AUTOMORPHIC FUNCTIONS FOR THE WHITEHEAD-LINK-COMPLEMENT GROUP

Matsumoto, K., Nishi, H. and Yoshida, M. Osaka J. Math. 43 (006), 839 876 AUTOMORPHIC FUNCTIONS FOR THE WHITEHEAD-LINK-COMPLEMENT GROUP Dedicated to Professor Takeshi Sasaki on his sixtieth birthday KEIJI MATSUMOTO, HARUKO NISHI and MASAAKI YOSHIDA (Received July 8, 005, revised December 5, 005) Abstract We construct automorphic functions on the real 3-dimensional hyperbolic space H 3 for the Whitehead-link-complement group W GL (Z[i]) and for a few groups commensurable with W. These automorphic functions give embeddings of the orbit spaces of H 3 under these groups, and arithmetical characterizations of them. Contents. Introduction... 840. A hyperbolic structure on the complement of the Whitehead link... 84 3. Discrete subgroups of GL (C), especially... 843 4. Symmetry of the Whitehead link... 846 4.. Symmetries of L.... 846 4.. Fixed loci.... 847 5. Orbit spaces under W S¼ 0 ( + i) and... 848 5.. The orbifold H 3Æ W.... 849 5.. The orbifold H 3 S¼ 0 ( + i).... 849 5.3. The orbifold H 3.... 850 6. Theta functions... 85 6.. Theta functions on D.... 85 6.. Embedding of H 3 into D and the pull-back of the theta functions. 853 6.3. Automorphic functions for ¼ T () and an embedding of H 3 ¼ T (). 854 6.4. Automorphic functions for and an embedding of H 3.... 854 7. Automorphic functions for W... 856 7.. Fundamental properties of and 3.... 856 7.. Isotropy subgroups.... 857 7.3. An arithmetical characterization of the Whitehead-link-complement group.... 859 8. Embeddings of the quotient spaces... 865 8.. Automorphic functions for W vanishing along F j.... 865 8.. An embedding of H 3 S¼ 0 ( + i).... 87 8.3. An embedding of H 3Æ W.... 873 8.4. An embedding of H 3 W.... 874 000 Mathematics Subject Classification. F55, 4P05, 57M5.

840 K. MATSUMOTO, H. NISHI AND M. YOSHIDA. Introduction Fig.. Whitehead link with its symmetry axes Fig. shows the Whitehead link L = L 0 L ½ in S 3 = R 3. The Whiteheadlink-complement S 3 L is known to admit a hyperbolic structure: there is a group W acting properly discontinuously on the 3-dimensional hyperbolic space H 3, and there is a homeomorphism h : H 3 W = S 3 L No one has ever tried to make the homeomorphism h explicit. In this paper we construct automorphic functions for W (analytic functions defined in H 3 which are invariant under W ), and express the homeomorphism h in terms of these automorphic functions. Since our embedding of H 3 W requires many automorphic functions (codimension of the embedding is high), we find several extensions of W, and give their embeddings, which have lower embedding dimensions. In particular, for the extension W ¼ such that W ¼ W ( = (ZZ) ) represents the group of symmetries (orientation-preserving ambient homotopies) of L S 3, we find five automorphic functions, say, h h 5, so that the map H 3 x (h (x) h 5 (x)) ¾ R 5 gives an embedding of H 3 W ¼. Its image is explicitly presented as part of an affine algebraic variety.

AUTOMORPHIC FUNCTIONS FOR THE WHITEHEAD LINK 84 Our automorphic functions are made from theta functions over the ring Z[i]. Our proofs heavily depends on properties of these theta functions, and on quadratic relations among them established in [], [3] and [5].. A hyperbolic structure on the complement of the Whitehead link Let H 3 be the upper half space model H 3 = (z t) ¾ C R t 0 of the 3-dimensional real hyperbolic space. The group GL (C) and an involution T act on H 3 as g ḡ t g + (g z + g )(g z + g ) det(g)t (z t) = g t + (g z + g )(g z + g ) g t + (g z + g )(g z + g ) T (z t) = ( z t) where g = (g jk ) ¾ GL (C). Let GL T (C) be the group generated by GL (C) and an involution T with relations T g = ḡ T for g ¾ GL (C). The Whitehead-link-complement S 3 L admits a hyperbolic structure (cf. [6], [7]): Let W be the discrete subgroup W of GL (C) generated by the two elements g = We have the homeomorphism i 0 and g = 0 + i H 3 W = S 3 L We call W the Whitehead-link-complement group. A fundamental domain, which will be denoted by FD, for W in H 3 is given in Fig. (cf. [7]); two pyramids are shown. Each face of the pyramids is a mirror of a reflection belonging to GL (Z[i]) T. The faces (together with the corresponding reflections) of the two pyramids and their patching rules are as follows:

84 K. MATSUMOTO, H. NISHI AND M. YOSHIDA Fig.. Fundamental domain FD of W in H 3 The faces of the two pyramids No. face reflection No. face reflection # #3 #5 #7 #9 Im(z) = 0, Re(z) 0, Re(z) = 0, 0 Im(z), Im(z) =, Re(z), Re(z) =, 0 Im(z), z +i T, # T, #4 i T, #6 0 T, #8 0 +t =, i 0 i T, #0 Im(z) = 0, 0 Re(z), Re(z) = 0, Im(z) 0, Im(z) =, 0 Re(z), Re(z) =, Im(z) 0, z i T, i 0 0 T, T, +t =, i 0 +i Patching rule face element of W its image face element of W its image i i # #5 # #6 0 0 i i #3 #4 #7 #8 0 0 0 #9 #0 + i The group W has two cusps. They are represented by the vertices of the pyramids: (z t) = ( +½) (0 0) ( i 0) ( 0) ( i 0) T, T.

AUTOMORPHIC FUNCTIONS FOR THE WHITEHEAD LINK 843 REMARK. The translation t := 0 is an element of W. Indeed one finds the relation g t g g g g g g g = in [7]. We can decide whether a given matrix is an element of W by Theorem 5 in 7.3. 3. Discrete subgroups of GL (C), especially Λ We define some discrete subgroups of GL (C): ¼ = GL (Z[i]) ¼ 0 ( + i) = g = (g jk ) ¾ ¼ g ¾ ( + i)z[i] S¼ 0 ( + i) = g ¾ ¼ 0 ( + i) det(g) = ¼( + i) = g ¾ ¼ g g g g ¾ ( + i)z[i] ¼() = g ¾ ¼ g g g g ¾ Z[i] W = T W T = ḡ g ¾ W Ŵ = W W W = Å W W «CONVENTION. Since we are interested only in the action of these groups on H 3, we regard these groups as subgroups of the projectified group PGL (C); in other words, every element of the groups represented by a scalar matrix is regarded as the identity. For any subgroup G in ¼ we denote G T the group generated by G and T in GL T (C). It is known ([5]) that the group ¼ T () is a Coxeter group generated by the eight reflections 0 i T T T T 0 0 0 0 + i + i i 0 T T T T i + i i i The mirrors of the reflections are four walls Im(z) = 0, Re(z) = 0, Re(z) =, Im(z) =, and four northern hemispheres with radius and centers i, + i, + i,, respectively, see Fig. 3. Note that the Weyl chamber bounded by these eight mirrors is an (ideal) octahedron in the hyperbolic space H 3. The group ¼ T () is well-studied in [5]. To relate ¼ T () with the Whiteheadlink-complement group W, we consider the smallest group which contains both ¼ T () and W : = Å ¼ T () W «Lemma.. ¼ T () is a normal subgroup of, and ¼ T () is isomorphic to the dihedral group of order eight.. [ W ] = 8, W is not a normal subgroup of : T W T = W.

