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COMPLEX HYPERBOLIC MANIFOLDS HOMOTOPY EQUIVALENT TO A RIEMANN SURFACE WILLIAM M. GOLDMAN, MICHAEL KAPOVICH, AND BERNHARD LEEB Abstract. We construct isometric actions of fundamental groups of closed Riemann surfaces on the complex hyperbolic plane, which realize all possible values of Toledo's invariant . For integer values of these actions are discrete embeddings. The quotient complex hyperbolic surfaces are disc bundles over closed Riemann surfaces, whose topological type is described in terms of . We relate our geometric construction to arithmetic constructions and discuss integrality properties of .

1. Introduction Let denote a closed oriented surface of genus g 2 and = 1() its fundamental group. Let G = PU(n; 1) denote the group of biholomorphic isometries of complex hyperbolic n-space HnC . In [15], Domingo Toledo considered the following invariant associated to a representation : ! PU(n; 1): Let f~ : ~ ! HnC be a -equivariant smooth mapping of the universal covering of . Then Toledo's invariant (; f~) is de ned as the (normalized) integral of the pull-back of the Kahler form ! on HnC : Z 1 (; f~) := 2 f~!

where is a fundamental domain for the action of on ~ . (; f~) is independent of f~ and depends continuously on the representation . Toledo proved that (1.1) 2 ? 2g 2g ? 2 He showed furthermore that j j = 2g ? 2 holds if and only if is an isomorphism onto a lattice in the stabilizer of a complex geodesic in HnC . By relating () to the rst Chern class of a certain line bundle we show that it satis es the integrality property: (1.2) 2 n +2 1 Z: The function : Hom(; PU(n; 1)) ?! n +2 1 Z is locally constant and the Toledo invariant takes one value on each connected component of the representation space. Xia [16] shows that, vice versa, the Toledo invariant Date : June 7, 1999. 1991 Mathematics Subject Classi cation. 53C55, 22E40. Key words and phrases. Complex hyperbolic space, at bundle, fundamental group of surface, holonomy, Euler class. This research was partially supported by National Science Foundation grants, the University of Maryland Institute for Advanced Computer Studies, the General Research Board of the University of Maryland, the Alfred P. Sloan Foundation, the Mathematical Sciences Research Institute, the Sonderforschungsbereich 256 in Bonn and the Volkswagen-Stiftung (Research-in-Pairs program at the Mathematisches Forschungsinstitut Oberwolfach). 1

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GOLDMAN, KAPOVICH, AND LEEB

distinguishes dierent connected components. The main results of this paper concern the values realized by Toledo's invariant: Theorem 1.1. For every genus g 2 and every satisfying (1.1) and (1.2) there exists a representation : ! PU(2; 1) with () = . When is not an integer, then is not faithful in our examples (but () is discrete when = 4=3). However, when is an even integer, we nd discrete embeddings: Theorem 1.2. For every genus g 2 and every even integer satisfying (1.1) there exists a convex-cocompact discrete and faithful representation : ! PU(2; 1) with () = . Furthermore, the complex hyperbolic surface M = H2C =() is dieomorphic to the total space of an oriented R 2 -bundle over with the Euler number (1.3) e( ) = () + j ()=2j Generally, for a discrete embedding ! PU(2; 1) the equality (1.3) will fail [8]. Simple examples of complex hyperbolic surfaces arise from representations preserving totally geodesic planes. There are two kinds of totally geodesic submanifolds W H2C of real dimension two: Complex geodesics (copies of H1C ) have constant sectional curvature ?1. Their stabilizers in PU(2; 1) are conjugate to U(1; 1). Totally real geodesic 2-planes (copies of H2R) have constant sectional curvature ?1=4. Their stabilizers in PU(2; 1) are conjugate to SO(2; 1). Let W be such a totally geodesic submanifold and G its stabilizer. For a surface group , there exist discrete embeddings : ?! G whose image is a cocompact lattice in G. When W is a complex line, then j ()j = 2g ? 2 is maximal; we say that is complex Fuchsian. When W is a totally real plane, then () = 0; then we say that is real Fuchsian. These two special cases are building blocks in our constructions. Quotients of H2C by real and complex Fuchsian groups provide two quite dierent kinds of complex hyperbolic surfaces. For any manifold M1 which is the quotient space of the complex Fuchsian group, contains as a totally geodesic complex curve. In contrast, the quotients M2 of the real Fuchsian groups are Stein manifolds (see Burns and Shnider [2] or Goldman [5],x5.4.7). Moreover, the surfaces M1 and M2 are not even homeomorphic. They are both dieomorphic to total spaces of oriented 2-plane bundles over . The Euler number of the bundle equals ()=2 for M1 and () for M2 . These two examples are discussed in detail in xx2.3{2.5. The examples in Theorem 1.2 amalgamate real and complex Fuchsian representations. The representations in Theorem 1.1 amalgamate pairs of holonomy representations of hyperbolic cone structures on closed surfaces. The present paper is an extended and corrected version of the preprint [6]. Acknowledgements. We thank Domingo Toledo for helpful discussions. Conversations with Igor Belegradek, Je Hakim, Steve Kudla, John Millson, Alan Reid and Eugene Xia have also been useful. Finally we would like to thank the referee for several important suggestions.

COMPLEX HYPERBOLIC MANIFOLDS

2. Representations of surface groups in PU(2,1) and their characteristic numbers

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2.1. Toledo's invariant. We refer to [5] and Toledo's papers [14, 15] for the necessary background for this section. (A general description of characteristic classes associated to representations may also be found in Dupont's paper [3].) Let G := PU(n; 1), X := HnC and is the fundamental group of a closed Riemann surface of genus g 2. In this paper we will use the unit ball model for HnC , whereby HnC C n PnC : For any 2 Hom(; G) we have the associated at HnC -bundle X over with holonomy . Its total space is the quotient of ~ X by the action of = 1 () given by

: (m; ~ x) 7! ( m; ~ ( )x) The Kahler form on X = HnC de nes a closed 2-form ! on X . (In the above description, ! satis es ! = X ! where X : ~ X ?! X and : ~ X ?! X are the natural projections.) For any section s : ! X , Z 1 (; s) = 2 s! is independent of s and is thus an invariant of . Conjugating by an antiholomorphic isometry of HnC yields representations with opposite characteristic number. (An example of an antiholomorphic isometry is the complex conjugation z 7! z.) Suppose 2 Hom(; G), and 2 Isom(HnC ) is an antiholomorphic isometry. Then ! = ?! and it follows that () = ? (). Recall that is called a discrete embedding if its image is a discrete subgroup of G and is injective. In case is a discrete embedding, the quotient M = HnC =() is a complex hyperbolic manifold. A section s of the at bundle X corresponds to a homotopy equivalence f : ?! M as follows. Lifting everything to the universal covering ~ , the section s of X de nes a section s~ : ~ ?! ~ X which is the graph of a -equivariant map f~ : ~ ?! X . Equivariance implies that f~ covers a map f : ?! M which induces the isomorphism 1 () ?! (1 ()) = 1 (M ): Since both and M are aspherical, f is a homotopy equivalence. Thus Z Z 1 1 () = 2 s ! = 2 f !M 2.2. Integrality. Let L ! HnC be a Hermitian complex line bundle, and let G be a Lie group acting on L by isomorphisms. Let denote the curvature form of L; it is a G-invariant 2-form on HnC . Given a representation : ! G and an equivariant smooth map f : ~ ! HnC , the group acts on the pull-back bundle f L ! ~ , and f L descends to a Hermitian

