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 τ. For integer values of τ these actions are discrete embeddings. The quotient complex hyperbolic surfaces are disc bundles over closed Riemann surfaces, whose topological type is describe in terms of τ. We relate our geometric construction to arithmetic constructions and discuss integrality properties of τ.

1. Introduction

Let Σ denote a closed oriented surface of genus g ≥ 2 and π = π_{1}(Σ) its fundamental group. Let G = PU(n, 1) denote the group of biholomorphic isometries of complex hyperbolic n-space H_{ℂ}^{n}. In [15], Domingo Toledo considered the following invariant associated to a representation ρ : π → PU(n, 1): Let f̃: Σ̃ → H_{ℂ}^{n} be a ρ-equivariant smooth mapping of the universal covering of Σ. Then Toledo’s invariant τ(ρ; f̃) is defined as the (normalized) integral of the pull-back of the Kähler form ω on H_{ℂ}^{n}:

τ(ρ; f̃) := \frac{1}{2π} ∫_{Ω} f̃^{∗}ω

where Ω is a fundamental domain for the action of π on Σ̃. τ(ρ; f̃) is independent of f̃ and depends continuously on the representation ρ. Toledo proved that

2 − 2g ≤ τ ≤ 2g − 2 (1.1)

He showed furthermore that |τ| = 2g −2 holds if and only if ρ is an isomorphism onto a lattice in the stabilizer of a complex geodesic in H_{ℂ}^{n}. By relating τ(ρ) to the first Chern class of a certain line bundle we show that it satisfies the integrality property:

τ ∈ \frac{2}{n + 1}ℤ (1.2)

The function

τ : Hom(π, PU(n, 1)) ⟶ \frac{2}{n + 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: February 21, 1999.

1991 Mathematics Subject Classification. 53C55, 22E40.

Key words and phrases. Complex hyperbolic space, flat 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).

2 GOLDMAN, KAPOVICH, AND LEEB

distinguishes different connected components. The main results of this paper are theorems 1.1, 1.2 about realization of values of Toledo’s invariant:

Theorem 1.1. For every genus g ≥ 2 and every τ satisfying (1.1) and (1.2) there exists a representation ρ : π → PU(2, 1) with τ(ρ) = τ.

When τ is not an integer, then ρ is not faithful in our examples (but ρ(π) is discrete when τ = ±4/3). However, when τ is an even integer, we find discrete embeddings:

Theorem 1.2. For every genus g ≥ 2 and every even integer τ satisfying (1.1) there exists a convex-cocompact discrete and faithful representation ρ : π → PU(2, 1) with τ(ρ) = τ. Furthermore, the complex hyperbolic surface M = H_{ℂ}^{2}/ρ(π) is diffeomorphic to the total space of an oriented ℝ^{2}-bundle ξ over Σ with the Euler number

e(ξ) = χ(Σ) + |τ(ρ)/2| (1.3)

Generally, for a discrete embedding π → PU(2, 1) the equality (1.3) will fail [8]. Simple examples of complex hyperbolic surfaces arise from representations pre-

serving totally geodesic planes. There are two kinds of totally geodesic submanifolds W ⊂ H_{ℂ}^{2} of real dimension two:

• Complex geodesics (copies of H_{ℂ}^{1}) have constant sectional curvature −1. Their stabilizers in PU(2, 1) are conjugate to U(1, 1).

• Totally real geodesic 2-planes (copies of H_{ℝ}^{2}) 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 π, there exist discrete embeddings ρ : π ⟶ G whose image is a cocompact lattice in G. When W is a complex line, then |τ(ρ)| = 2g − 2 is maximal; we say that ρ is complex Fuchsian. When W is a totally real plane, then τ(ρ) = 0; then we say that ρ is real Fuchsian. These two special cases are building blocks in our constructions.

Quotients of H_{ℂ}^{2} by real and complex Fuchsian groups provide two quite different kinds of complex hyperbolic surfaces. For any manifold M_{1} which is the quotient space of the complex Fuchsian group, contains Σ as a totally geodesic complex curve. In contrast, the quotients M_{2} of the real Fuchsian groups are Stein manifolds (see Burns and Shnider [2] or Goldman [5]). Moreover, the surfaces M_{1} and M_{2} are not even homeomorphic. They are both diffeomorphic to total spaces of oriented 2-plane bundles over Σ. The Euler number of the bundle equals χ(Σ)/2 for M_{1} and χ(Σ) for M_{2}. These two examples are discussed in detail in §§2.3–2.5.

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 version of the preprint [6] of the first two authors. Acknowledgements. We thank Domingo Toledo for helpful discussions. Conversations with Igor Belegradek, Jeff 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 3

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

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 := H_{ℂ}^{n} and π is the fundamental group of a closed Riemann surface Σ of genus g ≥ 2. In this paper we will use the unit ball model for H_{ℂ}^{n}, whereby

H_{ℂ}^{n} ⊂ ℂ^{n} ⊂ ℙ_{ℂ}^{n}.

For any ρ ∈ Hom(π, G) we have the associated flat H_{ℂ}^{n}-bundle X_{ρ} over Σ with holonomy ρ. Its total space is the quotient of Σ̃ × X by the action of π = π_{1}(Σ) given by

γ : (m̃, x) ↦ (γm̃, ρ(γ)x)

The Kähler form on X = H_{ℂ}^{n} defines a closed 2-form ω_{ρ} on X_{ρ}. (In the above description, ω_{ρ} satisfies Π^{∗}ω_{ρ} = Π_{X}^{∗}ω where

Π_{X} : Σ̃ × X ⟶ X

and

Π : Σ̃ × X ⟶ X_{ρ}

are the natural projections.) For any section s : Σ → X_{ρ},

τ(ρ; s) = \frac{1}{2π} ∫_{Σ} s^{∗}ω_{ρ}

is independent of s and is thus an invariant of ρ.

Conjugating by an antiholomorphic isometry of H_{ℂ}^{n} yields representations with opposite characteristic number. (An example of an antiholomorphic isometry is the complex conjugation z ↦ \overline{z}.) Suppose ρ ∈ Hom(π, G), and ι ∈ Isom(H_{ℂ}^{n}) is an antiholomorphic isometry. Then ι^{∗}ω = −ω and it follows that τ(ρ^{ι}) = −τ(ρ).

Recall that ρ is called a discrete embedding if its image is a discrete subgroup of G and ρ is injective. In case ρ is a discrete embedding, the quotient M = H_{ℂ}^{n}/ρ(π) is a complex hyperbolic manifold. A section s of the flat bundle X_{ρ} corresponds to a homotopy equivalence f : Σ ⟶ M as follows. Lifting everything to the universal covering Σ̃, the section s of X_{ρ} defines a section s̃: Σ̃ ⟶ Σ̃ × X which is the graph of a ρ-equivariant map f̃ : Σ̃ ⟶ X. Equivariance implies that f̃ covers a map f : Σ ⟶ M which induces the isomorphism

π_{1}(Σ) ⟶ ρ(π_{1}(Σ)) ≅ π_{1}(M).