844 K. MATSUMOTO, H. NISHI AND M. YOSHIDA Fig. 3. Weyl chamber of ¼ T () Proof.. We extend the reflection group ¼ T () by adding the reflection g T with mirror Im z =, and the -fold rotation with axis the geodesic arc joining the points (z t) = (0 0) and ( + i 0), which is given by i 0 R = i i These reflection and rotation preserve the Weyl chamber above, and generate a group isomorphic to the dihedral group of order eight. Since we have and 0 0 i 0 i i 0 0 = i ¾ ¼ T () 0 i = g this extended group coincides with.. By comparing the Weyl chamber of ¼ T () with the fundamental domain FD of W, we see that W has the same co-volume with ¼ T (). Thus [ W ] = 8. From the proof of this lemma, we have Corollary. The domain bounded by the four walls a : Im(z) = 0 b : Re(z) = 0 c : Im(z) = d : Re(z) =

AUTOMORPHIC FUNCTIONS FOR THE WHITEHEAD LINK 845 Fig. 4. Fundamental domain of and the big hemisphere #9 in is a fundamental domain of, see Fig. 4. The hemisphere part is folded by the rotation R above. We use this fundamental domain in 5.3. Lemma. We have = S¼ T 0 ( + i) and [S¼ 0( + i) W ] = 4. Proof. It is clear that S¼0 T ( + i). Since ¼ T 0 ( + i) ¼ T () = 6 and ¼ T 0 ( + i) S¼T 0 ( + i) = we have = S¼0 T ( + i). So far we defined many subgroups of ¼ T = GL T (Z[i]); their inclusion relation can be depicted as follows: ¼0 T ( + i) ¼ T ( + i) = S¼0 T ( + i) S¼ T ( + i) S¼ 0 ( + i) W = Å W W «¼ T () W W ¼() Ŵ = W W

846 K. MATSUMOTO, H. NISHI AND M. YOSHIDA When two groups are connected by a segment, the one below is a subgroup of the one above of index. More explanation about these groups will be given in 7.. 4. Symmetry of the Whitehead link In this section, we study the symmetries of the Whitehead link, and express each symmetry as an extension of the group W. 4.. Symmetries of L. The -rotations with axes F F and F 3 in Fig. are orientation preserving homeomorphisms of S 3 keeping L fixed; they form a group (ZZ). Here the axes are defined in Fig. ; F (resp. F ) meet L ½ (resp. L 0 ) at two points, and F 3 meets L ½ at two points and L 0 at two points. Recall that there is a homeomorphism S 3 L = H 3 W where the strings L 0 and L ½ correspond to the cusps of W represented by 0 := (0 0) and ½ := ( +½) ¾ H 3 respectively. Under this identification, we show Proposition. The three -rotations with axes F F and F 3 can be represented by the transformations respectively, of H 3 modulo W. z z + z z + and z z This assertion will be clear as soon as we study the fixed points of these transformations in the next subsection. Note that the three rotations modulo W (and the identity) form a group isomorphic to (ZZ), since [z z + ] ¾ W (see Remark ). We make some convention. The symbols and stand for the points in the W - orbits of + i = respectively. Let be the projection i Ô and = ¾ H 3 : H 3 (z t) z ¾ C: z-plane In the figures on the z-plane, a thick segment stands for a geodesic curve (in the upper half space H 3 ) on the hemispheres with center ( ( i) 0) and radius Ô (the big hemispheres #9 and #0 in ); its image under is the given segment. The eight geodesics in the fundamental domain FD shown in Fig., given as the intersections of walls # #9 #3 #9 #5 #9 #7 #9 # #0 #4 #0 #6 #0 #8 #0

AUTOMORPHIC FUNCTIONS FOR THE WHITEHEAD LINK 847 Fig. 5. Identification of eight geodesics in FD are identified modulo W freely later. as is seen in Fig. 5. This identification will be used 4.. Fixed loci. We study the fixed points of the transformations in Proposition in H 3 W. Recall that the translations [z z + i] and [z z + ] belong to W.. The transformation of H 3 W represented by [z z + ] fixes pointwise the following geodesics in FD: z = z = z = i z = + i In fact, for example, we have + = + i i + = i + i i mod mod i Thus the set of fixed points consists of two geodesics both starting and ending at ½ ¾ H 3, and passing through and, respectively. These can be easily understood by the diagram: ½ ½ ½ ½ This implies that this transformation represents the rotation with axis F.. The transformation [z z + ] fixes pointwise the following geodesics in FD: geodesic joining 0 and (i 0) through geodesic joining (i 0) and ( 0) through In fact, the former can be seen by the translation of the z-plane by i, and the identification of the eight geodesics shown in Fig. 5; and the latter by the same translation

848 K. MATSUMOTO, H. NISHI AND M. YOSHIDA Fig. 6. The fixed loci of [z z + ] [z z + ] [z z] and the transformation patching the big hemispheres #9 and #0 appeared in. Thus the set of fixed points consists of two geodesics both starting and ending at 0 ¾ H 3, and passing through and, respectively. These can be easily understood by the diagram: 0 0 0 0 This implies that this transformation represents the rotation with axis F. 3. The transformation [z z] fixes pointwise the following geodesics in FD: geodesic joining 0 and ( + i 0) through z = 0 z = z = + i z = i One can check these in the same way as the above two cases. These can be visualized as 0 0 0 ½ ½ ½ ½ 0 This implies that this transformation represents the rotation with axis F 3. The fixed loci in FD, as well as in H 3 W, of the rotations [z z + ], [z z + ] and [z z] are also called the axes F F and F 3 ; they are depicted in FD as in Fig. 6. A bullet stands for a vertical line: the inverse image of the point under. and that 5. Orbit spaces under W SΓ 0 ( + i) and Λ Note that W W S¼ 0 ( + i), W Æ W = S¼0 ( + i) Æ W = S¼0 ( + i) = S¼ 0 ( + i)w = (ZZ) [z z + ] ¾ W W [z z] ¾ S¼ 0 ( + i) W [z z] ¾ S¼ 0 ( + i)

AUTOMORPHIC FUNCTIONS FOR THE WHITEHEAD LINK 849 Fig. 7. A fundamental domain for W and the orbifold H 3Æ W By quotienting out the symmetry of the Whitehead link, we will see an essence of the Whitehead link. In fact, though the Whitehead link has at least five crossings, we will see that the quotient space has only one crossing; of course the ambient space necessarily has orbifold singularities. 5.. The orbifold H 3Æ W. Fig. 7 (left) shows a fundamental domain for W in FD; every wall has a counterpart to be identified with (under the order--rotations around the geodesics z = ( i), together with the patching rules of the walls tabulated in ). In the figure, a very thick segment stands for a vertical plane: the inverse image of the segment under. The quotient of S 3, where L lives, by the -rotation around the axis F is again a 3-sphere but with orbifold-singularities of index along a curve; in Fig. 7 (right), this curve is labeled by F and the numeral is attached. 5.. The orbifold H 3 SΓ 0 ( + i). Fig. 8 (left) shows a fundamental domain for S¼ 0 ( + i) in FD bounded by the four walls and the rectangle (part of the hemisphere #9 cut out by the four walls). Every wall has a counterpart to be identified with (under the order--rotations around the geodesics z = i ( + i), together with the displacement [z z + i]). The rectangle is divided into two squares; the upper square is folded (identified) by the rotation centered along the geodesic joining and (i 0), and the lower one is folded by the rotation centered along the geodesics joining and 0 = (0 0). The quotient of S 3, where L lives, by the -rotations around the axes F F and F 3 this is equivalent to the quotient of the orbifold H 3Æ W obtained in the previous subsection by the -rotation around the horizontal axis shown in Fig. 7 (right) is

850 K. MATSUMOTO, H. NISHI AND M. YOSHIDA Fig. 8. A fundamental domain for S¼ 0 ( + i) and the orbifold H 3 S¼ 0 ( + i) Fig. 9. A better picture of the fundamental domain for S¼ 0 (+i) corresponding to the left figure in Fig. 8 again a 3-sphere but with orbifold-singularities of index along three curves; in Fig. 8 (right), these curves are labeled by F F and F 3, and the numeral is attached to each of these. 5.3. The orbifold H 3 Λ. Fig. 0 (left) shows a fundamental domain for in FD bounded by the four walls a b c and d defined in Corollary, and the square (part of the hemisphere #9 cut out by the four walls). Every wall has no counterpart to be identified with. The square is folded (identified) by the rotation centered with the geodesic joining and 0 = (0 0). Thus the orbifold H 3 must be a 3-ball bounded by the -sphere divided by four (triangular) walls, which are shown in Fig. 0 (right).