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line bundle L ! . The curvature form of f L is given by f and therefore 21i f

descends to a 2-form on representing the rst Chern class c1(L ). Hence Z

= c1(L ) 2 Z:

We apply this remark now in two concrete situations. Let ! HnC be the canonical line bundle and ! HnC be the restriction to HnC of the tautological bundle C n+1 n f0g ! Pn (C ) over projective n-space. Then n+1 = . There are canonical actions of SU(n; 1) on and of PU(n; 1) on . Both bundles carry invariant Hermitian metrics which are unique up to scaling. The corresponding curvature forms are given by

() = 1 ! 2i respectively

() = n 2+i 1 ! where ! denotes the Kahler form on HnC . Given a representation : ! PU(n; 1) we consider the line bundle L ! associated to ! HnC . We obtain Z Z Z 1 n + 1 c1 (L) = = 2i f () = ? 4 f ! = ? n +2 1 (); so () 2 n +2 1 Z:

If lifts to a representation ~ : ! SU(n; 1) consider instead the line bundle L~ ! associated to ! HnC , obtaining c1(L~) = ? 21 (); so () 2 2 Z: 2.3. Example: Complex lines. Basic examples of representations ! PU(n; 1) arise from compact complex hyperbolic 1-manifolds embedded in complete complexhyperbolic surface. We represent complex hyperbolic space HnC by the subset of projective space of the inde nite Hermitian vector space C n;1 of index one and dimension n +1 corresponding to lines whose induced Hermitian form is negative de nite. Recall that a complex geodesic in HnC is a holomorphic totally geodesic complex curve. A complex geodesic has constant curvature ?1, and is equivalent to C = H1C embedded in HnC as f0g H1C . In terms of the unit ball model, in which HnC is represented by B n C n as 2 3 2 3 z1 z1 6 .. 7 4 ... 5 7?! 6 . 7 2 C n;1 : 4z 5 n zn 1


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Under this identi cation, a model complex geodesic is the image of 2 3 0 6 .. 7 6.7 H1C ?! f0g B 1 ?! C n;1 z 7?! (0; : : : ; 0; z) ! 66077 4z 5 1 where z 2 C and jzj < 1. The stabilizer of C is the image of the embedding U(n ? 1) U(1; 1) ,! PU(n; 1) obtained as the composition of the embedding U(n ? 1) U(1; 1) ,! U(n; 1) B 0 (B; A) 7! 0 A with the projectivization P : U(n; 1) ?! PU(n; 1). The kernel of the projectivization homomorphism P : U(n; 1) ?! PU(n; 1) consists of scalar matrices and is the center of U(n; 1). There are homomorphisms P SU(1; 1) ,! SU(n; 1) ,! U(n; 1) ?! PU(n; 1) The homomorphism SU(1; 1) ! PU(n; 1) is an embedding since the image of U(1; 1) in U(2; 1) intersects the kernel of P in the trivial subgroup. The biholomorphic isometries of H1C comprise the group PU(1; 1). The action of PU(1; 1) on H1C , however, does not quite extend to HnC . Rather, the group SU(1; 1) acts on HnC by the embedding SU(1; 1) ! PU(n; 1) above. The restriction homomorphism SU(1; 1) ! Isom(H1C ) de nes a double covering of SU(1; 1) onto PU(1; 1). The nontrivial element of its kernel is the re ection in H1C given by (z1 ; : : : ; zn?1; zn) 7?! (?z1 ; : : : ; ?zn?1 ; zn) which we call the inversion with respect to H1C . Although the inversion restricts to the identity on H1C , it acts nontrivially on HnC and on the normal bundle to H1C HnC . This inversion belongs to the maximal compact subgroup K of SU(1; 1) which is represented by diagonal matrices 3 2 In?1 0 0 4 0 05 0 0 ?1 where j j = 1. It acts on HnC by (z1 ; : : : ; zn) 7?! (z1; : : : ; zn?1; 2zn): Now we specialize to the case n = 2. Let 1 denote the unit normal bundle to H1C in H2C and UT denote the unit tangent bundle of H1C . Note that SU(1; 1) acts simply-transitively on 1 . Both the normal and tangent bundle of H1C have Hermitian connections induced by the Levi-Civita connection on H2C . These connections restrict to connections on the corresponding unit bundles. Lemma 2.1. There exists a 2-fold covering 1 ! UT which is equivariant with respect to the homomorphism of Lie groups p : SU(1; 1) ! PU(1; 1).

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Proof. As the group PU(1; 1) acts simply-transitively on UT , the 2-1 projection p : SU(1; 1) ! PU(1; 1) gives us the required 2-fold covering between corresponding circle bundles. Corollary 2.2. Consider an elliptic isometry : H2C ! H2C which lies in the image of the standard embedding SU(1; 1) ,! PU(2; 1), assume that the origin o is a xed point of . Then the angle of rotation of in the tangent line To H1C is twice the angle of rotation of in the normal line oH1C . Note that the curvature of the pull-back of the connection on UT via p to 1 is twice the curvature of the connection on UT . Since the induced metric on T H1C has sectional curvature -1, the curvature 2-form of T H1C equals i dA (where dA denotes the area form). Therefore the curvature 2-form of the normal bundle of H1C in H2C equals 2i dA. Suppose now that ? is a torsion-free uniform lattice in SU(1; 1), := H1C =?; (that is, ? is a complex Fuchsian group), and M := H2C =? where ? embeds in PU(2; 1) by the monomorphism SU(1; 1) ,! PU(2; 1). Corollary 2.3. The Euler number of the normal bundle of in M equals 12 (). The quotient M = H2C =(?) is a complete complex hyperbolic manifold containing a holomorphic totally geodesic submanifold H1C =(?) . Orthogonal projection onto H1C de nes a bration M : M ?! H1C =? whose bers are totally geodesic complex hyperplanes. The bration M is isomorphic to the normal bundle via the normal exponential mapping. Thus its Euler number equals ()=2. The Kahler form restricts to the area form on the holomorphic submanifold M , hence Toledo's invariant of the embedding ? ,! PU(2; 1) is ?() = 2g ? 2. We will need a relative version of Corollary 2.3. De nition 2.4. A biholomorphic isometry h : H2C ! H2C will be called a transvection if it has an invariant geodesic H2C and h acts trivially on the normal bundle of . (Note that an element is a transvection i it is conjugate to an element of SU(1; 1) with positive trace.) Suppose that W1 H2C is a totally geodesic subspace of the curvature ?1. Let PU(1; 1) = Isom(W1) be a nonabelian discrete nitely generated free purely hyperbolic subgroup, W1 = is an open hyperbolic surface. We make the following assumptions: The surface X = W1= has even number of ends; : = 1 (X ) ! SU(1; 1) PU(2; 1) is an embedding such that the image of any peripheral element is a transvection in H2C . In the next section we prove a technical lemma which implies that the rst condition is necessary and sucient for the existence of satisfying the second condition. Let W1 = be the convex core, M := H2C =() and let and 1 denote the normal bundle and the unit normal bundle to in M . The second hypothesis guarantees a parallel section of the restriction 1j@. Following Steenrod [12], the relative Euler number e(; ) is de ned as the obstruction to extending to a nonzero section of over all of .