Since both Σ and M are aspherical, f is a homotopy equivalence. Thus

τ(ρ) = \frac{1}{2π} ∫_{Σ} s^{∗}ω_{ρ} = \frac{1}{2π} ∫_{Σ} f^{∗}ω_{M}

2.2. Integrality. Let L → H_{ℂ}^{n} be a Hermitian complex line bundle, and let G be a Lie group acting on L by isomorphisms. Let Ω denote the curvature form of L; it is a G-invariant 2-form on H_{ℂ}^{n}.

Given a representation ρ : π → G and an equivariant smooth map f : Σ̃ → H_{ℂ}^{n}, the group π acts on the pull-back bundle f^{∗}L → Σ̃, and f^{∗}L descends to a Hermitian line bundle L_{ρ} → Σ. The curvature form of f^{∗}L is given by f^{∗}Ω and therefore \frac{1}{2πi}f^{∗}Ω descends to a 2-form α_{ρ} on Σ representing the first Chern class c_{1}(L_{ρ}). Hence

∫_{Σ} α_{ρ} = c_{1}(L_{ρ}) ∈ ℤ.

We apply this remark now in two concrete situations. Let κ → H_{ℂ}^{n} be the canonical line bundle and θ → H_{ℂ}^{n} be the restriction to H_{ℂ}^{n} of the tautological bundle

ℂ^{n+1} \ {0} → P_{n}(ℂ)

over projective n-space. Then θ^{n+1} ≅ κ.

There are canonical actions of SU(n, 1) on θ and of PU(n, 1) on κ. Both bundles carry group invariant Hermitian metrics which are unique up to scaling. The corresponding curvature forms are given by

Ω(θ) = \frac{1}{2i}ω

respectively

Ω(κ) = \frac{n + 1}{2i}ω

where ω denotes the Kähler form on H_{ℂ}^{n}.

Given a representation ρ : π → PU(n, 1) we consider the line bundle L_{ρ} → Σ associated to κ → H_{ℂ}^{n}. We obtain

c_{1}(L_{ρ}) = ∫_{Σ} α_{ρ} = \frac{1}{2πi} ∫_{Σ} f^{∗}Ω(κ) = −\frac{n + 1}{4π} ∫_{Σ} f^{∗}ω = −\frac{n + 1}{2}τ(ρ),

so

τ(ρ) ∈ \frac{2}{n + 1} · ℤ.

If ρ lifts to a representation ρ̃: π → SU(n, 1) consider instead the line bundle Lρ̃ → Σ associated to θ → H_{ℂ}^{n}, obtaining

c_{1}(Lρ̃) = −\frac{1}{2}τ(ρ),

so

τ(ρ) ∈ 2 · ℤ.

2.3. Example: Complex lines. Basic examples of representations π → PU(n, 1) arise from compact complex hyperbolic 1-manifolds embedded in complete complexhyperbolic surface. Recall that a complex geodesic in H_{ℂ}^{n} is a holomorphic totally geodesic complex curve. A complex geodesic has constant curvature −1, and is equivalent to C = H_{ℂ}^{1} embedded in H_{ℂ}^{n} by:

z ⟼ (0, …, 0, z) ⟷ [0 ; ⋮ ; 0 ; z ; 1]

COMPLEX HYPERBOLIC MANIFOLDS 5

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, A) ↦ [\begin{matrix}B & 0 \\ 0 & A \end{matrix}]

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

SU(1, 1) ↪ SU(n, 1) ↪ U(n, 1) ⟶^{ℙ} 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 H_{ℂ}^{1} comprise the group PU(1, 1). The action of PU(1, 1) on H_{ℂ}^{1}, however, does not quite extend to H_{ℂ}^{n}. Rather, according to the above discussion the group SU(1, 1) acts on H_{ℂ}^{n}. The restriction homomorphism

SU(1, 1) → Isom(H_{ℂ}^{1})

defines a double covering of SU(1, 1) onto PU(1, 1). The nontrivial element of its kernel is the reflection in H_{ℂ}^{1} given by

(z_{1}, …, z_{n−1}, z_{n}) ⟼ (−z_{1}, …, −z_{n−1}, z_{n})

which we call the inversion with respect to H_{ℂ}^{1}.

Now we specialize to the case n = 2. Let ν^{1} denote the unit normal bundle to H_{ℂ}^{1} in H_{ℂ}^{2} and UT denote the unit tangent bundle of H_{ℂ}^{1}. Note that SU(1, 1) acts simply-transitively on ν^{1}. Both normal and tangent bundle of H_{ℂ}^{1} have Hermitian connections induced by the Levi-Civita connection on H_{ℂ}^{2}. These connections restrict to connections on the corresponding unit bundles.

Lemma 2.1. There exists a 2-fold covering ν^{1} → UT which is equivariant with respect to the homomorphism of Lie groups p : SU(1, 1) → PU(1, 1).

Proof. The group PU(1, 1) acts simply-transitively on UT, thus 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 θ : H_{ℂ}^{2} → H_{ℂ}^{2} which lies in the image of the standard embedding SU(1, 1) ↪ PU(2, 1), assume that the origin o is a fixed point of θ. Then the angle of rotation of θ in the tangent line T_{o}H_{ℂ}^{1} is twice the angle of rotation of θ in the normal line ν_{o}H_{ℂ}^{1}.

Note that the curvature of the pull-back of the connection on UT via p to ν^{1} is twice the curvature of the connection on UT. The induced metric on T H_{ℂ}^{1} has sectional curvature -1, thus the curvature 2-form of T H_{ℂ}^{1} is i · dA (where dA denotes the area form). Therefore the curvature 2-form of the normal bundle of H_{ℂ}^{1} in H_{ℂ}^{2} equals \frac{i}{2} dA. Suppose now that Γ is a torsion-free uniform lattice in SU(1, 1),

Σ := H_{ℂ}^{1}/Γ,

(that is, Γ is a complex Fuchsian group), and

M := H_{ℂ}^{2}/Γ

6 GOLDMAN, KAPOVICH, AND LEEB

where we embed Γ to PU(2, 1) via the monomorphism of Lie groups SU(1, 1) ↪ PU(2, 1).

Corollary 2.3. The Euler number of the normal bundle ν_{Σ} of Σ in M equals \frac{1}{2}χ(Σ).

The quotient M = H_{ℂ}^{2}/ρ(Γ) is a complete complex hyperbolic manifold containing a holomorphic totally geodesic submanifold H_{ℂ}^{1}/ρ(Γ) ≈ Σ. Orthogonal projection onto H_{ℂ}^{1} defines a fibration Π_{M} : M ⟶ H_{ℂ}^{1}/Γ whose fibers are totally geodesic complex hyperplanes. The fibration Π_{M} is isomorphic to the normal bundle ν_{Σ} via the normal exponential mapping. Thus its Euler number equals χ(Σ)/2.

The Kähler form restricts to the area form on the holomorphic submanifold Σ ⊂ M, hence Toledo’s invariant of the embedding Γ ↪ PU(2, 1) is −χ(Σ) = 2g − 2.