AUTOMORPHIC FUNCTIONS FOR THE WHITEHEAD LINK 85 Fig. 0. A fundamental domain for and the boundary of H 3 On the other hand the orbifold H 3 should be equivalent to the quotient of the orbifold H 3 S¼ 0 (+i) obtained in the previous subsection by the reflection represented by T : z z. The mirror of the reflection in the orbifold H 3 S¼ 0 ( + i) is shown in Fig. as the union of four triangles, they are labeled by a b c and d for the obvious reason. 6. Theta functions In 6., 6., 6.3, we introduce some results for theta functions defined on a Hermitian symmetric domain D, and restrict them on H 3 embedded in D; refer to [], [], [3] and [5]. In 6.4, the final subsection, we give an embedding of H 3. 6.. Theta functions on D. The symmetric domain D of type I is defined as D = ¾ M (C) i is positive definite The group U (C) = g ¾ GL 4 (C) g Jg = J = O I O I and an involution T act on D as g = (g + g )(g + g ) T = t where g = (g jk ) ¾ U (C), and g jk are matrices. Theta functions ¾ b a () on D are defined as a ¾ () = e[(n + a)(n + a) + Re(nb )] b n¾z[i]

85 K. MATSUMOTO, H. NISHI AND M. YOSHIDA Fig.. The mirror of the reflection in the orbifold H 3 S¼ 0 (+i) is shown as the union of four parts where ¾ D, a b ¾ Q[i] and e[x] = exp[ix]. By definition, we have the following theta-transformation-formulas. FACT.. If b ¾ (Z[i]( + i)), then ¾ a ib () = a ¾ b (). If b ¾ (Z[i]), then ¾ b a () = a ¾ b ().. For k ¾ Z and m n ¾ Z[i], we have i k a a ¾ () = i k ¾ () b b a + m a ¾ () = e[ Re(mb )]¾ () b + n b 3. We have a ¾ b ag (g g ) = ¾ b(g () for g ¾ ) ¼

AUTOMORPHIC FUNCTIONS FOR THE WHITEHEAD LINK 853 a ā ¾ (T ) = ¾ () b b It is shown in [3] that theta functions ¾ a b () satisfy the following quadratic relations. Proposition. We have a + c a c 4¾ ()¾ () b + d b d = e[ Re(( + i)(b + d)e )]¾ e + ( + i)a f + ( + i)b e f ¾ +i Z[i] Z[i] Especially, a 4¾ b () = e[ Re(( + i)be )]¾ e + ( + i)a f + ( + i)b e f ¾ +i Z[i] Z[i] e + ( + i)c ()¾ f + ( + i)d e ()¾ f 6.. Embedding of H 3 into D and the pull-back of the theta functions. We embed H 3 into D by : H 3 (z t) i t + z z ¾ D; t z accordingly, we define the homomorphism ÆÔ g det(g) O : GL (C) g O g ÆÔ det(g) which we denote by the same symbol, sorry. They satisfy (g (z t)) = (g) (z t) for any g ¾ GL (C) (T (z t)) = T (z t) ¾ U (C) We denote the pull back of ¾ a b () under the embedding : H3 D by ¾ a b (z t). The following is shown in [] and [5]. FACT.. For a b ¾ (Z[i]), each ¾ a b (z t) is real valued. If Re(ab ) + Im(ab ) ¾ Z[i] then ¾ a b (z t) is identically zero.. If b = (0 0) then ¾ a b (z t) is non-negative. 3. For a b ¾ (Z[i]( + i)), each ¾ a b (z t) is invariant under the action of ¼ T (). 4. The function ¾ = ¾ 00 (z t) is positive and invariant under the action of ¼ T. () ()

854 K. MATSUMOTO, H. NISHI AND M. YOSHIDA 6.3. Automorphic functions for Γ T () and an embedding of H 3 Γ T (). Set p p p ¾ = ¾ (z t) = ¾ (z t) p q ¾ Z[i] q q q and x 0 = ¾ x = ¾ + i + i + i + i x = ¾ + i 0 0 + i x 3 = ¾ 0 + i + i 0 One of the main results in [5] is Theorem. The map : H 3 (z t) x 0 (x x x 3 ) ¾ R 3 induces an isomorphism between H 3 ¼ T () and the octahedron Oct = (t t t 3 ) ¾ R 3 t + t + t 3 minus the six vertices ( 0 0) (0 0) (0 0 ). There are essentially ten non-zero ¾ a b () for a b ¾ (Z[i]). Their restrictions on H 3 are expressed in terms of x 0 x 3 in [5]; we cite these expression as FACT 3. + i + i 0 0 ¾ = ¾ = 0 0 + i + i x 0 + x x x3 + i 0 0 0 ¾ = ¾ = 0 0 0 + i x 0 x + x x3 0 + i 0 0 ¾ = ¾ = 0 0 + i 0 x 0 x x + x 3 6.4. Automorphic functions for Λ and an embedding of H 3 Λ. Once an embedding of H 3 ¼ T () is obtained, in terms of x j, for a supergroup of ¼ T (), an embedding of H 3 can be obtained by polynomials of the x j s invariant under the finite group ¼ T (); this is a routine process. Since we have = ¼ T () g g, we study the actions of the generators g and g of the Whiteheadlink-complement group W on the theta functions ¾ a b for a b ¾ ( + i)z[i]. The theta-transformation-formulas (Fact ) leads to the following.

AUTOMORPHIC FUNCTIONS FOR THE WHITEHEAD LINK 855 Proposition 3. The generators g and g induce linear transformations of x x and x 3 : ¼ x x x 3 ½ g = ¼ ½¼ x x x 3 ½ ¼ ½ ¼ x x g = x 3 ½¼ x x x 3 Theorem. The functions x + x x x x 3 and x x x 3 are invariant under the action of. The map : H 3 (z t) ( 3 4 ) = + 3 3 ¾ R 4 where j = x j x 0, induces an embedding of H 3 into the subdomain of the variety 3 = 4 (homeomorphic to a 3-ball with two holes) bounded by the four triangular faces, which are the images (under H 3 ¼ T () x ¾ H 3 ) of a : x x + x 3 = x 0 b : x + x + x 3 = x 0 c : x x = 0 d : x + x = 0 Proof. Since = ¼ T () g g, we have the first half of this theorem. The definition of the group in 3, the fundamental domain of in 5.3, and Theorem lead to the latter half. ½ () The two matrices appeared in Proposition 3 generate a subgroup REMARK. of GL 3 (Z) isomorphic to the dihedral group of order eight. () By Proposition 3, we have x x g = x + x x x x + x x x g = x + x x x x + x The group generated by these matrices is isomorphic to the dihedral group of order eight. Proposition 4. The functions 0 + i + i + i + i 0 ¾ ¾ + ¾ 0 0 0 0 0 0 are invariant under the action of. Proof. Since ¾ +i+i 00 implies the identities of real valued functions: ¾ 0 + i 0 0 ¾ +i0 00 = and ¾ + i + i 0 0 ¾ + i 0 0 0 and ¾ 0+i are non-negative by Fact, Fact 3 00 Õ x 0 x x + x 3