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Proposition 2.5. The relative Euler number e(; ) equals Proof. Let p : () ! T () denote the projection from the normal to the tangent 1 (). 2

bundle of given above. It is clear that the relative Euler number is independent of the choice of parallel section (since all parallel sections are homotopic). Thus we may assume that the projection p() is tangent to the boundary of . Then e(T (); p()) is the Euler characteristic of (it follows for instance from taking the double of along its boundary and making a symmetric extension of to and its mirror image in the double). On the other hand, since p is a 2-fold covering we conclude that e(; ) = 21 ().

2.4. Lifting representations from PU(1; 1) to SU(1; 1). Gluing manifolds with geometric structures requires that the structures along the common interface agree. In our particular case, we glue along quotients by cyclic groups of hyperbolic elements. We restrict ourselves to the case of complex-hyperbolic plane H2C . To construct surfaces M as in Theorem 1.2 we will glue quotients of domains in H2C by subgroups ?1 stabilizing a complex geodesic W1 and quotients of domains by subgroups ?2 stabilizing a totally real geodesic 2-plane W2. The interface along which we glue is the quotient of a tube around a (real) geodesic W0 = W1 \ W2 by a cyclic subgroup. This cyclic subgroup belongs to the intersection G0 := G1 \ G2, where Gj is the stabilizer of Wj in PU(2; 1). In suitable coordinates we may take W1 = H1C , W2 = H2R and W0 = H1R. The corresponding stabilizers are G1 = U(1; 1), G2 = SO(2; 1) and G0 = SO(1; 1) respectively. Figure 1 depicts a \hybrid" surface which is the union of two hyperbolic halfplanes. Figure 2 schematically depicts an embedding of the \hybrid" surface into H2C . The upper half has curvature ?1 and represents the Poincare model; the lower half has curvature ?1=4 and represents the Klein-Beltrami model. Geometrically this surface is realized in H2C as a subset of the union of the totally geodesic subspace H1C (the upper half) and H2R (the lower half) which meet in H1R (the equator). Geodesics orthogonal to H1R are drawn in the two halves, as well as the hypercycles parallel to H1R. Although G1 is connected, neither G2 nor G0 is connected. In particular any hyperbolic element represented by a matrix in SO(2; 1) with negative trace does not lie in the identity component. A typical such matrix is: 2 3 1 0 0 40 ? cosh(t) ? sinh(t) 5 0 ? sinh(t) ? cosh(t) The corresponding isometry of H2C is biholomorphic and restricts to an orientationpreserving isometry of H1C (since its restriction to H1C is holomorphic). However, its restriction to H1R reverses orientation: it is a glide-re ection, the composition of transvection along the geodesic H1R with \inversion" in H1C . A hyperbolic element of SU(1; 1) is a transvection if and only if it preserves orientation on an invariant real 2-plane; equivalently it lies in a hyperbolic one-parameter subgroup of SU(1; 1). Proposition 2.6. Let PU(1; 1) be a nitely generated nonabelian purely hyperbolic discrete group of automorphisms of H1C such that H1C = has an even number of ends. Then there exists a lift : ?! SU(1; 1) such that for each g 2 corresponding to a simple loop around an end, (g) is a transvection.

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Figure 1. Hybrid surface.

Proof. If H1C = is a closed surface then the subgroup lifts isomorphically to SU(1; 1) (see Petersson [11] or [4]); otherwise is a free group. The convex core of H1C = is a compact surface whose boundary consists of closed geodesics c1; : : : ; cm (see [1]). Choose lifts of the generators of to SU(1; 1) to obtain a homomorphism ~ : ?! SU(1; 1) lifting the inclusion ,! SU(1; 1). Denote the elements of corresponding to the ends of H1C = by g1 ; : : : ; gm. For each j = 1; : : : ; m write ~(gj ) = fj g^j where fj = I 2 SU(1; 1) and g^j lies in a one-parameter subgroup. Recall that the normal bundle of H1C in H2C has a canonical Hermitian connection induced by the Levi-Civita connection on H2C . This connection projects to the connection r on the normal bundle of in H2C =. Choose a basepoint and a loop l which starts at the basepoint and traverses the boundary components c1 ; : : : ; cm. The holonomy of the connection r around l Q equals the product (of unit complex numbers) j fj of the holonomies around each cj . (Compare Kobayashi-Nomizu [10], xII.) By integrating the connection form around l


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+

W1

W2−

2

HC

Figure 2. Schematic view of the embedding of hybrid surface into H2C .

this equals the integral of the curvature form Z Z i dA = exp(i()): exp r = exp 2 Since () m( mod 2), (2.1)

m Y j =1

fj = (?1)m

Since g1; : : : ; gm?1 lie in a set of free generators for , there exists a representation : ?! SU(1; 1) such that (gj ) = fj ~(gj ) for j < m and (g) = ~(g) for g 2 ? f 1; : : : ; mg. so that (gj ) are transvections all for j < m. Apply (2.1) when m is even to conclude that (gm) is also a transvection. This concludes the proof of the proposition. The referee has suggested the following alternative point of view on the ideas in Proposition 2.6. For a Fuchsian representation into PU(1; 1) whose quotient is a triply-perforated sphere (a \pair of pants"), the three generators corresponding to the boundary lift to elements of SU(1; 1) with negative trace, as can be deduced from the law of cosines for hyperbolic right-angled hexagons (see, for example, Theorem 7.19.2

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of Beardon [1]). By changing the lifting, one can assume that two of the generators have positive trace and the other generator has negative trace. If is a compact hyperbolic surface with totally geodesic boundary, then decomposes into pants. Furthermore the number of pants in the decomposition equals ?() which has the same parity as the number m of components of @ . Since the loops corresponding to m ? 1 components of @ can be included in the set of free generators of H1(; Z), we can lift the generators of the fundamental groups of all but one of the boundary components C1 ; : : : ; Cm?1 to have positive trace. We show that the the sign of the trace of (Cm) will be (?1)m = (?1)() , by induction on ?(). Decompose into pants, and choose one pants P in the decomposition which intersects @ . Apply the induction hypothesis to 0 = ? P , whose Euler characteristic equals 1+ (). Now 0 has either one less or one more boundary component than (depending on how many components of @P meet @ ). Suppose rst that 0 has one more boundary component than . By induction we assume that one component of @ 0 \ @P lifts to an element of positive trace, the other component of @ 0 \ P lifts to an element whose trace is (?1)m+1 and the components of @ ? P all lift to elements of positive trace. It follows from the above paragraph that when P is attached to 0 , the new boundary component of P will lift to an element of trace (?1)m , retaining the lifts of all other boundary components. Similarly, if 0 has one less boundary component, then by induction we suppose that all the components of @ 0 have been lifted to elements of positive trace and that C = @ 0 \ @P has been lifted to an element of sign (?1)m+1 . By the above description of the pants, we can attach P and nd a lift of the representation of 1 () such that one boundary component of @ \ @P has positive trace, the other one has trace of sign (?1)m and all other lifts of components of @ are retained. 2.5. Example: Totally real geodesic subspaces. At the opposite extreme from holomorphic totally geodesic submanifolds are totally real totally geodesic submanifolds. Every such submanifold is equivalent by an automorphism to real hyperbolic space H2R H2C | the subset consisting of points with real coordinates. This subspace is the xed-point-set of the anti-holomorphic isometric involution (z1; z2 ) 7?! (z 1 ; z2 ) The induced Riemannian metric on H2R has constant negative curvature ?1=4. The stabilizer of H2R is the image of the embedding SO(2; 1) ,! PU(2; 1) obtained by composing the embedding SO(2; 1) ,! U(2; 1) with projectivization. The bers of the real orthogonal projection R : H2C ?! H2R are totally real geodesic subspaces which are orthogonal to H2R. (Compare x3.3.6 of Goldman [5].) Since H2R is totally real, the almost complex structure de nes an SO(2; 1)-equivariant isomorphism between its tangent bundle and its normal bundle. Let ? SO(2; 1)0 be a discrete torsion-free subgroup such that the convex core of H2R=? is compact. The composition : 1 () = ? ,! SO(2; 1) U(2; 1) ! PU(2; 1) is a discrete embedding. The quotient M = H2C =(?) is a complete complex hyperbolic surface containing a totally real totally geodesic submanifold H2R=(?) =