We will need a relative version of Corollary 2.3.

Definition 2.4. A biholomorphic isometry h : H_{ℂ}^{2} → H_{ℂ}^{2} will be called a transvection if it has an invariant geodesic γ ⊂ H_{ℂ}^{2} and h acts trivially on the normal bundle of γ. (Note that an element is a transvection iff it is conjugate to an element of SU(1, 1) with positive trace.)

Suppose that W_{1} ⊂ H_{ℂ}^{2} is a totally geodesic subspace of the curvature −1. Let Υ ⊂ PU(1, 1) = Isom(W_{1}) be a nonabelian discrete finitely generated free purely hyperbolic subgroup, W_{1}/Υ is an open hyperbolic surface. We make the following assumptions:

• Condition 1. The surface X = W_{1}/Υ has even number of ends;

• Condition 2. λ : Υ = π_{1}(X) → SU(1, 1) ⊂ PU(2, 1) is an embedding such that

the image of any peripheral element is a transvection in H_{ℂ}^{2}. Let Σ ⊂ W_{1}/Υ be the convex core, M := H_{ℂ}^{2}/λ(Υ) and let ν and ν^{1} denote the normal bundle and the unit normal bundle to Σ in M. The second hypothesis guarantees a parallel section α of the restriction ν^{1}|_{∂Σ}. Following Steenrod [12], the relative Euler number e(ν, α) is defined as the obstruction to extending α to a nonzero section of ν over all of Σ.

Proposition 2.5. The relative Euler number e(ν, α) equals \frac{1}{2}χ(Σ).

Proof. Let p : ν(Σ) → T (Σ) denote the projection from the normal to the tangent bundle of Σ 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(α) is tangent to the boundary of Σ. Then e(T (Σ), p(α)) is the Euler characteristic of Σ (it follows for instance from taking the double of Σ along its boundary and making a symmetric extension of α to Σ and its mirror image in the double). On the other hand, since p is a 2-fold covering we conclude that e(ν, α) = \frac{1}{2}χ(Σ).

In the next section we prove a technical lemma which implies that the first condition is necessary and sufficient for the existence of α satisfying the second condition.

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 H_{ℂ}^{2}.

To construct surfaces M as in Theorem 1.2 we will glue quotients of domains in H_{ℂ}^{2} by subgroups Γ_{1} stabilizing a complex geodesic W_{1} and quotients of domains by subgroups Γ_{2} stabilizing a totally real geodesic 2-plane W_{2}. The interface along

COMPLEX HYPERBOLIC MANIFOLDS 7

which we glue is the quotient of a tube around a (real) geodesic W_{0} = W_{1} ∩ W_{2} by a cyclic subgroup. This cyclic subgroup belongs to the intersection G_{0} := G_{1} ∩ G_{2}, where G_{j} is the stabilizer of W_{j} in PU(2, 1). In suitable coordinates we may take W_{1} = H_{ℂ}^{1}, W_{2} = H_{ℝ}^{2} and W_{0} = H_{ℝ}^{1}. The corresponding stabilizers are G_{1} = U(1, 1), G_{2} = SO(2, 1) and G_{0} = SO(1, 1) respectively.

(Figure 1 depicts a surface which is the union of two hyperbolic half-planes. The upper half has curvature −1 and represents the Poincaré model; the lower half has curvature −1/4 and represents the Klein-Beltrami model. Geometrically this surface is realized in H_{ℂ}^{2} as a subset of the union of the totally geodesic subspace H_{ℂ}^{1} (the upper half) and H_{ℝ}^{2} (the lower half) which meet in H_{ℝ}^{1} (the equator). Geodesics orthogonal to H_{ℝ}^{1} are drawn in the two halves, as well as the hypercycles parallel to H_{ℝ}^{1}.)

Figure 1. Tube arond a real geodesic.

Although G_{1} is connected, neither G_{2} nor G_{0} 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:

1  0  0
0 − cosh(t) − sinh(t)
0 − sinh(t) − cosh(t)

The corresponding isometry of H_{ℂ}^{2} is biholomorphic and restricts to an orientationpreserving isometry of H_{ℂ}^{1} (since its restriction to H_{ℂ}^{1} is holomorphic). However, its restriction to H_{ℝ}^{1} reverses orientation: it is a glide-reflection, the composition of translation along the geodesic H_{ℝ}^{1} with “inversion” in H_{ℂ}^{1}. 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 Υ ⊂ PU(1, 1) be a finitely generated nonabelian purely hyperbolic discrete group of automorphisms of H_{ℂ}^{1} such that H_{ℂ}^{1}/Υ has an even number of ends. Then there exists a lift λ : Υ ⟶ SU(1, 1) such that for each g ∈ Υ corresponding to a simple loop around an end, λ(g) is a transvection.

Proof. If H_{ℂ}^{1}/Υ is a closed surface then the subgroup Υ lifts isomorphically to SU(1, 1) (see Petersson [11] or [4]); otherwise Υ is a free group. The convex core Σ of H_{ℂ}^{1}/Υ is a compact surface whose boundary consists of closed geodesics c_{1}, …, c_{m} (see [1]).

Choose lifts of the generators of Υ to SU(1, 1) to obtain a homomorphism λ̃ : Υ ⟶ SU(1, 1) lifting the inclusion Υ ↪ SU(1, 1). Denote the elements of Υ corresponding to the ends of H_{ℂ}^{1}/Υ by g_{1}, …, g_{m}. For each j = 1, …, m write λ̃(g_{j}) = f_{j}ĝ_{j} where

f_{j} = ±I ∈ SU(1, 1)

and ĝ_{j} lies in a one-parameter subgroup.

Recall that the normal bundle of H_{ℂ}^{1} in H_{ℂ}^{2} has a canonical Hermitian connection induced by the Levi-Civita connection on H_{ℂ}^{2}. This connection projects to the connection ∇ on the normal bundle ν of Σ in H_{ℂ}^{2}/Υ.

Choose a basepoint and a loop l which starts at the basepoint and traverses the boundary components c_{1}, …, c_{m}. The holonomy of the connection ∇ around l equals the product (of unit complex numbers) ∏_{j} f_{j} of the holonomies around each c_{j} (compare [10]). By integrating the connection form around l this equals the integral of the curvature form

exp (∫_{Σ} κ_{∇}) = exp (∫_{Σ} \frac{i}{2}dA) = exp(πiχ(Σ)).

Since χ(Σ) ≡ m( mod 2),

∏_{j=1}^{m} f_{j} = (−1)^{m} (2.1)

Since g_{1}, …, g_{m−1} lie in a set Δ of free generators for Υ, there exists a representation λ : Υ ⟶ SU(1, 1) such that

λ(g_{j}) = f_{j}λ̃(g_{j})

for j < m and

λ(g) = λ̃(g)

for g ∈ Δ − {γ_{1}, …, γ_{m}}. so that λ(g_{j}) are transvections all for j < m. Apply (2.1) when m is even to conclude that λ(g_{m}) is also a transvection. This concludes the proof of the proposition.