856 K. MATSUMOTO, H. NISHI AND M. YOSHIDA + i + i + i 0 ¾ + ¾ ¾ 0 0 + i + i 0 0 0 0 + i 0 ¾ 0 0 Õ = Ô Õx 0 + x x x 3 + x0 x + x x 3 = Õ x 0 + x x x 3 x 0 x + x x 3 They are invariant under the action of by Proposition 3. 7. Automorphic functions for W We would like to give an explicit embedding of H 3 W. Though we already found an embedding of H 3, since W is a subgroup of, we must find new functions invariant under the action of W, which are not invariant under. In this section, we construct such automorphic functions and 3 for W by utilizing theta functions with characteristics in Z[i]. We define these functions and show their fundamental properties in 7.. We show in 7. that the groups S¼ 0 ( + i), W = Å W W «and W can be regarded as isotropy subgroups of some of these functions. An arithmetical characterization of the Whitehead-link-complement group W is given in 7.3. 7.. Fundamental properties of Φ Φ and Φ 3. Set 0 + i 0 y = ¾ y = ¾ z = ¾ + i 0 + i 0 0 z = ¾ + i + i We define functions and 3 as = x 3 z z = (x x )y + (x + x )y 3 = x x y y Theorem 3. The functions and 3 are invariant under the action of W. Only the signs of them change by the action of g = I + ( p q r s ) ¾ ¼() as follows: g = e[re(( + i)p + ( i)s)] g = e[re(r( i))] 3 g = 3 Under the action of T, the function is invariant, and 3 becomes 3. REMARK 3. The function is transformed into (x x )y (x + x )y by the action of T. This function is not invariant under the action of W but invariant under the action of W = ḡ g ¾ W = T W T. By Fact, we can easily get the following proposition, which is a key to prove Theorem 3. Proposition 5. We have y 0 g = y 0 y y z g = z 0 z 0 z

y AUTOMORPHIC FUNCTIONS FOR THE WHITEHEAD LINK 857 y z z g = y 0 0 y g = z 0 0 By the action of g = I + ( p q r s ) ¾ ¼(), the functions y y z and z change as z y g = e[re(r( i))]y y g = e[re(r( i))]y z g = e[re(r)]z z g = e[re(( + i)p + r + ( i)s)]z By the action of elements T, = 0 signs of y y z z change as follows:, = 0 and 3 = 0 0 in ¼ T (), the T 3 y + + y + z + + z + + Proof of Theorem 3. Proposition 5 implies that the product z z is invariant under the action of g and that its sign changes by the action of g. Proposition 3 implies the same for x 3. Thus = x 3 z z is invariant under the action of W. Remark () and Proposition 5 show that (x x )y and (x + x )y are invariant under the action of g and that they are interchanged by the action of g. Thus their fundamental symmetric polynomials and 3 are invariant under the action of W. Proposition 5 leads to transformation formulas for and 3 with respect to ¼ T (), since x x x 3 are invariant under the action of ¼ T (). REMARK 4. Representatives of S¼ 0 (+i)w can be given by I 3. The elements [z z + ] [z z + ] and [z z] appeared in 4. are equivalent to and 3 modulo W, respectively. These can be verified by using Theorem 5. 7.. Isotropy subgroups. Let Iso j be the subgroup of = S¼0 T (+i) consisting of elements which leave j invariant. Theorem 4. We have S¼ 0 ( + i) = Iso 3 W = Iso Iso 3 W = Iso Iso Iso 3 W : W = W : W = W : Ŵ = W : Ŵ = where W = Å W W «and Ŵ = W W. The Whitehead-link-complement group W is a normal subgroup of S¼ 0 ( + i); the quotient group S¼ 0 ( + i)w is isomorphic to Z Z.

858 K. MATSUMOTO, H. NISHI AND M. YOSHIDA REMARK 5. () The square of any element of S¼ 0 ( + i) belongs to W. () The Whitehead-link-complement group W is not a normal subgroup of, since T W T = W = W. Proof. We first show that S¼ 0 ( + i) = Iso 3. Note that the group S¼ 0 ( + i) is generated by W and ¼(). Theorem 3 shows that 3 is invariant under the action of W and ¼(). Thus we have S¼ 0 (+i) Iso 3. Theorem 3 also shows that 3 T = 3, which means that T ¾ Iso 3. Since [ : S¼ 0 ( + i)] =, we have S¼ 0 ( + i) = Iso 3. We next show that W = Iso Iso Iso 3. It is clear that W Iso Iso Iso 3. By Theorem 3, only the signs of and change by the action of S¼ 0 (+i) = Iso 3, we have [Iso 3 : Iso Iso 3 ] = and [Iso 3 : Iso Iso 3 ] =. Since the element belongs to Iso but not to Iso, we have [Iso 3 : Iso Iso Iso 3 ] = 4 The fact [S¼ 0 ( + i) : W ] = 4 shows that W is equal to Iso Iso Iso 3. Since W is a subgroup of S¼ 0 ( + i) consisting of elements keeping and invariant (only the signs of and change by the action of S¼ 0 ( + i)), W is a normal subgroup of S¼ 0 ( + i) with S¼ 0 ( + i)w ³ Z. We finally show that W = Iso Iso 3. Since W = T W T and is invariant under the actions of W and T by Theorem 3, we have W Iso. And we have W S¼ 0 (+ i) = Iso 3. Thus W Iso Iso 3. Since W g = 0 i = g 0 we have g =, which implies g ¾ W and W W. Thus we have The fact [S¼ 0 ( + i) : W ] = 4 shows that S¼ 0 ( + i) = Iso 3 Iso Iso 3 W W Iso Iso 3 = W W : W = Now it is clear that W : W = W : Ŵ = W : Ŵ =. Proposition 6. The functions (x x )y and (x + x )y are invariant under the action of Ŵ = W W. The group Ŵ is a normal subgroup of of index 6. Proof. The function is the sum of these two functions, which are invariant under the action of W. The function T is the difference of these functions, which are invariant under the action of W. Thus + T and T are invariant under the action of Ŵ.

