 3 g = Φ 3. Φ 1 T = Φ 1, Φ 3 T = Φ 3. Remark 3 Φ 2 T = (x 2 x 1 )y 1 (x 2 + x 1 )y 2. This is not invariant under W but invariant under W. 22

23 Let Iso j be the isotropy subgroup of Λ = SΓ T 0 (1 + i) for Φ j. Theorem 5 We have SΓ 0 (1+i) = Iso 3, W = Iso1 Iso 3, W = Iso 1 Iso 2 Iso 3. By this theorem, we have SΓ 0 (1+i)/W (Z/2Z) 2. ( ) p q Theorem 6 An element g = SΓ r s 0 (1+i) satisfying Re(s) 1 mod 2 belongs to W if and only if Re(p) + Im(s) ( 1) Re(q)+Im(q) (Im(p) + Re(s)) 2 (( 1)Re(r) + 1)Im(q) + (Re(q) + Im(q))(Re(r) + Im(r)) 2 mod2. The element g W belongs to W if and only if Re(p + q) + Re(r) ( 1)Re(q)+Im(q) Im(r) 2 1 mod 2. The element g W belongs to Ŵ = W W if and only if r 2Z[i]. 23

24 11 Embeddings of the quotient spaces Put f 00 = (x 2 2 x 2 1)y 1 y 2 = Φ 3, f 01 = (x 2 2 x 2 1)z 1 z 2 z 3 z 4, f 11 = x 3 z 1 z 2 = Φ 1, f 12 = x 1 x 2 z 1 z 2, f 13 = x 3 (x 2 2 x 2 1)z 3 z 4, f 14 = x 1 x 2 (x 2 2 x 2 1)z 3 z 4, f 20 = (x 2 x 1 )z 2 z 3 + (x 2 + x 1 )z 1 z 4, f 21 = z 1 z 2 {(x 2 x 1 )z 1 z 3 + (x 2 + x 1 )z 2 z 4 }, f 22 = (x 2 2 x 2 1){(x 2 x 1 )z 1 z 4 + (x 2 + x 1 )z 2 z 3 }, f 30 = (x 2 x 1 )y 1 + (x 2 + x 1 )y 2 = Φ 2, f 31 = (x 2 x 1 )z 1 z 3 (x 2 + x 1 )z 2 z 4, f 32 = z 3 z 4 { (x 2 x 1 )z 1 z 4 + (x 2 + x 1 )z 2 z 3 }, where (x 0, x 1, x 2, x 3 ) and (z 1, z 2, z 3, z 4 ) are Θ 0, 0, Θ 0, 0 Θ 1+i, 1+i, Θ 1+i, 0, Θ 1+i, 1+i 0, 1+i 0, 1+i, 1+i, 0 0, 1, Θ 1+i, 1, Θ 0, i, Θ 1+i, i. 1, 0 1, 1+i 1, 0 1, 1+i 24

25 Proposition 5 We have 4z 2 1 = (x 0 + x 1 + x 2 + x 3 )(x 0 x 1 x 2 + x 3 ), 4z 2 2 = (x 0 + x 1 x 2 x 3 )(x 0 x 1 + x 2 x 3 ), 4z 2 3 = (x 0 + x 1 x 2 + x 3 )(x 0 x 1 + x 2 + x 3 ), 4z 2 4 = (x 0 + x 1 + x 2 x 3 )(x 0 x 1 x 2 x 3 ). Proposition 6 f jp are W -invariant. These change the signs by the actions of γ 1, γ 2 and γ 3 as in the table γ 1 γ 2 γ 3 f 0j f 1j + f 2j + f 3j + Theorem 7 The analytic sets V 1, V 2, V 3 of the ideals I 1 = f 11, f 12, f 13, f 14, I 2 = f 21, f 22, I 3 = f 31, f 32 are F 2 F 3, F 1 F 3, F 1 F 2. Corollary 2 The analytic set V jk of the ideals I j, I k is F l for {j, k, l} = {1, 2, 3}. 25