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 H_{ℝ}^{2} ⊂ H_{ℂ}^{2} — the subset consisting of points with real coordinates. This subspace is the fixed-point-set of the anti-holomorphic isometric involution

(z_{1}, z_{2}) ⟼ (\overline{z}_{1}, \overline{z}_{2})

COMPLEX HYPERBOLIC MANIFOLDS 9

The induced Riemannian metric on H_{ℝ}^{2} has constant negative curvature −1/4. The stabilizer of H_{ℝ}^{2} 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 fibers of the real orthogonal projection Π_{ℝ} : H_{ℂ}^{2} ⟶ H_{ℝ}^{2} are totally real geodesic subspaces which are orthogonal to H_{ℝ}^{2}. Since H_{ℝ}^{2} is totally real, the almost complex structure defines an SO(2, 1)-equivariant isomorphism between its tangent bundle and its normal bundle.

Let Γ ⊂ SO(2, 1)^{0} be a discrete torsion-free subgroup such that the convex core Σ of H_{ℝ}^{2}/Γ is compact. The composition

ρ : π_{1}(Σ) ≅ Γ ↪ SO(2, 1) ⊂ U(2, 1) → PU(2, 1)

is a discrete embedding. The quotient M = H_{ℂ}^{2}/ρ(Γ) is a complete complex hyperbolic surface containing a totally real totally geodesic submanifold

H_{ℝ}^{2}/ρ(Γ) ≅ Σ

The corresponding embedding

Σ ↪ H_{ℝ}^{2}/Γ ⊂ M

induces the homomorphism ρ : π_{1}(Σ) ≅ Γ of fundamental groups. From now on identify Σ with its image under this embedding.

Suppose Σ has no boundary. The above discussion implies the fibration M → Σ induced by the orthogonal projection H_{ℂ}^{2} ⟶ H_{ℝ}^{2} is fiber-diffeomorphic to the tangent bundle of Σ. Hence this fibration has Euler number

e(T Σ) = χ(Σ) = 2 − 2g

We will also need a relative version of this statement in the case ∂Σ ≠ ∅. Let ν(Σ) be the normal bundle of Σ in M, α is a unit normal parallel vector-field along ∂Σ in M.

Proposition 2.7. The relative Euler number e(ν(Σ), α) equals the Euler characteristic χ(Σ).

Proof. We repeat the argument from the proof of Proposition 2.3, just instead of 2-fold covering we use the isomorphism J between the normal and tangent bundle. This isomorphism will map α to a parallel section J(α) along ∂Σ. Hence e(ν(Σ), α) = e(T (Σ), J(α)) = χ(Σ).

Since Σ is embedded as a totally real submanifold, the restriction ω|_{Σ} = 0. Thus Toledo’s integrand is identically zero which implies that τ(ρ) = 0.

3. Geometric preliminaries

3.1. Orthogonal projections in Hadamard manifolds. Let W be a simplyconnected complete manifold with sectional curvature K_{W} ≤ −1/4. Let W_{1} and W_{2} be totally geodesic submanifolds of W with non-empty intersection W_{0}. Then W_{0} is a totally geodesic submanifold as well. The angle of intersection of W_{1} and W_{2} at x ∈ W_{0} is defined to be:

∠_{x}(W_{1}, W_{2}) := min{∠(v_{1}, v_{2}) | v_{1} ∈ T_{x}W_{1}, v_{2} ∈ T_{x}W_{2}, v_{1} ⊥ W_{0}, v_{2} ⊥ W_{0}}
∈ [0, π/2]

10 GOLDMAN, KAPOVICH, AND LEEB

Since the W_{j} are totally geodesic, parallel translation along W_{0} preserves tangency to W_{j}. Therefore the angle ∠_{x}(W_{1}, W_{2}) is independent of the point x ∈ W_{0} and we simply write ∠(W_{1}, W_{2}) for ∠_{x}(W_{1}, W_{2}).

Denote by Π_{j} : W → W_{j} the orthogonal projection onto W_{j} (j = 0, 1, 2). For a subset S ⊂ X in a metric space X let Nbd_{r}(S) denote the closed r-neighborhood of S.

Proposition 3.1. Let ϵ(α) := 2 · cosh^{−1}(csc(α/2)) where α := ∠(W_{1}, W_{2}). Then for all ϵ > ϵ(α) we have:

(3.1) W = Π_{1}^{−1}(Nbd_{ϵ}(W_{0})) ∪ Π_{2}^{−1}(Nbd_{ϵ}(W_{0}))

and the distance between boundary components of

Π_{1}^{−1}(Nbd_{ϵ}(W_{0})) ∩ Π_{2}^{−1}(Nbd_{ϵ}(W_{0}))

is positive.

Proof. Pick a point p ∈ W. Consider the projections p_{j} = Π_{j}(p) to W_{j} (j = 0, 1, 2) and let q = Π_{0}(p_{1}) ∈ W_{0}. It suffices to show that d(p_{1}, q) ≤ ϵ or d(p_{2}, q) ≤ ϵ.

Denote by u_{j} ∈ T_{q}(W_{j}) the unit tangent vector at q pointing towards p_{j}. Then u_{1} ⊥ W_{0}. Let u′_{2} ∈ T_{q}W_{2} be the component of u_{2} orthogonal to W_{0}. Then

|⟨u_{1}, u_{2}⟩| = |⟨u_{1}, u′_{2}⟩| ≤ |u′_{2}| cos(α) ≤ cos(α).

and we get α ≤ ∠(p_{1}qp_{2}) ≤ ∠(p_{1}qp) + ∠(pqp_{2}). Hence, we may assume without loss of generality that q ∈ W_{0} satisfies

∠(p_{1}qp) ≥ \frac{α}{2}. (3.2)

Let H_{ℝ}^{2} be the hyperbolic plane with sectional curvature −1/4. Consider a triangle △(p̃_{1}q̃p̃) in H_{ℝ}^{2} having the same side lengths as △(p_{1}qp). (Compare Figure 2.) Since W has sectional curvature bounded above by −1/4, angle comparison with H_{ℝ}^{2} implies (see [9])

∠(p̃_{1}q̃p̃) ≥ ∠(p_{1}qp) ≥ \frac{α}{2} and ∠(q̃p̃_{1}p̃) ≥ ∠(qp_{1}p) = \frac{π}{2} .

(by (3.2)). Hence

d(p_{1}, q) = d(p̃_{1}, q̃) ≤ ϵ(α)

where ϵ = ϵ(α) is the length of the finite side of the right ideal triangle in H_{ℝ}^{2} with angle α/2. For such a triangle (see, for example Beardon [1], Theorem 7.9.1)

cosh (\frac{ϵ}{2}) = csc (\frac{α}{2})

whence the first assertion of the proposition follows. In particular for each ϵ > ϵ(α) we have:

∂Π_{1}^{−1}(Nbd_{ϵ}(W_{0})) ∩ ∂Π_{2}^{−1}(Nbd_{ϵ}(W_{0})) = ∅

To show that the distance between these hypersurfaces is positive choose δ such that ϵ(α) < δ < ϵ. Recall that the nearest-point projection in Hadamard spaces does not increase the distance. Hence for j = 1, 2 the set

U_{j,δ} := Π_{j}^{−1}(W_{0} − Nbd_{δ}(W_{0}))

contains the (ϵ − δ)-neighborhood of the hypersurface ∂Π_{j}^{−1}(Nbd_{ϵ}(W_{0})) and as we already proved U_{1,δ} ∩ U_{2,δ} = ∅. The second assertion of the proposition follows.