The corresponding embedding

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,! H2R=? M induces the homomorphism : 1 () = ? of fundamental groups. From now on identify with its image under this embedding. Suppose has no boundary. The above discussion implies the bration M ! induced by the orthogonal projection H2C ?! H2R is dieomorphic as a ber bundle to the tangent bundle of . Hence this bration has Euler number e(T ) = () = 2 ? 2g We will also need a relative version of this statement in the case @ 6= ;. Let () be the normal bundle of in M , is a unit normal parallel vector- eld along @ in M . Proposition 2.7. The relative Euler number e( (); ) equals the Euler characteristic (). Proof. We repeat the argument from the proof of Proposition 2.3, replacing the 2fold covering by the isomorphism J between the normal and tangent bundle. This isomorphism will map to a parallel section J () along @ . Hence e( (); ) = e(T (); J ()) = () Since is embedded as a totally real submanifold, !j = 0. Thus Toledo's integrand is identically zero so () = 0. 3. Geometric preliminaries

3.1. Orthogonal projections in Hadamard manifolds. Let W be a simplyconnected complete manifold with sectional curvature KW ?1=4. Let W1 and W2 be totally geodesic submanifolds of W with non-empty intersection W0. Then W0 is a totally geodesic submanifold as well. The angle of intersection of W1 and W2 at x 2 W0 is de ned to be: \x(W1; W2 ) := minf\(v1; v2) j v1 2 Tx W1; v2 2 TxW2 ; v1 ? W0; v2 ? W0 g 2 [0; =2] Since the Wj are totally geodesic, parallel translation along W0 preserves tangency to Wj . Therefore the angle \x(W1 ; W2) is independent of the point x 2 W0 and we simply write \(W1 ; W2) for \x(W1 ; W2). Denote by j : W ! Wj the orthogonal projection onto Wj (j = 0; 1; 2). For a subset S X in a metric space X let Nbdr (S ) denote the closed r-neighborhood of S. Proposition 3.1. Let () := 2 cosh?1 (csc(=2)) where := \(W1; W2). Then for all > () we have: (3.1) W = ?1 1 (Nbd(W0 )) [ ?2 1(Nbd(W0)) and the distance between boundary components of ?1 1 (Nbd(W0)) \ ?2 1(Nbd(W0 )) is positive.

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Proof. Pick a point p 2 W . Consider the projections pj = j (p) to Wj (j = 0; 1; 2) and let q = 0(p1 ) 2 W0 . It suces to show that d(p1; q) or d(p2; q) . Denote by uj 2 Tq (Wj ) the unit tangent vector at q pointing towards pj . Then u1 ? W0. Let u02 2 Tq W2 be the component of u2 orthogonal to W0 . Then jhu1; u2ij = jhu1; u02ij ju02j cos() cos(): and we get \(p1qp2) \(p1qp) + \(pqp2). Hence, we assume without loss of generality that q 2 W0 satis es \(p1qp) 2 : (3.2) Let H2R be the hyperbolic plane with sectional curvature ?1=4. Consider a triangle 4(~p1q~p~) in H2R having the same side lengths as 4(p1qp). (Compare Figure 3.) Since W has sectional curvature bounded above by ?1=4, angle comparison with H2R implies (see [9]) \(~p1q~p~) \(p1qp) 2 and \(~qp~1p~) \(qp1p) = 2 : (by (3.2)). Hence d(p1; q) = d(~p1; q~) () where = () is the length of the nite side of the right ideal triangle in H2R with angle =2. For such a triangle (see, for example Beardon [1], Theorem 7.9.1) cosh 2 = csc 2 whence the rst assertion of the proposition follows. In particular for each > () we have: @ ?1 1 (Nbd(W0 )) \ @ ?2 1 (Nbd(W0)) = ; To show that the distance between these hypersurfaces is positive choose such that () < < . Recall that the nearest-point projection in Hadamard spaces does not increase the distance. Hence for j = 1; 2 the set Uj; := ?j 1 (W0 ? Nbd (W0)) contains the ( ? )-neighborhood of the hypersurface @ ?j 1 (Nbd(W0 )) and as we already proved U1; \ U2; = ;. The second assertion of the proposition follows.

3.2. A model for the neck. In this section we discuss the topology of special neighborhoods of closed real geodesics in complex-hyperbolic surfaces; these neighborhoods are called necks. First we describe in nitesimal data associated with each neck. Let h be a transvection in H2C along a real geodesic W0. Consider the quotient M := H2C =hhi: it contains the closed geodesic := W0=hhi, let ` be the length of . The choice of the generator h of the cyclic group hhi corresponds to the choice of an orientation on . Hence, pick a point x 2 and a unit tangent vector v1 2 Tx( ) Tx(M ) pointing in the positive direction. Then choose a unit tangent vector v2 2 Tx(M ) which is Hermitian orthogonal to v1 . The pair (v1 ; v2) can be canonically completed to an orthonormal basis of Tx(M ) over R: let p p w1 := ?1v1 ; w2 := ?1v2 :


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~ p

~ q ~ p1

Figure 3. The comparison triangle

We call (v1 ; v2) a normal frame at x. Note that our assumptions on imply that the normal frame canonically extends to a smooth parallel frame eld along . Such a normal frame and the number > 0 uniquely determine a neck N (; `) in the manifold M around as follows: Any lift w~1 of w1 to T (H2C )jW0 is tangent to the unique complex geodesic W1 in H2C containing W0 . Any lift v~2 of v2 to T (H2C )jW0 is tangent to a totally geodesic real-hyperbolic plane W2 in H2C containing W0 . The vectors w~1, v~2 determine half-planes in W1+ W1; W2+ W2 bounded by W0. (The vectors point outside these half-planes). We denote by Xj the union of the open half-plane Wj+ and the closed -neighborhood of W0 in Wj . Then Sj := Xj ? Wj+ is a strip bounded by W0 and a hypercycle hj equidistant from W0. Denote by Yj the inverse image of Xj under the orthogonal projection j : H2C ! Wj . The next lemma follows directly from Proposition 3.1. Lemma 3.2. Let p p > 0 = (=2) = 2 ln(1 + 2) = ln(3 + 2 2): (3.3) Then Y1 [ Y2 = W and moreover the distance between the boundary components of Y1 \ Y2 is positive. Let 1 := ln(6) > 0 and suppose in the following that > 1. The 1-parameter group O(1; 1)0 of transvections along the geodesic W0 acts on the intersection Y () = Y = Y1 \ Y2. We call the intersection Y a tube. We describe the topology of the tube Y in an O(1; 1)0-invariant way. The group O(1; 1)0 acts freely on the smooth manifold with boundary H2C [ @1H2C ? @1W0 which can be thought of as the total space of an R -principal bundle E whose base is the closed 3-ball B 3 . Let Y be the closure of Y in H2C [ @1H2C ? @1 W0. It is invariant under the O(1; 1)0-action. The frontier @Y of Y in H2C [ @1H2C ? @1W0 projects