AUTOMORPHIC FUNCTIONS FOR THE WHITEHEAD LINK 859 For g ¾ S¼ 0 ( + i) we have seen that gwg = W, which implies gw g. Thus we have gŵ g = Ŵ. On the other hand, we have T W T = W and T W T = W ; these imply T Ŵ T = Ŵ. REMARK 6. The functions in Proposition 6 give a representation of S¼ 0 ( + i). The representation matrices are 0 0 0 ; 0 this shows that the quotient group S¼ 0 ( + i)ŵ is isomorphic to the dihedral group of order eight. 7.3. An arithmetical characterization of the Whitehead-link-complement group. The Whitehead-link-complement group W is defined as the group generated by two elements g and g. It is hard to decide whether a given -matrix is in W. In this subsection, we give a criterion for elements of SL (Z[i]) to belong to W. The functions j play a key role. The main theorem of this subsection is the following. Theorem 5. An element g = ( p q r s ) ¾ S¼ 0 ( + i) satisfying Re(s) mod belongs to W = Å W W «if and only if Re(p) + Im(s) ( ) Re(q)+Im(q) (Im(p) + Re(s)) + ( ) Re(r) + Im(q) (Re(q) + Im(q))(Re(r) + Im(r)) The element g ¾ W belongs to W if and only if mod Re(p + q) + Re(r) ( )Re(q)+Im(q) Im(r) mod The element g ¾ W belongs to Ŵ = W W if and only if r ¾ Z[i]. Note that, by multiplying i I, we can always normalize g so that () Re(s) mod The rest of this subsection is devoted to a proof of this theorem. We study the action of g ¾ S¼ 0 ( + i) on and. For any element g ¾ S¼ 0 ( + i), since r ¾ ( + i)z[i] and det(g) =, we have p s ¾ ( + i)z[i], i.e., Re(p) Im(p) mod Re(s) Im(s) mod

860 K. MATSUMOTO, H. NISHI AND M. YOSHIDA By Fact () and (3) (in 6.) we may regard () (g s r ) as q p when we compute the action of g ¾ S¼ 0 ( + i) on ¾ a b s with characteristic b ¾ (Z[i]). In order to prove the first statement of theorem, we give some lemmas which can be proved by Fact and straightforward calculations. Lemma 3. We have p x 3 g = e[re(r)]x 3 g = r q s ¾ S¼ 0 ( + i) Lemma 4. given by For g ¾ S¼ 0 ( + i) satisfying Re(s) mod, the function z g is Re(r) e Re(r + s) + Im(s) e z (z t) if r ¾ Z[i] z (z t) if r ¾ Z[i] Lemma 5. For g ¾ S¼ 0 ( + i) satisfying Re(s) mod, the function z g is given by Re(p + r + s) Im(p s) e + Re(q) z if r ¾ Z[i] q ¾ ( + i)z[i] Re(p s) + Im(p + r + s) e + Im(q) z if r ¾ Z[i] q ¾ ( + i)z[i] Re(p + r) Im(p) e z if r ¾ Z[i] q ¾ ( + i)z[i] Re(p) + Im(p + r) e z if r ¾ Z[i] q ¾ ( + i)z[i] Lemmas 3, 4, 5 yield the following proposition. Å Proposition «7. An element g ¾ S¼ 0 ( + i) satisfying Re(s) mod belongs to W W = Iso Iso 3 if and only if Re(p + s) Im(p s) + Re(q) mod if q ¾ ( + i)z[i] r ¾ Z[i] Re(p + s) Im(p s) mod if q ¾ ( + i)z[i] r ¾ Z[i]

AUTOMORPHIC FUNCTIONS FOR THE WHITEHEAD LINK 86 Re(p + r + s) + Im(p + r + s) + Im(q) 0 mod if q ¾ ( + i)z[i] r ¾ Z[i] Re(p + r + s) + Im(p + r + s) 0 mod if q ¾ ( + i)z[i] r ¾ Z[i] This proposition yields the first statement of Theorem 5. We next give a necessary and sufficient condition for g ¾ S¼ 0 ( + i) to belong to Iso Iso 3. Fact and straightforward calculations imply the following. Lemma 6. For an element g ¾ S¼ 0 ( + i), if q ¾ ( + i)z[i] then x g = e[re(q)]x x g = e[re(p + q + r + s)]x if q ¾ ( + i)z[i] then x g = e[re(p + q)]x x g = e[re(q + s)]x Lemma 6 yields the following. Lemma 7. x x g = A x x g = x + x x + x p r q s ¾ S¼ 0 ( + i) where matrix A is given by if q ¾ ( + i)z[i] Re(q) ¾ Z Re(p + q + r + s) ¾ Z if q ¾ ( + i)z[i] Re(q) ¾ Z Re(p + q + r + s) ¾ Z if q ¾ ( + i)z[i] Re(q) ¾ Z Re(p + q + r + s) ¾ Z if q ¾ ( + i)z[i] Re(q) ¾ Z Re(p + q + r + s) ¾ Z if q ¾ ( + i)z[i] Re(p + q) ¾ Z Re(q + s) ¾ Z if q ¾ ( + i)z[i] Re(p + q) ¾ Z Re(q + s) ¾ Z if q ¾ ( + i)z[i] Re(p + q) ¾ Z Re(q + s) ¾ Z if q ¾ ( + i)z[i] Re(p + q) ¾ Z Re(q + s) ¾ Z

86 K. MATSUMOTO, H. NISHI AND M. YOSHIDA Fact and straightforward calculations imply the following. Lemma 8. By the action of an element g ¾ S¼ 0 ( + i) satisfying Re(s) mod, y is transformed into Re(r) + Im(r) Re(r) + Im(r) e and y is transformed into e Re(p) + e e Im(p) + e Re(p) + Im(p) + y if r ¾ Z[i] e Re(r) + Im(r) Re(r) + Im(r) Re(r) + Im(r) Re(r) + Im(r) Lemma 8 implies the following Lemma. y if r ¾ Z[i] y if r ¾ Z[i] q ¾ ( + i)z[i] y if r ¾ Z[i] q ¾ ( + i)z[i] y if r ¾ Z[i] q ¾ ( + i)z[i] y if r ¾ Z[i] q ¾ ( + i)z[i] Lemma 9. y y g = A y y g = p r q s ¾ S¼ 0 ( + i) Re(s) mod where matrix A is given by if r ¾ ( + i)z[i] P ¾ Z if r ¾ ( + i)z[i] r ¾ Z[i] P ¾ Z if if r ¾ Z[i] r ¾ Z[i] Re(r) + Im(r) Re(r) + Im(r) ¾ Z ¾ Z P + P + if r ¾ ( + i)z[i] P ¾ Z if r ¾ ( + i)z[i] r ¾ Z[i] P ¾ Z if r ¾ Z[i] Re(r) + Im(r) ¾ Z P + Re(r) + Im(r) Re(r) + Im(r) Re(r) + Im(r) ¾ Z ¾ Z ¾ Z

AUTOMORPHIC FUNCTIONS FOR THE WHITEHEAD LINK 863 if r ¾ Z[i] Re(r) + Im(r) ¾ Z P + Re(r) + Im(r) ¾ Z where = ( ) Re(q)+Im(q) and P = Re(p + q) + Im(q). Proposition 8. Iso if and only if An element g ¾ S¼ 0 ( + i) satisfying Re(s) mod belongs to Re(p + q) + Re(r) ( )Re(q)+Im(q) Im(r) mod Proof. Since only the sign of changes by the action of g ¾ S¼ 0 ( + i), if x x x +x is transformed into A x x x +x by the action of g then y y is transformed into A y y by the action of g, where A = in Lemmas 7 and 9. Thus g ¾ S¼ 0 ( + i) belongs to Iso if and only if the sign of the transformation matrix A for the action of g on x x x +x coincides with that on y y. () the case A =. By Lemma 7, g ¾ S¼ 0 ( + i) satisfies q ¾ ( + i)z[i] Re(q) + Re(p + q + r + s) ¾ Z i.e., By Lemma 9, we have Re(q) + Im(q) 0 mod Re(p + r) mod r ¾ Z[i] P = Re(p + q) + Im(q) ¾ Z The coincident condition for the signs is Thus we have Re(q) Re(r) + Im(r) Re(p + q) + () the case A =. By Lemma 7, g ¾ S¼ 0 ( + i) satisfies Re(r) + Im(r) Re(r) Im(r) Re(p + r) + mod mod q ¾ ( + i)z[i] Re(q) + Re(p + q + r + s) ¾ Z i.e., Re(q) + Im(q) 0 mod Re(p + r) 0 mod