Figure 2. The comparison triangle

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 infinitesimal data associated with each neck. Let h be a transvection in H_{ℂ}^{2} along a real geodesic W_{0}. Consider the quotient M := H_{ℂ}^{2}/⟨h⟩: it contains the closed geodesic γ := W_{0}/⟨h⟩, let ℓ be the length of γ. The choice of the generator h of the cyclic group ⟨h⟩ corresponds to the choice of an orientation on γ. Hence, pick a point x ∈ γ and a unit tangent vector v_{1} ∈ T_{x}(γ) ⊂ T_{x}(M) pointing in the positive direction. Then choose a unit tangent vector v_{2} ∈ T_{x}(M) which is Hermitian orthogonal to v_{1}. The pair (v_{1}, v_{2}) can be canonically completed to an orthonormal basis of T_{x}(M) over ℝ: let

w_{1} := \sqrt{−1}v_{1}, w_{2} := \sqrt{−1}v_{2}.

We call (v_{1}, v_{2}) a normal frame at x. Note that our assumptions on γ imply that the normal frame canonically extends to a smooth parallel frame field along γ.

Such a normal frame and the number ϵ > 0 uniquely determine a neck N(ϵ, ℓ) in the manifold M around γ as follows:

• Any lift w̃_{1} of w_{1} to T (H_{ℂ}^{2})|_{W_{0}} is tangent to the unique complex geodesic W_{1} in H_{ℂ}^{2} containing W_{0}.

• Any lift ṽ_{2} of v_{2} to T (H_{ℂ}^{2})|_{W_{0}} is tangent to a totally geodesic real-hyperbolic plane W_{2} in H_{ℂ}^{2} containing W_{0}.

• The vectors w̃_{1}, ṽ_{2} determine half-planes in

W_{1}^{+} ⊂ W_{1}, W_{2}^{+} ⊂ W_{2}

bounded by W_{0}. (The vectors point outside these half-planes). We denote by X_{j} the union of the open half-plane W_{j}^{+} and the closed ϵ-neighborhood of W_{0} in W_{j}. Then S_{j} := X_{j} − W_{j}^{+} is a strip bounded by W_{0} and a hypercycle h_{j} equidistant from W_{0}. Denote by Y_{j} the inverse image of X_{j} under the orthogonal projection Π_{j} : H_{ℂ}^{2} → W_{j}. The next lemma follows directly from Proposition 3.1.

12 GOLDMAN, KAPOVICH, AND LEEB

Lemma 3.2. Let

ϵ > ϵ_{0} = ϵ(π/2) = 2 ln(1 + \sqrt{2}) = ln(3 + 2\sqrt{2}). (3.3)

Then Y_{1} ∪ Y_{2} = W and moreover the distance between the boundary components of Y_{1} ∩ Y_{2} is positive.

Let ϵ_{1} := ln(6) > ϵ_{0} and suppose in the following that ϵ > ϵ_{1}. The 1-parameter group O(1, 1)^{0} of transvections along the geodesic W_{0} acts on the intersection Y (ϵ) = Y = Y_{1} ∩ Y_{2}. 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 H_{ℂ}^{2}∪∂_{∞}H_{ℂ}^{2}−∂_{∞}W_{0} 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 \overline{Y} be the closure of Y in H_{ℂ}^{2} ∪ ∂_{∞}H_{ℂ}^{2} − ∂_{∞}W_{0}. It is invariant under the O(1, 1)^{0}-action. The frontier ∂Y of Y in H_{ℂ}^{2} ∪ ∂_{∞}H_{ℂ}^{2} − ∂_{∞}W_{0} projects to a union of two disjoint smoothly properly embedded 2-discs in B^{3}. Hence, by the smooth Schönflies Theorem in dimension three, the projection of \overline{Y} is diffeomorphic to D^{2} × [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 fibration of ∂Y by totally geodesic 2-planes in an O(1, 1)^{0}-invariant manner to a smooth fibration of \overline{Y} by closed 2-discs. As base for this bundle we can use the strip S_{1} ∪ S_{2}. The above discussion shows:

Proposition 3.3. The quotient Ȳ /⟨h⟩ is diffeomorphic to the total space of a trivial disk bundle over the annulus A. The restriction of this fibration to the two components of ∂Y/⟨γ⟩ corresponds to the restriction of the nearest-point projections Π_{j} to Π^{−1}(h_{j}), j = 1, 2.

We call the quotient N(ϵ, ℓ) = N := Y/⟨h⟩ a model neck. Its isometry type depends on two parameters: ϵ (the width) and ℓ (the length).

3.3. Collars of geodesics. In this section we determine ℓ = ℓ(D) > 0 such that a closed geodesic of length ℓ 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 find 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 W_{k} be a complete simply connected Riemannian 2-manifold of constant curvature −k^{−2} and Γ_{k} a purely hyperbolic discrete group of isometries. Suppose that f, g are nontrivial elements of Γ_{k} which are translations along geodesics A_{f}, A_{g} in W_{k}. The geodesics A_{f}, A_{g} are called the axes of f, g. We assume that:

• The geodesics A_{f}, A_{g} are disjoint.

• The geodesics A_{f}/⟨f⟩, A_{g}/⟨g⟩ have the same length ℓ.

Lemma 3.4.

sinh (\frac{ℓ}{2k} ) sinh (\frac{d(A_{f}, A_{g})}{2k}) ≥ 1/2

Proof. This assertion is a quantitative version of the Margulis Lemma for the hyperbolic plane and slightly modifies Corollary 11.6.10 of Beardon [1]. Beardon considers the case of curvature −1 but the formulas remain valid for constant curvature K < 0 if the lengths are multiplied by \sqrt{−K}.

COMPLEX HYPERBOLIC MANIFOLDS 13

Let S be a complete hyperbolic surface and γ a simple closed geodesic in S. Let S_{γ} → S denote the locally isometric covering of S corresponding to the subgroup of π_{1}(S) generated by γ. Let γ̂ denote the simple closed geodesic in S_{γ} covering γ. We say that the ϵ-neighborhood Nbd_{ϵ}(γ) of γ in S is injective if the projection

Nbd_{ϵ}(γ̂) → Nbd_{ϵ}(γ)

is a diffeomorphism. Equivalently, α is contained in γ for every geodesic arc α in S with both end-points on γ and ℓ(α) ≤ 2ϵ. Suppose that Γ_{k} is a finitely-generated group as in Lemma 3.4, Σ_{k} is the compact convex core of the surface S_{k} = W_{k}/Γ_{k}. We assume that all boundary geodesics of Σ_{k} have the same length ℓ.