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to a union of two disjoint smoothly properly embedded 2-discs in B 3 . Hence, by the smooth Schon ies Theorem in dimension three, the projection of Y is dieomorphic to D2 [1; 2]. Since E is a trivial bundle it has a smooth trivialization. It can be chosen to extend the canonical trivializations of @Y whose horizontal sections are totally geodesic 2-planes. This allows us to extend the bration of @Y by totally geodesic 2-planes in an O(1; 1)0-invariant manner to a smooth bration of Y by closed 2-discs. As base for this bundle we can use the strip S1 [ S2. The above discussion shows: Proposition 3.3. The quotient Y =hhi is dieomorphic to the total space of a trivial disk bundle over the annulus A. The restriction of this bration to the two components of @Y=h i corresponds to the restriction of the nearest-point projections j to ?1 (hj ), j = 1; 2. We call the quotient N (; `) = N := Y=hhi a model neck. Its isometry type depends on two parameters: (the width) and ` (the length). 3.3. Collars of geodesics. In this section we determine ` = `(D) > 0 such that a closed geodesic of length ` in a hyperbolic surface of curvature ?k?2 is contained in an regular tubular neighborhood (a \collar") of width at least D and distinct neighborhoods are disjoint. This will enable us to nd a neck along which to glue the two complex hyperbolic surfaces with holomorphic and totally real spines respectively. For simplicity we consider only surfaces where all boundary geodesics have the same length. Let Wk be a complete simply connected Riemannian 2-manifold of constant curvature ?k?2 and ?k a purely hyperbolic discrete group of isometries. Suppose that f; g are nontrivial elements of ?k which are transvections along geodesics Af ; Ag in Wk . The geodesics Af ; Ag are called the axes of f; g. We assume that: The geodesics Af ; Ag are disjoint. The geodesics Af =hf i; Ag =hgi have the same length `.

Lemma 3.4.

sinh ` sinh d(Af ; Ag ) 1=2 2k 2k Proof. This assertion quanti es the Margulis Lemma for the hyperbolic plane and slightly modi es Corollary 11.6.10 of Beardon [1]. Beardon considers the case of curvature ?1 but the formulas p remain valid for constant curvature K < 0 if the lengths are multiplied by ?K . Let S be a complete hyperbolic surface and a simple closed geodesic in S . Let S ! S denote the locally isometric covering of S corresponding to the subgroup of 1 (S ) generated by . Let ^ denote the simple closed geodesic in S covering . We say that the -neighborhood Nbd( ) of in S is injective if the projection Nbd(^ ) ! Nbd( ) is a dieomorphism. Equivalently, is contained in for every geodesic arc in S with both end-points on and `() 2. Suppose that ?k is a nitely-generated group as in Lemma 3.4, k is the compact convex core of the surface Sk = Wk =?k . We assume that all boundary geodesics of k have the same length `. Lemma 3.4 implies:


Lemma 3.5. Let > 0 and suppose that ` satis es sinh 2`k sinh k 12 ; (3.4)

15

Then the {neighborhood of each boundary geodesic of k is injective and {neighborhoods of distinct geodesics are disjoint. Remark 3.6. In the case of equality in (3.4) ` k sinh sinh = 1; 2k k 2 then `1 < `2 . If = 1 = ln(6) then sinh(`1 =2) = 6=35. 3.4. Neck neighborhoods. Now we combine the results of the previous three sections to construct neck neighborhoods around certain geodesics in complex-hyperbolic surfaces. De nition 3.7. Let M be a complex-hyperbolic surface and M be a simple closed geodesic of length `. We say that admits an (; l)-neck neighborhood Neck( ) of width in M if it has a neighborhood in M which is isometric to the model neck N (; `). Suppose that Wk W = H2C is a totally geodesic plane of the curvature ?k?2 and ?k is a nitely-generated purely loxodromic discrete subgroup of PU(2; 1) stabilizing Wk . We assume that Sk = Wk =?k is not compact and k Sk is the compact convex core so that the length of each boundary geodesic equals the same number `. We also assome that each element of ?k representing a boundary geodesic of k is a transvection. To construct the in nitesimal data for the neck around each boundary geodesic of k choose a normal frame at H2C =?k (see subsection 3.2): Pick a base point x at . The vector v1ptangent to at x is determined by the choice of orientation on , we let w1 := ?1v1. If k = 1 then the vector w1 is tangent to 1 and is directed inwards 1 . If k = 2 then the second vector v2 is chosen tangent to 2 and directed inwards 2 . Lemma 3.8. Suppose that ` < `1, where `1 is given by sinh(`1=2) = 6=35 Then for = 1 = ln(6) we have: Each boundary geodesic k H2C =?k is contained in a neck neighborhood Neck( ) with the normal frame as above. If ; @ k are distinct boundary geodesics then Neck( ) \ Neck( ) = ;. p Proof. The condition = 1 > 0 = 2 ln(1 + 2) implies the existence of a model neck of the width (Lemma 3.2). Let be a boundary geodesic of k and let M ! M = H2C =?k ; k; ! k denote the coverings de ned by the subgroups 1 ( ) 1 (M ); 1( ) 1 (k ):

16


De ne ^ as the simple closed geodesic on k; M covering . Lift of the normal frame along from M to a normal frame along ^ in M determines an neck neighborhood Neck(^ ) M . We rst show that the restriction of the projection M ! M to Neck(^ ) is injective. Let P~ : M ! ~ k denote the nearest-point projection to the lift ~ k of k into M . Then Neck(^ ) is contained in the inverse image P~ ?1(Nbd(^ )) where the -neighborhood is taken within the surface ~ k . Recall that by Lemma 3.5 and Remark 3.6 the geodesic in k has an injective -neighborhood. Thus the restriction of the covering k; ! k to Nbd(^ ) is injective and the restriction of the covering M ! M to P~ ?1(Nbd(^ )) is injective as well. This implies that we have a neck neighborhood Neck( ) of the geodesic in M . Finally we show that neck neighborhoods of distinct geodesics are disjoint. Let P : M = H2C =?k ! k denote the orthogonal projection. Then again Neck( ) P ?1(Nbd( )) for every boundary geodesic k . Let ; @ k are distinct boundary components. Then their -neighborhoods Nbd( ); Nbd( ) in k are disjoint by Lemma 3.5. Hence P ?1(Nbd( )) \ P ?1(Nbd( )) = ; and Neck( ) \ Neck( ) = ; as desired. 4. Proof of the Main Theorem We begin with an oriented closed surface with a piecewise hyperbolic structure of the following kind. There is a collection B of disjoint simple closed geodesics on such that on each component of ? B the metric either has constant curvature ?1 or constant curvature ?1=4. Let k (k = 1; 2) denote the subsurface of ? B where the curvature equals ?k?2 . The surfaces k could be disconnected. We assume that each loop 2 B is adjacent to both 1 ; 2 . Thus we orient B so that the surface 1 lies to the left from B . Our construction involves two assumptions: The boundary of each component of 1 has an even number of components. All components of B have the same length ` < `1 = sinh?1(6=35). Theorem 4.1. There exists a complete complex hyperbolic surface M together with a piecewise totally geodesic isometric embedding f : ! M such that f is a homotopy equivalence. The Toledo invariant of M equals ?(1 ). M is dieomorphic to the total space of an oriented disc bundle which has the Euler number (1 )=2 + (2 ). Outline of proof. We embed 2 (respectively 1 ) as a totally real (respectively holomorphic) totally geodesic submanifold inside a complete complex hyperbolic surface X2 (respectively X1 ) so that: 1. The embeddings fk : k ! Xk , k = 1; 2 are homotopy equivalences. 2. The homotopy-inverse to fk is the nearest-point projection ^ k : Xk ! fk (k ). 3. The numbers ; ` are chosen so that -neighborhoods of distinct boundary components of k are injective and disjoint. 4. Each component fk ( ki) of fk (@ k ) is contained in a neck Nki := Neck( ki) and necks around distinct geodesics are disjoint. 5. If two annular components A1i Nbd(@ 1 ); A2i Nbd(@ 2 )