864 K. MATSUMOTO, H. NISHI AND M. YOSHIDA By Lemma 9, we have r ¾ Z[i] and Re(r) + Im(r) + P + Re(r) + Im(r) = Re(p + q) + Im(q + r) + + ( )Re(q)+Im(q) Re(r) Re(p + r) + Im(r) Re(p) mod The coincident condition for the signs is Re(q) Thus we have Re(r) + Im(r) Re(p + q) + + (3) the case A =. By Lemma 7, g ¾ S¼ 0 ( + i) satisfies Re(r) + Im(r) Re(r) Im(r) Re(p + r) + mod mod q ¾ ( + i)z[i] Re(p + q) + Re(q + s) ¾ Z i.e., By Lemma 9, we have Re(q) + Im(q) mod Re(p) mod r ¾ Z[i] P = Re(p + q) + Im(q) ¾ Z The coincident condition for the signs is Re(p + q) + (4) the case A =. By Lemma 7, g ¾ S¼ 0 ( + i) satisfies Re(r) + Im(r) mod q ¾ ( + i)z[i] Re(p + q) + Re(q + s) ¾ Z i.e., Re(q) + Im(q) mod Re(p) 0 mod By Lemma 9, we have r ¾ Z[i] and Re(r) + Im(r) Re(r) + Im(r) + P + Re(p + q) + Im(q + r) Re(p) 0 mod

AUTOMORPHIC FUNCTIONS FOR THE WHITEHEAD LINK 865 The coincident condition for the signs is Re(p + q) + Re(r) + Im(r) mod This proposition yields the second statement of Theorem 5. We show the last statement of Theorem 5. The element g ¾ W satisfying Re(s) mod belongs to Ŵ if and only if the transformation matrix A for the action of g on x x x +x is or Thus we have the condition r ¾ Z[i], which is kept under the multiplication i I to g. 8. Embeddings of the quotient spaces In the previous section, we constructed automorphic functions and 3 for W. The map H 3 (z t) 4 x 3 0 x0 3 x0 4 induces a map H 3 W R 7, which is generically injective but not quite. In 8., we construct, for each j = 3, automorphic functions f j f j for W such that their common zero is F k F l, where j k l = 3. Here the curves F F F 3 H 3 are defined as the W -orbits of the fixed loci of the transformations 3, respectively ( 4., Remark 4 in 7.). These functions give, in 8., 8.3 and 8.4, embeddings of the quotient spaces H 3 S¼ 0 ( + i) H 3 W and H 3 W, respectively. 8.. Automorphic functions for W vanishing along F j. We use W -invariant functions as follows: f 00 = x x y y = 3 f 0 = x x z z z 3 z 4 f = x 3 z z = f = x x z z f 3 = x 3 x x z3 z 4 f 4 = x x x x z3 z 4 f 0 = (x x )z z 3 + (x + x )z z 4 f = z z (x x )z z 3 + (x + x )z z 4 f = x x (x x )z z 4 + (x + x )z z 3

866 K. MATSUMOTO, H. NISHI AND M. YOSHIDA f 30 = (x x )y + (x + x )y = f 3 = (x x )z z 3 (x + x )z z 4 f 3 = z 3 z 4 (x x )z z 4 + (x + x )z z 3 where Set z 3 = ¾ 0 i 0 z 4 = ¾ + i i + i f i j = f i j Æ x deg( f i j ) 0 where deg( f ) denotes the total degree of the polynomial f with respect to x i y j z k. Proposition 9. The functions f j p are invariant under the action of W. These functions change the signs by the actions of and 3 as in the table 3 f 0 j + + + f j + f j + f 3 j + This proposition can be obtained by Proposition 5 and the following lemma. Lemma 0. We have z3 0 g = z 4 0 z3 z 4 z3 g = z 4 0 0 z3 z 4 By the action of T 3 ¾ ¼ T (), the signs of z 3 z 4 change as T 3 z 3 + + + + z 4 + Proposition and Fact 3 yield the following proposition, which is a key to study the zero locus of f j p.

AUTOMORPHIC FUNCTIONS FOR THE WHITEHEAD LINK 867 Proposition 0. We have 0 4z = 4¾ 0 00 = ¾ ¾ 00 0+i +¾ +i0 0+i 00 = (x 0 + x + x + x 3 )(x 0 x x + x 3 ) +i 4z = 4¾ +i 00 = ¾ ¾ 00 0+i +¾ +i0 0+i 00 = (x 0 + x x x 3 )(x 0 x + x x 3 ) 0i 4z3 = 4¾ 0 00 0+i 0+i = ¾ ¾ +¾ 00 +i0 00 = (x 0 + x x + x 3 )(x 0 x + x + x 3 ) +ii 4z4 = 4¾ +i 00 0+i 0+i = ¾ ¾ +¾ 00 +i0 00 = (x 0 + x + x x 3 )(x 0 x x x 3 ) ¾ ¾ 00 +i0 ¾ ¾ 00 +¾ +i0 00 +¾ +i0 ¾ 00 +i0 +i+i +i0 +i+i ¾ 0+i +i+i ¾ +i+i +i+i ¾ +i+i ¾ +i0 0+i +i0 0+i +i+i +i0 +i+i ¾ 0+i REMARK 7. The functions z +z, z z, z 3 +z 4, z 3 z 4, z z 3 +z z 4 and z z 4 +z z 3 are invariant under the action of. They can be expressed in terms of 4 and x 0 : z + z = z 3 + z 4 = x 0 + 3 z z = 6 3 x0 + z 3 z 4 = 6 3 x 0 + 3 + 8 4 x 0 + x 4 0 x 0 + 4 3 8 4 x 0 + x 4 0 x 0 + 4 z z 3 + z z 4 = 8 3 + 3x 0 3 + x 4 0 x 0 + 4 z z 4 + z z 3 = 8 3 x 0 + 3 + x 4 0 x 0 + + 4 REMARK 8. Proposition 0 implies z z = x 0x 3 x x

868 K. MATSUMOTO, H. NISHI AND M. YOSHIDA z 3 z 4 = x 0x 3 + x x z 3 z = z z 4 = x x z z 4 = z 3 z = x 0x 3 z z 3 z z 4 = x 0x 3 x 0 x x + x 3 z z 4 + z z 3 = x x x 0 x x + x 3 These functions are invariant under the action of g and their signs change by the action g. The product of x 3 (resp. x x ) and each of these is invariant under the action and can be expressed in terms of 4 and x 0. Theorem 6. The analytic sets V V and V 3 of the ideals I = f f f 3 f 4 I = f f I 3 = f 3 f 3 are (set-theoretically) equal to F F 3 F F 3 and F F, respectively. Corollary. The analytic set V jk of the ideals I j I k is set-theoretically equal to F l for j k l = 3. Proof of Theorem 6. Since the sets F j are the fixed loci of j modulo W and f kl are invariant under the action of W, it is clear that V j F k F l for j k l = 3. We first show V F F 3. Since we have f = x 3 f = x x 3 = 3 = 3 = f 3 = x 3 x x f 4 = x x x x (x 0 + x + x + 3 x 3 ) 3 = 3 = (x 0 + x + x + 3 x 3 ) 3 = 3 = 3 = (x 0 + x + x + 3 x 3 ) (x 0 + x + x + 3 x 3 ) they are invariant also under the action of ¼ T (). So we express the common zeros of them in terms of x j. The twelve edges of the octahedron Oct = x = [x 0 x x x 3 ] ¾ P 3 (R) x + x + x 3 x 0