Lemma 3.4 implies:

Lemma 3.5. Let ϵ > 0 and suppose that ℓ satisfies

(3.4) sinh (\frac{ℓ}{2k} ) sinh (\frac{ϵ}{k} ) ≤ \frac{1}{2},

Then the ϵ–neighborhood of each boundary geodesic of Σ_{k} is injective and ϵ–neighborhoods of distinct geodesics are disjoint.

Remark 3.6. In the case of equality in (3.4)

sinh (\frac{ℓ_{k}}{2k} ) sinh (\frac{ϵ}{k} ) = \frac{1}{2},

then ℓ_{1} < ℓ_{2}. If ϵ = ϵ_{1} = ln(6) then sinh(ℓ_{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.

Definition 3.7. Let M be a complex-hyperbolic surface and γ ⊂ M be a simple closed geodesic of length ℓ. We say that γ admits an (ϵ, l)-neck neighborhood Neck_{ϵ}(γ) of width ϵ in M if it has a neighborhood in M which is isometric to the model neck N(ϵ, ℓ).

Suppose that W_{k} ⊂ W = H_{ℂ}^{2} is a totally geodesic plane of the curvature −k^{−2} and Γ_{k} is a finitely-generated purely loxodromic discrete subgroup of PU(2, 1) stabilizing W_{k}. We assume that S_{k} = W_{k}/Γ_{k} is not compact and Σ_{k} ⊂ S_{k} is the compact convex core so that the length of each boundary geodesic equals the same number ℓ. We also assome that each element of Γ_{k} representing a boundary geodesic γ of Σ_{k} is a transvection.

To construct the infinitesimal data for the neck around each boundary geodesic γ of Σ_{k} choose a normal frame at γ ⊂ H_{ℂ}^{2}/Γ_{k} (see subsection 3.2):

• Pick a base point x at γ.

• The vector v_{1} tangent to γ at x is determined by the choice of orientation on γ, we let w_{1} := \sqrt{−1}v_{1}. If k = 1 then the vector w_{1} is tangent to Σ_{1} and is directed inwards Σ_{1}.

• If k = 2 then the second vector v_{2} is chosen tangent to Σ_{2} and directed inwards Σ_{2}.

Lemma 3.8. Suppose that ℓ < ℓ_{1}, where ℓ_{1} is given by

sinh(ℓ_{1}/2) = 6/35

Then for ϵ = ϵ_{1} = ln(6) we have:

14 GOLDMAN, KAPOVICH, AND LEEB

• Each boundary geodesic γ ⊂ Σ_{k} ⊂ H_{ℂ}^{2}/Γ_{k} is contained in a neck neighborhood Neck_{ϵ}(γ) with the normal frame as above.

• If γ, β ⊂ ∂Σ_{k} are distinct boundary geodesics then Neck_{ϵ}(γ) ∩ Neck_{ϵ}(β) = ∅.

Proof. The condition ϵ = ϵ_{1} > ϵ_{0} = 2 ln(1 + \sqrt{2}) implies the existence of a model neck of the width ϵ (Lemma 3.2). Let γ be a boundary geodesic of Σ_{k} and let

M_{γ} → M = H_{ℂ}^{2}/Γ_{k}, Σ_{k,γ} → Σ_{k}

denote the coverings defined by the subgroups

π_{1}(γ) ⊂ π_{1}(M), π_{1}(γ) ⊂ π_{1}(Σ_{k})

Define γ̂ as the simple closed geodesic on Σ_{k,γ} ⊂ M_{γ} covering γ. Lift of the normal frame along γ from M to a normal frame along γ̂ in M_{γ} determines an neck neighborhood Neck_{ϵ}(γ̂) ⊂ M_{γ}. We first show that the restriction of the projection M_{γ} → M to Neck_{ϵ}(γ̂) is injective. Let P̃ : M_{γ} → Σ̃_{k} denote the nearest-point projection to the lift Σ̃_{k} of Σ_{k} into M_{γ}. Then Neck_{ϵ}(γ̂) is contained in the inverse image P̃^{−1}(Nbd_{ϵ}(γ̂)) where the ϵ-neighborhood is taken within the surface Σ̃_{k}. Recall that by Lemma 3.5 and Remark 3.6 the geodesic γ in Σ_{k} has an injective ϵ-neighborhood. Thus the restriction of the covering Σ_{k,γ} → Σ_{k} to Nbd_{ϵ}(γ̂) is injective and the restriction of the covering M_{γ} → M to P̃^{−1}(Nbd_{ϵ}(γ̂)) is injective as well. This implies that we have a neck neighborhood Neck_{ϵ}(γ) of the geodesic γ in M. Finally we show that neck neighborhoods of distinct geodesics are disjoint. Let P : M = H_{ℂ}^{2}/Γ_{k} → Σ_{k} denote the orthogonal projection. Then again

Neck_{ϵ}(γ) ⊂ P^{−1}(Nbd_{ϵ}(γ))

for every boundary geodesic γ ⊂ Σ_{k}. Let β, γ ⊂ ∂Σ_{k} are distinct boundary components. Then their ϵ-neighborhoods Nbd_{ϵ}(β), Nbd_{ϵ}(γ) in Σ_{k} are disjoint by Lemma 3.5. Hence P^{−1}(Nbd_{ϵ}(γ)) ∩ P^{−1}(Nbd_{ϵ}(β)) = ∅ and Neck_{ϵ}(γ) ∩ Neck_{ϵ}(β) = ∅ as desired.

4. Proof of the main theorem

We begin with an oriented closed surface Σ with a piecewise hyperbolic structure of the following kind. There is a collection B of disjoint simple closed geodesics on Σ such that on each component of Σ − B the metric either has constant curvature −1 or constant curvature −1/4. Let Σ_{k} (k = 1, 2) denote the subsurface of Σ − B where the curvature equals −k^{−2}. The surfaces Σ_{k} could be disconnected. We assume that each loop β ∈ B is adjacent to both Σ_{1}, Σ_{2}. Thus we orient B so that the surface Σ_{1} lies to the left from B.

Our construction involves two assumptions:

• The boundary of each component of Σ_{1} has an even number of components.

• All components of B have the same length ℓ < ℓ_{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 : Σ → M such that

• f is a homotopy equivalence.

• The Toledo invariant of M equals −χ(Σ_{1}).

• M is diffeomorphic to the total space of an oriented disc bundle which has the Euler number χ(Σ_{1})/2 + χ(Σ_{2}).

COMPLEX HYPERBOLIC MANIFOLDS 15

Outline of proof. We embed Σ_{2} (respectively Σ_{1} ) as a totally real (respectively holomorphic) totally geodesic submanifold inside a complete complex hyperbolic surface X_{2} (respectively X_{1}) so that:

1. The embeddings f_{k} : Σ_{k} → X_{k}, k = 1, 2 are homotopy equivalences.

2. The homotopy-inverse to f_{k} is the nearest-point projection Π̂_{k} : X_{k} → f_{k}(Σ_{k}).

3. The numbers ϵ, ℓ are chosen so that ϵ-neighborhoods of distinct boundary components of Σ_{k} are injective and disjoint.