17

are adjacent in the surface then there is a canonical isometry i between the corresponding necks N1i ; N2i. To guarantee the properties 3-5, we need \enough room" around the boundary geodesics of fk (@ k ) in Xk . This is achieved by the choice of ` and as above. Now the construction of M proceeds as follows. Remove from the manifolds Xk (k = 1; 2) all points in ^ ?k 1(@ k ) that do not belong to the union of necks Nki. The resulting complex hyperbolic manifolds with boundaries M1 and M2 are glued together along their necks via the isometries i. Then we prove that the resulting complex hyperbolic surface M satis es the assertions of Theorem 4.1. The following two sections contain the details of the proof. 4.1. Construction of the complex hyperbolic surface. We consider a surface as in the previous section. For each component kj of the surface k we have the action of ?kj = 1(kj ) on its Nielsen region ~ kj in the hyperbolic plane H2(?k?2 ) with curvature ?k?2 . Recall that ~ kj is the convex hull of the limit set of ?kj = 1 (kj ) and is a convex domain with totally geodesic boundary. Furthermore kj = ~ kj =?kj is a compact geodesically convex hyperbolic surface with geodesic boundary | the convex core of a the complete hyperbolic manifold kj = Wk =?kj . According to Proposition 2.6 for each k; j there exists an equivariant totally geodesic embedding of the hyperbolic plane H2(?k?2 ) in H2C which satis es the following properties: 1. For k = 1 the image is the complex hyperbolic line H1C = W1, for k = 2 the image is a totally real hyperbolic plane H2R = W2. 2. Each component of B is represented by a transvection h in H2C . 3. The canonical orientation on W1 agrees with the orientation on 1 . Thus representatives h of all the components of B are conjugate in SU(2; 1). We let Xk be the disjoint union of quotients of H2C =?kj . It is clear that we have totally geodesic embeddings fk : k ,! Xk which are homotopy equivalences. The property 2 implies that for each boundary geodesic of k we can choose a normal frame at fk ( ) Xk , see section 3.4: These normal frames determine necks Neck( ) Xk around the geodesics fk ( ) so that necks around distinct geodesics are disjoint (see Lemma 3.8). We de ne submanifolds Mk Xk as in the previous section, and denote by Mkj the component of Mk containing kj . Each neck has a canonical bration over an annulus, this R 2 - bration extends to the bration of [ Mk ? Neck( ) @ k

given by the nearest-point projection to fk (k ), see section 3.2. Suppose now that k @ k (k = 1; 2) are boundary geodesics identi ed in the surface . Then there exists a unique biholomorphic isometry : Neck( 1 ) ! Neck( 2 ) that preserves the normal frames and carries base points to base points. We let M be the complex hyperbolic surface obtained via performing these gluings. Clearly the mappings fk can be combined to de ne a piecewise totally geodesic embedding (4.1) f : ,! M which we call a spine. Similarly R 2 - brations of the components Mk de ne a smooth ^ : M ! . This bration is a homotopy inverse to f . Moreover, R 2 - bration ^ f = id. In the next section we shall compute the Euler number of this bration.

18


Lemma 4.2. The complex hyperbolic structure on M is geodesically complete. Proof. Assume that M is not complete and : [0; T ) ! M is a nonextendable nite

geodesic ray. Each component Mkj of Mk is metrically complete by the construction and there is > 0 such that boundary components of each neck of M are -separated according to Proposition 3.1. Therefore there are in nitely many disjoint intervals [t?n ; t+n ] on [0; T ) such that the segment ([t?n ; t+n ]) connects dierent boundary components of necks in M . Each segment has length at least . This contradicts the niteness of T . As a consequence of the lemma, the corresponding holonomy representations ! PU(2; 1) are discrete and faithful. The pleated surface f~ : ~ ! H2C has bending angles equal to =2 and the bending locus consists of 2-separated geodesics. According to (3.4), we can make arbitrarily large if we choose l suciently small. Since H2C has pinched negative curvature, it follows that for suciently large , the image of f~ is quasi-convex and Hausdor close to its convex hull. Thus, the action of G = 1(M ) on H2C by deck transformations is convex cocompact. The action of G on H2C preserves a nonempty closed convex subset on which G acts cocompactly. Therefore the manifold M = (H2C [ (G))=G is compact. This compacti cation of M is consistent with the open disk bundle structure of M and therefore M is dieomorphic to the closed disk bundle over having the same Euler number as the bundle M ! . Corollary 4.3. The manifold (G)=G is dieomorphic to the total space of an S1{ bundle over the surface which has the same Euler number as the R 2 - bration of M. 4.2. Calculation of Euler number and Toledo invariant. The contribution of the totally real components of f to the integral Z 1 = 2 f !M is zero. Thus the integral equals ?(1 ) (see Section 2.3). This nishes the proof of the rst and second assertions of Theorem 4.1. We construct a continuous vector eld V along the piecewise totally geodesically embedded spine surface (4.1) as follows: Along the folding loops f (B ) we choose V to be a parallel unit vector eld orthogonal to both f (k ). This is possible because the holonomy along all loops f (B ) is trivial. Next we extend V to sections of the normal bundles of the surfaces f (k ) which are smooth on f (k ) and have isolated zeros. By the computation in x2.3 and x2.5, the sum of the multiplicities for the zeros of V equals (1 )=2 on the complex portion f (1) and (2 ) on the totally real part f (2 ). We use the vector eld V to homotope f () o itself to a nearby surface g() which intersects f () transversally in nitely many points. The multiplicities of the intersection points sum to the Euler number e(M ! ) of the oriented D2- bration M ?! , which equals (1 )=2 + (2 ) as desired. This concludes the proof of Theorem 4.1.


19

Proof of Theorem 1.1. If = 2g ? 2; 0; 2 ? 2g then discrete embeddings of into SU(1; 1) or SO(2; 1) give the desired representations. Thus let be an even integer satisfying 0 < < 2g ? 2: Then t = g ? 1 ? 2 satis es 0 < t < g ? 1: Let denote a closed orientable surface of genus g. Then there exist two disjoint simple closed curves decomposing as = 1 [ 2 where 1 is a surface of genus g ? t ? 1 with two boundary components and 2 is a surface of genus t with two boundary components. Thus the complex hyperbolic surface M in Theorem 4.1 achieves Toledo invariant . The negative values of can be obtained by conjugating these representations by an anti-holomorphic isometry of H2C . Finally, e(M ! ) ? () = j j=2 follows from direct computation. Using degree one maps between hyperbolic surfaces and complex Fuchsian representations one can easily construct nonfaithful discrete representations of 1 (g ) realizing all even integer values of Toledo's invariant. These representations will have images in SU(1; 1).