AUTOMORPHIC FUNCTIONS FOR THE WHITEHEAD LINK 869 (recall that H 3 ¼ T () is realized as Oct minus the vertices, in Theorem ) and the segments x ¾ Oct x = x 3 = 0 x ¾ Oct x = x 3 = 0 x ¾ Oct x 0 + x + x + x 3 = x x = 0 x ¾ Oct x 0 x x + x 3 = x x = 0 x ¾ Oct x 0 + x x x 3 = x + x = 0 x ¾ Oct x 0 x + x x 3 = x + x = 0 come into the game. Theorem shows that V is the union of the inverse images of, which coincides with F F 3. We next show V 3 F F. Since we have the products f 3 T = (x x )z z 3 + (x + x )z z 4 f 3 T = z 3 z 4 (x x )z z 4 + (x + x )z z 3 f 3 = ( f 3 )( f 3 T ) = (x x ) z z 3 (x + x ) z z 4 f 3 = ( f 3 )( f 3 T ) = z 3 z 4 (x x ) z z 4 (x + x ) z z 3 are invariant under the action of ¼ T (). We express the common zero of f 3 and f 3 in terms of x j. By Proposition 0, we have f 3 = 4 x x x 4 3 + x x 3x 0 x x x 3 x 0 x + x x 0 x x + x 3 x3 + x x (x + x 0 x )(x 0 + x + x )(x x + x 0 )(x 0 x x ) f 3 = 64 x x x0 + x x x3 x 0 x + x x3 3 = (x 0 + x + x + 3 x 3 ) Thus V 3 is a subset of the union of the common zeroes of f 3 and the factors of f 3. We study the restriction of f 3 on the algebraic set of each factor of f 3. In the octahedron Oct, the factors x0 + x x x 3 and x 0 x + x x 3 vanish only on [x 0 x x x 3 ] = [ 0 0 ] [ 0 0] and [ 0 0 ] [ 0 0], respectively. The functions f 3 vanishes on these points. On x j = 0, f 3 reduces to x 0xk x 3 x0 xk + x 3 where j k =. On x 0 + x + x + 3 x 3 = 0 ( 3 = ), f 3 reduces to 3 x 0 (x 3 x ) (x 0 + x )(x 0 + x )(x 0 + x + x )

870 K. MATSUMOTO, H. NISHI AND M. YOSHIDA Thus the common zero of f 3 and f 3 in the fundamental domain FD of H 3 W in Fig. is the union of F F and the geodesic joining (z t) = (0 0) ( +i 0) through = ( +i) ÆÔ which is the inverse image of of [x 0 x x x 3 ] ¾ Oct x = x 3 = 0. We have only to show that f 3 does not vanish on [x 0 x x x 3 ] ¾ Oct x = x 3 = 0. Since (z z z 3 z 4 ) = 3 = (x 0 + x + x + 3 x 3 ) the function z z z 3 z 4 never vanish in the interior of FD. Thus we have z z z 3 z 4 0 or z z z 3 z 4 0 in the interior of FD. Since f 3 reduces to x (z z 3 + z z 4 ) on the set [x 0 x x x 3 ] ¾ Oct x = x 3 = 0 included in F and f 3 vanishes on this set, the sign of z z 3 is different from that of z z 4, which implies z z z 3 z 4 0 in the interior of FD. On the other hand, f 3 reduces to x (z z 3 z z 4 ) on the set [x 0 x x x 3 ] ¾ Oct x = x 3 = 0. Since the sign of z z 3 is different from that of z z 4 in the interior of FD, z z 3 z z 4 never vanish in the interior of FD. Hence f 3 never vanishes in the interior of [x 0 x x x 3 ] ¾ Oct x = x 3 = 0 x = 0. We finally show V F F 3. Since we have the products f T = z z (x x )z z 3 (x + x )z z 4 f T = x x (x x )z z 4 + (x + x )z z 3 f = ( f )( f T ) = z z (x x ) z z 3 (x + x ) z z 4 f = ( f )( f T ) = x x (x x ) z z 4 + (x + x ) z z 3 are invariant under the action of ¼ T (). We express the common zero of them in terms of x j. By Proposition 0, we have f = 6 f 3 3 = 3 = (x 0 + x + x + 3 x 3 ) f = 4 x x x x x 0 + x x x 3 x 0 x + x x 3 Thus V is a subset of the union of the common zeroes of f and the factors of f. We study the restriction of f on the algebraic set of each factor of f. Since we have studied the restriction of f 3 on the algebraic set of each factor of f 3, we have only to consider the restriction of z z = 6 3 = 3 = (x 0 + x + x + 3 x 3 )

AUTOMORPHIC FUNCTIONS FOR THE WHITEHEAD LINK 87 on the algebraic set of each factor of f and that of f on the sets x x = 0. We can see that the common zero of f and f in FD is the union of F F 3 and the geodesic joining (z t) = ( 0) (i 0) through = ( + i) ÆÔ which is the inverse image of of [x 0 x x x 3 ] ¾ Oct x = x 3 = 0. In order to show that f does not vanish on [x 0 x x x 3 ] ¾ Oct x = x 3 = 0, follow the proof of the non-vanishing of f 3 on [x 0 x x x 3 ] ¾ Oct x = x 3 = 0. 8.. An embedding of H 3 SΓ 0 ( + i). Theorem 7. The map ³ 0 : H 3 S¼ 0 ( + i) (z t) ( 4 f 0 ) ¾ R 5 is injective, where f 0 = f 0 x 6 0. Its image Image(³ 0) is determined by the image Image() under : H 3 (z t) ( 4 ) and the relation 56 f 0 = x x = 4 3 = 3 = (x 0 + x + x + 3 x 3 ) 3 x 0 + 3 + 3 8x 0 4 + x 4 0 x 0 + 4 as a double cover of Image() branching along its boundary. If we replace f 0 by f 00, the map is injective as well, but the expression of the image becomes a bit more complicated for f 00. Proof. Proposition 0 and Remark 7 give the expression f 0 in terms of 4 and x 0. Since the function f 0 is invariant under the action of S¼ 0 ( + i) and changes its sign by the action of T, the map ³ 0 induces a double cover Image(³ 0 ) ( 4 f 0 ) ( 4 ) ¾ Image() which ramifies along the zero locus of f 0 : H 3 S¼ 0 ( + i) Ý H 3 ³ 0 Image(³0 ) Ý Image() The natural map (studied in 5. and 5.3) H 3 S¼ 0 ( + i) H 3 is a double cover of a 3-ball (minus two points) by a 3-sphere (minus two points) branching along the boundary of the 3-ball. Thus we have only to show that the function f 0 vanishes only along the boundary a b c d of the 3-ball H 3 (see Theorem ).

87 K. MATSUMOTO, H. NISHI AND M. YOSHIDA By Remark (), we have (x +x )(g T ) = (x +x ). Thus x +x vanishes on the mirror (z t) ¾ H 3 Im(z) = of the reflection g T. By Theorem, x +x vanishes only on d in the fundamental domain in Fig. 0 (left). Similarly, x x vanishes only on the mirror c in the fundamental domain in Fig. 0 (left). By Theorem and Proposition 0, z z z 3 z 4 vanishes only on a b in the fundamental domain in Fig. 0 (left). We briefly observe the image Image(³ 0 ). The two cusps ½ and 0, and the points and (defined in 4.) are mapped to ½ := (0 0 0 0) 0 := ( 0 0 0 0) := (0 0 0 0 0) := 6 0 0 0 The images F F and F 3 of the axes F F and F 3 (see 4.) are union of curves joining these points: F : ½ F : 0 F 3 : 0 ½ Four of these curves come to each cusp. We parameterize these curves (0 t ) and present them as follows: and F F F 3 ½ (0 0 ( t) 0 0) as t 0 t ½ t 4 6 ( t ( t) t) 0 as t 0 4 0 ½ (t 0 ( t) 0 0) as t 0 t ½ t 4 6 ( t ( t) t) 0 as t 0 4 0 (( t) 0 0 0 ( t) t ( t) ) as t 0 t 0 4 + t t t 0 0 0 as t 0 4 F 3 0 (( t) 0 0 0 ( t) t ( t) ) as t 0 ½ 0 (( t) 0 t 0 0) as t 0 These curves can be illustrated as in Fig.. Each of the two cusps ½ and 0 is shown as a hole. These holes can be deformed into sausages as in Fig. 3. Note that this is just the orbifold H 3 S¼ 0 ( + i) shown in Fig. 8 (right), if we replace the curves L ½ andl 0 by their tubular neighborhoods.