26 Theorem 8 The map ϕ 0 : H 3 /SΓ 0 (1+i) (z, t) (λ 1,..., λ 4, η 01 ) R 5 is injective, where η 01 = f 01 /x 6 0. Its image Image(ϕ 0 ) is determined by the image Image(λ) under λ : H 3 (z, t) (λ 1,..., λ 4 ) and the relation 256f 2 01 = (λ 2 1 4λ 2 ) ε 3 =±1 (λ 2 3 2(x λ 1 )λ 3 + ε 3 8x 0 λ 4 +x 4 0 2x 2 0λ 1 + λ 2 1 4λ 2 ), as a double cover of Image(λ) branching along its boundary. F 1, F 2 and F 3 can be illustrated as in Figure 8. Each of the two cusps and 0 is shown as a hole. These holes can be deformed into sausages as in Figure 9. F 1 F 3 F 3 F 2 F 1 F 3 0 F 2 Figure 8: Orbifold singularities in Image(ϕ 0 ) and the cusps and 0 26

27 L L 0 F 1 F 2 F 3 Figure 9: The cusp-holes are deformed into two sausages Theorem 9 The map ϕ 1 : H 3 / W (z, t) (ϕ 0, η 11,..., η 14 ) R 9 is injective, where η 1j = f 1j /x deg(f 1j) 0. The products f 1p f 1q (1 p q 4) can be expressed as polynomials of x 0, λ 1,..., λ 4 and f 01. The image Image(ϕ 0 ) together with these relations determines the image Image(ϕ 1 ) under the map ϕ 1. The boundary of a small neighborhood of the cusp ϕ 1 (0) is a torus, which is the double cover of that of the cusp ϕ 0 (0); note that two F 2 -curves and two F 3 -curves stick into ϕ 0 (0). The boundary of a small neighborhood of the cusp ϕ 1 ( ) remains to be a 2-sphere; note that two F 1 -curves and two F 3 -curves stick into ϕ 0 ( ), and that four F 1 -curves stick into ϕ 1 ( ). 27

28 ϕ 1 (0) 2 F 2 F 3 ϕ 1 ( ) F 1 Figure 10: The double covers of the cusp holes Theorem 10 The map ϕ : H 3 /W (z, t) (ϕ 1, η 21, η 22, η 31, η 32 ) R 13 is injective, where η ij = f ij /x deg(f ij) 0. The products f 2q f 2r f 3q f 3r and f 1p f 2q f 3r (p = 1,..., 4, q, r = 1, 2) can be expressed as polynomials of x 0, λ 1,..., λ 4 and f 01. The image Image(ϕ 1 ) together with these relations determines the image Image(ϕ) under the map ϕ. The boundary of a small neighborhood of the cusp ϕ( ) is a torus, which is the double cover of that of the cusp ϕ 1 ( ); recall that four F 1 -curves stick into ϕ 1 ( ). The boundary of a small neighborhood of the cusp ϕ(0) is a torus, which is the unbranched double cover of that of the cusp ϕ 1 (0), a torus. 28

29 Eventually, the sausage and the doughnut in Figure 10 are covered by two linked doughnuts, tubular neighborhoods of the curves L 0 and L of the Whitehead link. F 1 F 2 L 0 L F 3 29

30 References [F] E. Freitag, Modulformen zweiten Grades zum rationalen und Gaußschen Zahlkörper, Sitzungsber. Heidelb. Akad. Wiss., 1 (1967), [I] J. Igusa, Theta Functions, Springer-Verlag, Berlin, Heidelberg, New York [MSY] K. Matsumoto, T. Sasaki and M. Yoshida, The monodromy of the period map of a 4-parameter family of K3 surfaces and the Aomoto-Gel fand hypergeometric function of type (3,6), Internat. J. of Math., 3 (1992), [M1] K. Matsumoto, Theta functions on the bounded symmetric domain of type I 2,2 and the period map of 4-parameter family of K3 surfaces, Math. Ann., 295 (1993), [M2] K. Matsumoto, Algebraic relations among some theta functions on the bounded symmetric domain of type I r,r, to appear in Kyushu J. Math. [MY] K. Matsumoto and M. Yoshida, Invariants for some real hyperbolic groups, Internat. J. of Math., 13 (2002), [T] W. Thurston, Geometry and Topology of 3-manifolds, Lecture Notes, Princeton Univ., 1977/78. 30

31 [W] N. Wielenberg, The structure of certain subgroups of the Picard group. Math. Proc. Cambridge Philos. Soc. 84 (1978), no. 3, [Y] M. Yoshida, Hypergeometric Functions, My Love, Aspects of Mathematics, E32, Friedr Vieweg & Sohn, Braunschweig,