4. Each component f_{k}(γ_{ki}) of f_{k}(∂Σ_{k}) is contained in a neck N_{ki} := Neck_{ϵ}(γ_{ki}) and necks around distinct geodesics are disjoint.

5. If two annular components

A_{1i} ⊂ Nbd_{ϵ}(∂Σ_{1}), A_{2i} ⊂ Nbd_{ϵ}(∂Σ_{2})

are adjacent in the surface Σ then there is a canonical isometry ξ_{i} between the corresponding necks N_{1i}, N_{2i}.

To guarantee the properties 3-5, we need “enough room” around the boundary geodesics of f_{k}(∂Σ_{k}) in X_{k}. This is achieved by the choice of ℓ and ϵ as above.

Now the construction of M proceeds as follows. Remove from the manifolds X_{k} (k = 1, 2) all points in Π̂_{k}^{−1}(∂Σ_{k}) that do not belong to the union of necks N_{ki}. The resulting complex hyperbolic manifolds with boundaries M_{1} and M_{2} are glued together along their necks via the isometries ξ_{i}. Then we prove that the resulting complex hyperbolic surface M satisfies 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 Σ as in the previous section. For each component Σ_{kj} of the surface Σ_{k} we have the action of Γ_{kj} = π_{1}(Σ_{kj}) on its Nielsen region Σ̃_{kj} in the hyperbolic plane H^{2}(−k^{−2}) with curvature −k^{−2}. Recall that Σ̃_{kj} is the convex hull of the limit set of Γ_{kj} = π_{1}(Σ_{kj}) and is a convex domain with totally geodesic boundary. Furthermore Σ_{kj} = Σ̃_{kj}/Γ_{kj} is a compact geodesically convex hyperbolic surface with geodesic boundary — the convex core of a the complete hyperbolic manifold Σ_{kj} = W_{k}/Γ_{kj}. According to Proposition 2.6 for each k, j there exists an equivariant totally geodesic embedding of the hyperbolic plane H^{2}(−k^{−2}) in H_{ℂ}^{2} which satisfies the following properties:

1. For k = 1 the image is the complex hyperbolic line H_{ℂ}^{1} = W_{1}, for k = 2 the image is a totally real hyperbolic plane H_{ℝ}^{2} = W_{2}.

2. Each component β of B is represented by a hyperbolic translation h_{β} in H_{ℂ}^{2}.

3. The canonical orientation on W_{1} agrees with the orientation on Σ_{1}.

Thus representatives h_{β} of all the components of B are conjugate in SU(2, 1). We let X_{k} be the disjoint union of quotients of H_{ℂ}^{2}/Γ_{kj}. It is clear that we have totally geodesic embeddings f_{k} : Σ_{k} ↪ X_{k} which are homotopy equivalences. The property 2 implies that for each boundary geodesic β of Σ_{k} we can choose a normal frame at f_{k}(β) ⊂ X_{k}, see section 3.4:

These normal frames determine necks Neck_{ϵ}(β) ⊂ X_{k} around the geodesics f_{k}(β) so that necks around distinct geodesics are disjoint (see Lemma 3.8). We define submanifolds M_{k} ⊂ X_{k} as in the previous section, and denote by M_{kj} the component of M_{k} containing Σ_{kj}. Each neck has a canonical fibration over an annulus, this ℝ^{2}-fibration extends to the fibration of

M_{k} − ⋃_{β⊂∂Σ_{k}} Neck_{ϵ}(β)

16 GOLDMAN, KAPOVICH, AND LEEB

given by the nearest-point projection to f_{k}(Σ_{k}), see section 3.2. Suppose now that β_{k} ⊂ ∂Σ_{k} (k = 1, 2) are boundary geodesics identified in the surface Σ. Then there exists a unique biholomorphic isometry ξ : Neck_{ϵ}(β_{1}) → Neck_{ϵ}(β_{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 f_{k} can be combined to define a piecewise totally geodesic embedding

f : Σ ↪ M (4.1)

which we call a spine. Similarly ℝ^{2}-fibrations of the components M_{k} define a smooth ℝ^{2}-fibration Π̂ : M → Σ. This fibration is a homotopy inverse to f. Moreover, Π̂ ∘ f = id. In the next section we shall compute the Euler number of this fibration.

Lemma 4.2. The complex hyperbolic structure on M is geodesically complete.

Proof. Assume that M is not complete and σ : [0, T) → M is a nonextendable finite geodesic ray. Each component M_{kj} of M_{k} is metrically complete by the construction and there is δ > 0 such that boundary components of each neck of M are δ-separated according to Proposition 3.1. Therefore there are infinitely many disjoint intervals [t_{n}^{−}, t_{n}^{+}] on [0, T) such that the segment σ([t_{n}^{−}, t_{n}^{+}]) connects different boundary components of necks in M. Each segment has length at least δ. This contradicts the finiteness of T.

As a consequence of the lemma, the corresponding holonomy representations π → PU(2, 1) are discrete and faithful. The pleated surface f̃: Σ̃ → H_{ℂ}^{2} has bending angles equal to π/2 and the bending locus consists of 2ϵ-separated geodesics. According to (3.4), we can make ϵ arbitrarily large if we choose l sufficiently small. Since H_{ℂ}^{2} has pinched negative curvature, it follows that for sufficiently large ϵ, the image of f̃ is quasi-convex and Hausdorff close to its convex hull. Thus, the action of G = π_{1}(M) on H_{ℂ}^{2} by deck transformations is convex cocompact. I.e. the action of G on H_{ℂ}^{2} preserves a nonempty closed convex subset on which G acts cocompactly.

Therefore the manifold

\overline{M} = (H_{ℂ}^{2} ∪ Ω(G))/G

is compact. This compactification of M is consistent with the open disk bundle structure of M and therefore \overline{M} is diffeomorphic to the closed disk bundle over Σ having the same Euler number as the bundle M → Σ.

Corollary 4.3. The manifold Ω(G)/G is diffeomorphic to the total space of an 𝕊^{1}– bundle over the surface Σ which has the same Euler number as the ℝ^{2}-fibration of M.

4.2. Calculation of Euler number and Toledo invariant. The contribution of the totally real components of f to the integral

τ = \frac{1}{2π} ∫_{Σ} f^{∗}ω_{M}

is zero. Thus the integral τ equals −χ(Σ_{1}) (see Section 2.3). This finishes the proof of the first and second assertions of Theorem 4.1.

We construct a continuous vector field 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 field orthogonal to both f(Σ_{k}). This is possible because the holonomy along all loops f(B) is trivial. Next we extend V to sections of the

COMPLEX HYPERBOLIC MANIFOLDS 17

normal bundles of the surfaces f(Σ_{k}) which are smooth on f(Σ_{k}) and have isolated zeros. By the computation in §2.3 and §2.5, the sum of the multiplicities for the zeros of V equals χ(Σ_{1})/2 on the complex portion f(Σ_{1}) and χ(Σ_{2}) on the totally real part f(Σ_{2}). We use the vector field V to homotopy f(Σ) off itself to a nearby surface g(Σ) which intersects f(Σ) transversally in finitely many isolated points. The multiplicities of the intersection points sum to the Euler number e(M → Σ) of the oriented D^{2}-fibration M ⟶ Σ, which equals

χ(Σ_{1})/2 + χ(Σ_{2})

as desired. This concludes the proof of Theorem 4.1.