AUTOMORPHIC FUNCTIONS FOR THE WHITEHEAD LINK 873 Fig.. Orbifold singularities in Image(³ 0 ) and the cusps and 0 ½ Fig. 3. The cusp-holes are deformed into two sausages Recall that four of the orbifold-singular-loci stick into each cusp-hole, of which boundary is a -sphere, and that the double cover of a -sphere branching at four points is a torus. 8.3. An embedding of H 3Æ W. Theorem 8. The map ³ : H 3Æ W (z t) (³ 0 f f 4 ) ¾ R 9 is injective, where f i j = f i j Æ x deg( f i j ) 0. The products f p f q ( p q 4) can be ex-

874 K. MATSUMOTO, H. NISHI AND M. YOSHIDA pressed as polynomials of x 0, 4 and f 0. The image Image(³ 0 ) together with these relations determines the image Image(³ ) under the map ³. Proof. Each f p is Æ invariant under the action of W and its sign changes under the action of S¼ 0 (+i) W. By Proposition 0 and Remark 7, f f 4 and f f f 3 f 4 can be expressed in terms of j and x 0 ; they are invariant under the action of. The product f p f q (p =, q = 3 4) is invariant under the action of S¼ 0 ( + i) by Proposition 9 and they can be expressed in terms of x 0 j, and f 0. Thus if one of the values of f f 4 is not zero at a point (z t) ¾ H 3, then this non-zero value together with the image ³ 0 (z t) determines the vector ( f (z t) f 4 (z t)). Thus we have the commutative diagram H 3Æ W Ý H 3 S¼ 0 ( + i) ³ Image(³ ) Ý ³ 0 Image(³0 ) of ³ 0 ³ and the two (vertical) double covers. Since ³ 0 is an isomorphism and the left vertical map ramifies exactly along F F 3 ( 5., 5.), the map ³ is injective thanks to Theorem 6. Though the embedding dimension is too high to see the shape of the image directly unfortunately, the theorem above and the argument in 5 asserts the following: The boundary of a small neighborhood of the cusp ³ (0) is a torus, which is the double cover of that of the cusp ³ 0 (0); note that two F -curves and two F 3 -curves stick into ³ 0 (0). The boundary of a small neighborhood of the cusp ³ (½) remains to be a -sphere; note that two F -curves and two F 3 -curves stick into ³ 0 (½), and that four F -curves stick into ³ (½). Topologically, the sausage L 0 in Fig. 3 (and Fig. 8 (right)) is covered by a doughnut, a tubular neighborhood of the curve L 0 in Fig. 7 (right). 8.4. An embedding of H 3 W. Theorem 9. The map ³ : H 3 W (z t) (³ f f f 3 f 3 ) ¾ R 3 is injective, where f i j = f i j x deg( f i j ) 0. The products f q f r f 3q f 3r and f p f q f 3r (p = 4, q r = ) can be expressed as polynomials of x 0, 4 and f 0. The image Image(³ ) together with these relations determines the image Image(³) under the map ³.

AUTOMORPHIC FUNCTIONS FOR THE WHITEHEAD LINK 875 Proof. By Proposition 0, the products f q f r f 3q f 3r and f p f q f 3r (p = 4, q r = ) are invariant under the action of S¼ 0 ( + i) by Proposition 9. They can be expressed in terms of x 0, 4 and f 0. For example, f = 4 x + x z z4 + z z 3 f3 = x + x z z3 + z z 4 x x z z 3 z z 4 x x z z 4 z 3 z + f0 f 0 f 3 f 3 = z 3 z 4 x x z z x + x z + z + z 3 + z 4 f0 f f 3 = z z f 3 f f 3 = x + x x0 f 3 f 4 f 4 z + z + x x x0 f + f f f 3 = 64 x 0 + x x x 3 x 0 x + x x 3 f4 ; (Remark 7 and 8 help us to find these expressions.) So the values of f, f, f 3 and f 3 at any point in H3 are determined by those of x 0 4 and f 0. Moreover, if one of the values of f, f, f 3 and f 3 is not zero at a point (z t) ¾ H 3, then this non-zero value together with the image ³ (z t) determines the vector Thus we have the commutative diagram ( f (z t) f (z t) f 3 (z t) f 3 (z t)) H 3 W Ý H 3Æ W ³ Image(³) Ý ³ Image(³ ) of ³ ³ and the two (vertical) double covers. Since ³ is an isomorphism and the left vertical map ramifies exactly along F ( 5.), the map ³ is injective thanks to Corollary. Though the embedding dimension is too high to see the shape of the image directly unfortunately, the theorem above and the argument in 5 asserts the following: The boundary of a small neighborhood of the cusp ³(½) is a torus, which is the double cover of that of the cusp ³ (½); recall that four F -curves stick into ³ (½). The boundary of a small neighborhood of the cusp ³(0) is a torus, which is the unbranched double cover of that of the cusp ³ (0), a torus. Eventually, the two sausages in Fig. 3 (and Fig. 8 (right)) are covered by two linked doughnuts, tubular neighborhoods of the curves L 0 and L ½ in Fig.. Note that, in the (high dimensional) ambient space, the two tori look as if they are not linked, however they are linked in the Image(³).

876 K. MATSUMOTO, H. NISHI AND M. YOSHIDA References [] E. Freitag: Modulformen zweiten Grades zum rationalen und Gaußschen Zahlkörper, Sitzungsber. Heidelb. Akad. Wiss. (967), 49. [] K. Matsumoto: Theta functions on the bounded symmetric domain of type I and the period map of 4-parameter family of K 3 surfaces, Math. Ann. 95 (993), 383 408. [3] K. Matsumoto: Algebraic relations among some theta functions on the bounded symmetric domain of type I rr, Kyushu J. Math. 60 (006), 63 77. [4] K. Matsumoto, T. Sasaki and M. Yoshida: The monodromy of the period map of a 4-parameter family of K 3 surfaces and the Aomoto-Gel fand hypergeometric function of type (3 6), Internat. J. of Math. 3 (99), 64. [5] K. Matsumoto and M. Yoshida: Invariants for some real hyperbolic groups, Internat. J. of Math. 3 (00), 45 443. [6] W. Thurston: Geometry and Topology of 3-manifolds, Lecture Notes, Princeton Univ., 977/78. [7] N. Wielenberg: The structure of certain subgroups of the Picard group, Math. Proc. Cambridge Philos. Soc. 84 (978), 47 436. [8] M. Yoshida: Hypergeometric Functions, My Love, Aspects of Mathematics, E3, Friedr Vieweg & Sohn, Braunschweig, 997. Keiji Matsumoto Department of Mathematics Hokkaido University Sapporo 060-080 Japan e-mail: matsu@math.sci.hokudai.ac.jp Haruko Nishi Department of Mathematics Kyushu University Fukuoka 8-858 Japan e-mail: nishi@math.kyushu-u.ac.jp Masaaki Yoshida Department of Mathematics Kyushu University Fukuoka 80-8560 Japan e-mail: myoshida@math.kyushu-u.ac.jp