5. Examples with nonintegral Toledo invariant In this section we construct representations of the fundamental group = 1 () that realize all possible fractional values of the invariant (). We start with the case of surfaces of genus 2. Proposition 5.1. Suppose that is a closed oriented surface of genus 2. Then there are representations j : = 1 () ! PU(2; 1) so that (j ) = 2j=3 (j = 1; 2). Proof. Consider a family of regular quadrilaterals Q ; 0 < < =2 in H1C H2C such that all angles of Q equal . Let A ; B be transvections in PU(2; 1) pairing the opposite sides of Q . The commutator C = [A ; B ] is an elliptic element xing a corner of Q . It rotates by the angle 4 inside H1C and by the angle + 2 in the normal direction. Indeed, according to Corollary 2.2 the speed of rotation in the normal direction is half of the speed of rotation in H1C , thus it is either + 2 or 2. Since lim C = id; !=2 the rotation angle equals + 2. We now restrict to the special value = =6, then the angles of rotation in H1C and in the normal direction are 2=3 and 2 + = 4=3;

20


respectively. The elliptic element C has order 3. We similarly de ne hyperbolic elements A0 ; B0 2 SO(2; 1) whose commutator C0 has order 3. The eigenvalues of the derivative of C0 at its xed point are e2=3i and e?2=3i . We consider the corresponding (discrete and non-faithful) representation : F2 = h; i ! SU(1; 1) PU(2; 1) of the free group on two generators given by () = A and ( ) = B . The normalizer of the order 3 element C in PU(2; 1) contains an involution with which it anticommutes: C ?1 = C?1. The involution interchanges the eigenspaces of the derivative of C at its xed point, i.e. the direction tangent to H1C and the normal direction. Similarly we have the representation (5.1) 0 : F2 = h; i ! SO(2; 1) PU(2; 1) given by 0() = A0 and 0( ) = B0 , which preserves a totally real geodesic plane. We use the representation as a building block to construct a representation 2 : = 1() = h1; 1; 2; 2 j [1; 1][2 ; 2] = 1i ! PU(2; 1) by sending 1 7?! (); 1 7?! ( ); 2 7?! ()?1 2 7?! ( )?1: Now we compute Toledo's invariant for the representation 2 . Identifying the sides of Q via the transformations A ; B gives a hyperbolic cone structure on the 2torus T with a single cone point k with total angle 4 = 2=3, see [13]. (This cone structure is actually an orbifold structure.) The punctured surface S = T ? fkg has an incomplete hyperbolic structure whose developing map d : S~ ! H1C maps a fundamental domain in S~ to Q . The homomorphism is the holonomy of this hyperbolic cone structure. We decompose along a simple loop c into two punctured tori j so that 1(j ) is identi ed with the subgroup in 1 () generated by j ; j . We choose orientation preserving dieomorphisms hj : j ! S which map a neighborhood of c to a neighborhood of k. These dieomorphisms give rise to equivariant developing maps f1 = d h~ 1 : ~ 1 ! W1 ; f2 = d h~ 2 : ~ 2 ! (W1 ): The complex lines W1 and (W1) are orthogonal and intersect in the xed point of the elliptic isometry 2 ([1; 1 ]) = 2 ([2; 2])?1 : The equivariant maps f1 ; f2 extend uniquely to a 2-equivariant map f~ : ~ ! H2C . There is a fundamental domain for the action of on ~ which maps onto the union Q [ (Q ) preserving orientation. Thus Z ? 1 (2 ) = 2 f~! = 1 area(S ) = 1 2 ? 4 = 34 :

Now we nd a representation 1 with (1 ) = 23 by a similar construction. We leave 1 j1(1 ) = 2j1(1 )


21

as above and replace 2 j1(2 ) by setting 1 on 1(2 ) by 1 : 2 7! g0()g?1 (5.2) 1 : 2 7! g0( )g?1: where 0 is de ned as in (5.1) and g is de ned as follows. Since ([; ]) and 0([; ])?1 have the same eigenvalues, there exists an isometry g of H2C which conjugates 0([; ]) to ([; ])?1. By (5.2), 1 ([1; 1]) = 1([2 ; 2])?1; so 1 is a representation 1 : ! PU(2; 1). Now (1 ) = 23 ; because the totally real part 1 j1(2 ) does not contribute to the Toledo invariant. This completes the proof of the proposition. Note that no representation : ! PU(2; 1) with non-integer Toledo invariant preserves a complex geodesic. The stabilizer of a complex geodesic in PU(2; 1) is conjugate to SU(1; 1) and therefore lifts canonically to SU(2; 1). Proposition 5.2. The image of 2 is a discrete subgroup of PU(2; 1). Proof. We shall embed the image of 2 in an arithmetic lattice ? PU(2; 1). Consider the Hermitian form p H (z) = z1z1 + z2 z2 ? 3z3 z3 on C 3 and let G be the group of all linear transformations C 3 ?! C 3 unitary with respect to H . The group G is the group of R -points in an R -algebraic group. Let GO be the subgroup of G with entries from the subring p O = Z[ 3; i] C p where i = ?1. Let be the embedding O ,! C de ned by Galois conjugation p p 3 7?! ? 3 i 7?! i: Then transforms GO into the unitary group GO of the de nite Hermitian form p H (z) = z1 z1 + z2 z2 + 3z3 z3 : Let G denote the unitary group fo H with entries from C . The graph (I; ) of embeds O as a lattice in C C and realizes GO as an arithmetic lattice in the product G G . Since G is compact, the projection G G ?! G is proper, hence GO is a lattice in G. To realize 2 in G, we rst note that the involution is contained in GO. Next we will show that a (2; 4; 12)-triangle group ?2;4;12 could be embedded in PU(1; 1) so that the image is commensurable with GO \ PU(1; 1). (Figure 4 depicts the tessellation of H1C corresponding to the re ection group which contains ?2;4;12 as a subgroup of index 2). Finally, the group (1 (1 )) is contained in ?2;4;12 as a nite index subgroup. Here is the detailed construction. Consider a triangle H1C with a vertex p at the origin, with angle =12, another vertex q with angle =4 and a third vertex r

22


!, qr!, rp !, are rotations with right angle. The products of re ections in the lines pq P; Q; R in the points p; q; r respectively. These isometries of H1C satisfy P 12 = Q4 = R2 = PQR = I and any triple (P; Q; R) satisfying these relations, where P (respectively Q, R) represents elliptic rotation through angle =6 (respectively =2, ) corresponds to a triangle . Explicitly, such a group is represented by the matrices exp( ? i= 12) 0 P= 0 exp(i=12) and p p p 1 3) i 3(1 + 1 ? i (2 + Q = p ?ip3(1 + p3) 1 + i(2 + p3)3) 2 and p

2 6

R = 64 where

1 2

?i(1 + 3) p 1 ? i(2 + 3)

p3 2

p

1 + i(2 + 3)

p

i(1 + 3):

p

3

7 7: 5

! = e2i=3 = (?1 + i 3)=2 exp(3i=4) = 1p? i 2 p p 3) + i ( ? 1 + 3) : (1 + p exp(i=12) = ? exp(3i=4)! = 2 2 Since the projection from 2 2 matrices to PU(1; 1) is not faithful, an isometry of even order 2m lifts to a matrix whose 2m-th power is ?I (not I ). One can easily check that the above matrices satisfy relations P 12 = Q4 = R2 = PQR = ?I: Thus ?2;4;12 GO. We now realize (1(1 )) in ?2;4;12. We nd the generators ~ = (); ~ = ( ) for a nite index subgroup of ?2;4;12 as follows. Begin with the free product Z=4 ? Z=2 = hX i hY i = hX; Y j X 2 = Y 4 = I i and consider the homomorphism : Z=2 ? Z=4 ?! Z=4 X 7?! 2 mod 4 Y 7?! 1 mod 4: Then Y XY and XY 2 generate the kernel of . To see this, replace hX; Y j X 2 = Y 4 = I i by the equivalent presentation hY; A; B j Y 4 = I; Y AY ?1 = B ?1 ; Y BY ?1 = Ai;


BY 2

23

where X corresponds to and A (respectively B ) corresponds to Y XY (respec2 tively XY ). In the new presentation maps Y to a generator of Z=4 and annihilates A; B , so A; B generate ker(). Their commutator is easily computed to be: (5.3) [Y XY; XY 2] = (Y X )4: Now apply this to the homomorphism Z=2 ? Z=4 ?! ?2;4;12 which maps X to Q, Y to R and XY to P . The homomorphism projects to an epimorphism ~ : ?2;4;12 ?! Z=4 and we obtain elements ~, ~ 2 ker(~) as ~ = QRQ; ~ = RQ2 : Since (?2;4;12) = ?1=4, the index four subgroup ker(~) satis es (ker(~)) = ?1 and thus is free of rank two. It follows that ~; ~ generate ker(~)). Now de ne the homomorphism on the free group F2 = h; i p p (1 + 3) ! ? 3( ! + i ) () = ~ = QRQ = ?! + i (1 + p3)! : and p p 3) ! ? 3( ! ? i ) (1 + 2 ~ p ( ) = = RQ = ?! ? i (1 + 3)!

Then the commutator

([; ]) = P ?4 =

! 0 0 !

is the desired elliptic element of order 3. Applying the construction of the proof of Proposition 5.1, we see that 2 GO and thus the image of 2 lies in the arithmetic lattice GO and is therefore discrete.

Figure 4. The (2,4,12) Schwarz triangle group

24


Proof of Theorem 1.1. We modify the construction in the proof of Proposition 5.1 by amalgamating representations coming from holonomy representations of hyperbolic cone structures on surfaces of higher genus. Let be a closed surface of genus g 2 which is split along a separating simple loop c into two subsurfaces 1 and 2 of genera g1 = g ? 1 and g2 = 1, respectively. We pick singular hyperbolic surfaces Tj (j = 1; 2) of genus gj with one cone point kj of angle ; such surfaces can be constructed by suitably identifying the sides of a regular hyperbolic 4gj -gon with angles =4gj . Since 1 (Tj ? fkj g) is a free group, the holonomy representation holj : 1 (Tj ? fkj g) ! PU(1; 1) for the singular hyperbolic structure on Tj lifts to a representation f j : 1 (Tj ? fkj g) ! SU(1; 1) PU(2; 1) hol Let W be the complex geodesic preserved by SU(1; 1). An element representing a small simple loop around kj is carried by holj to a rotation in W by angle around f j ( ) acts as a rotation by angle + =2 in a point o 2 W . The lifted holonomy hol the normal direction to W at the point o. For the genus one case this was already discussed in the proof of Proposition 5.1. The angle of rotation in the normal direction is either =2 or + =2. To exclude the rst possibility, we note that the hyperbolic cone structure of Tj can be continuously deformed so that increases to 2. The limit is a nonsingular hyperbolic structure whose holonomy can be lifted to SU(1; 1), see [11, 4]. Hence the assertion follows by continuity. Now we restrict to the special value 2=3 and proceed as in the proof of Propof 1 and hol f 2 we construct a representation with Toledo sition 5.1. By combining hol f 1 with the appropriate conjugate of the invariant ?() ? 2=3. By combining hol 0 representation we get a representation with Toledo invariant ?() ? 4=3. Thus we can realize the extremal positive non-integer values ?() ? 2=3; ?() ? 4=3 for Toledo's invariant. To realize the values 2k ? 2=3 and 2k ? 4=3 for k = 1; : : : ; g ? 2 choose a degree-one map of onto a surface 0 of genus k + 1. Take representations 0 : 0 ! PU(2; 1) with Toledo's invariant 2k ? 2=3 and 2k ? 4=3 as constructed above and compose them with the induced map 1 () ! 1 (0 ) of fundamental groups. This yields all positive fractional values for Toledo's invariant. The negative values are obtained by conjugating the above representations with an antiholomorphic isometry of H2C . References 1. Beardon, A. F. , \The Geometry of Discrete Groups," Graduate Texts in Mathematics 91, Springer-Verlag, New York | Heidelberg | Berlin (1983). 2. Burns, D. and Shnider, S. , Spherical hypersurfaces in complex manifolds, Inv. Math., vol. 33 (1976), 223-246 3. Dupont, J. , Simplicial de Rham cohomology and characteristic classes of at bundles, Topology, vol. 15 (1976), 233{245 4. Goldman, W. M. , Topological components of spaces of representations, Inv. Math., vol. 93 (1988), 557{607 , \Complex Hyperbolic Geometry", Oxford Mathematical Monographs, Oxford Univer5. sity Press (1999). 6. and Kapovich, M., Complex hyperbolic surfaces homotopy equivalent to a Riemann surface, MSRI Preprint, 1992.

7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

COMPLEX HYPERBOLIC MANIFOLDS 25 Griths, P. A. and Harris, J., \Principles of Algebraic Geometry," John Wiley & Sons, New York (1978) Kapovich, M. and Leeb, B., (in preparation) Karcher, H., Riemannian comparison constructions, in: S.S. Chern ed., \Global Dierential Geometry", MAA studies in Mathematics, vol. 27 (1989), 170{ 222. Kobayashi, S. and Nomizu, K., \Foundations of Dierential Geometry I," Interscience Tracts in Pure and Applied Mathematics, 15, John Wiley & Sons, New York (1969). Petersson, H., Zur analytischen Theorie der Grenzkreisgruppen III, Math. Ann. 115 (1938), 518{572. Steenrod, N., \The Topology of Fibre Bundles", Princeton University Press, 1951. Thurston, W., Shapes of polyhedra and triangulations of the sphere, 511{549, Geometry and Topology Monographs I: The Epstein Birthday Schrift (I. Rivin, C. Rourke, C. Series eds.), (1998) International Press. Toledo, D. , Harmonic maps from surfaces to certain Kahler manifolds, Math. Scand., vol. 45 (1979), 13{ 26. , Representations of surface groups on complex hyperbolic space, J. Di. Geom., vol. 29 (1989), 125{133. Xia, E., The moduli of at U (p; q) structures over Riemann surfaces, (submitted).

Department of Mathematics, University of Maryland, College Park, MD 20742 USA

E-mail address :

[email protected]

(Goldman)

Department of Mathematics, University of Utah, Salt Lake City, Utah 84112 USA

E-mail address :

[email protected]

(Kapovich)

Math. Inst., Eberhard-Karls-Universitat Tubingen, D-72076 Tubingen, Germany

E-mail address :

[email protected]

(Leeb)