Proof of Theorem 1.1. If τ = 2g − 2, 0, 2 − 2g then discrete embeddings of π into SU(1, 1) or SO(2, 1) give the desired representations. Thus let τ be an even integer satisfying

0 < τ < 2g − 2.

Then

t = g − 1 − \frac{τ}{2}

satisfies

0 < t < g − 1.

Let Σ denote a closed orientable surface of genus g. Then there exist two disjoint simple closed curves decomposing Σ as

Σ = Σ_{1} ∪ Σ_{2}

where Σ_{1} is a surface of genus g − t − 1 with two boundary components and Σ_{2} is a surface of genus t with two boundary components.

Thus the complex hyperbolic surface M in Theorem 4.1 achieves Toledo invariant τ. The negative values of τ can be obtained by conjugating these representations by an anti-holomorphic isometry of H_{ℂ}^{2}. Finally,

e(M → Σ) − χ(Σ) = |τ|/2

follows from direct computation.

Using degree one maps between hyperbolic surfaces and complex Fuchsian representations one can easily construct nonfaithful discrete representations of π_{1}(Σ_{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 π = π_{1}(Σ) that realize all possible fractional values of the invariant τ(ρ). We start with the case of surfaces of genus 2.

Proposition 5.1. Suppose that Σ is a closed oriented surface of genus 2. Then there are representations ρ_{j} : π = π_{1}(Σ) → PU(2, 1) so that τ(ρ_{j}) = 2j/3 (j = 1, 2).

Proof. Consider a family of regular quadrilaterals

Q_{δ}, 0 < δ < π/2

in H_{ℂ}^{1} ⊂ H_{ℂ}^{2} such that all angles of Q_{δ} equal δ. Let A_{δ}, B_{δ} be hyperbolic translations in PU(2, 1) pairing the opposite sides of Q_{δ}. The commutator C_{δ} = [A_{δ}, B_{δ}] is an

18 GOLDMAN, KAPOVICH, AND LEEB

elliptic element fixing a corner of Q_{δ}. It rotates by the angle 4δ inside H_{ℂ}^{1} and by the angle π + 2δ 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 H_{ℂ}^{1}, thus it is either π + 2δ or 2δ. Since

lim_{δ→π/2} C_{δ} = id,

the rotation angle equals π + 2δ. We now restrict to the special value δ = π/6, then the angles of rotation in H_{ℂ}^{1} and in the normal direction are 2π/3 and

2δ + π = 4π/3,

respectively. The elliptic element C_{δ} has order 3. We similarly define hyperbolic elements A′_{δ}, B′_{δ} ∈ SO(2, 1) whose commutator C′_{δ} has order 3. The eigenvalues of the derivative of C′_{δ} at its fixed point are e^{2π/3i} and e^{−2π/3i}. We consider the corresponding (discrete and non-faithful) representation

ϕ : F_{2} = ⟨α, β⟩ → SU(1, 1) ⊂ PU(2, 1)

of the free group on two generators given by ϕ(α) = A_{δ} and ϕ(β) = B_{δ}. The normalizer of the order 3 element C_{δ} in PU(2, 1) contains an involution θ with which it anticommutes: θC_{δ}θ^{−1} = C_{δ}^{−1}. The involution θ interchanges the eigenspaces of the derivative of C_{δ} at its fixed point, i.e. the direction tangent to H_{ℂ}^{1} and the normal direction. Similarly we have the representation

ϕ′ : F_{2} = ⟨α, β⟩ → SO(2, 1) ⊂ PU(2, 1)

given by ϕ′(α) = A′_{δ} and ϕ′(β) = B′_{δ}, which preserves a totally real geodesic plane. We use the representation ϕ as a building block to construct a representation

ρ_{2} : π = π_{1}(Σ) = ⟨α_{1}, β_{1}, α_{2}, β_{2} | [α_{1}, β_{1}][α_{2}, β_{2}] = 1⟩ → PU(2, 1)

by sending

α_{1} ⟼ ϕ(α),
β_{1} ⟼ ϕ(β),
α_{2} ⟼ θϕ(α)θ^{−1}
β_{2} ⟼ θϕ(β)θ^{−1}.

Now we compute Toledo’s invariant for the representation ρ_{2}. By identifying the sides of Q_{δ} via the transformations A_{δ}, B_{δ} we get a hyperbolic cone structure on the 2-torus T which has a single cone point k with total angle 4δ = 2π/3, cf. [13]. (This cone structure is actually an orbifold structure.) The punctured surface S = T − {k} has an incomplete hyperbolic structure whose developing map d : S̃ → H_{ℂ}^{1} maps a fundamental domain in S̃ to Q_{δ}. The homomorphism ϕ is the holonomy of this hyperbolic cone structure. We decompose Σ along a simple loop c into two punctured tori Σ_{j} so that π_{1}(Σ_{j}) is identified with the subgroup in π_{1}(Σ) generated by α_{j}, β_{j}. We choose orientation preserving diffeomorphisms h_{j} : Σ_{j} → S which map a neighborhood of c to a neighborhood of k. These diffeomorphisms give rise to equivariant developing maps

f_{1} = d ∘ h̃_{1} : Σ̃_{1} → W_{1}, f_{2} = θ ∘ d ∘ h̃_{2} : Σ̃_{2} → θ(W_{1})

The complex lines W_{1} and θ(W_{1}) are orthogonal and intersect in the fixed point of the elliptic isometry ρ_{2}([α_{1}, β_{1}]) = ρ_{2}([α_{2}, β_{2}])^{−1}. The equivariant maps f_{1}, f_{2} extend uniquely to a ρ_{2}-equivariant map f̃: Σ̃ → H_{ℂ}^{2}. There is a fundamental domain Ω for the action of π on Σ̃ which maps onto the union Q_{δ} ∪ θ(Q_{δ}) preserving orientation. Thus

τ(ρ_{2}) = \frac{1}{2π} ∫_{Ω} f̃^{∗}ω = \frac{1}{π} · area(S) = \frac{1}{π} (2π − 4δ) = \frac{4}{3}.

To realize the value \frac{2}{3} for the Toledo invariant, we leave

ρ_{1}|_{π_{1}(Σ_{1})} = ρ_{2}|_{π_{1}(Σ_{1})}

as above and replace ρ_{2}|_{π_{1}(Σ_{2})} by setting

ρ_{1} : α_{2} ↦ gϕ′(α)g^{−1}

and

ρ_{1} : β_{2} ↦ gϕ′(β)g^{−1}.

Since ϕ([α, β]) and ϕ′([α, β]) have the same eigenvalues, we may choose g ∈ PU(2, 1) so that

ρ_{1}([α_{1}, β_{1}]) = ρ_{1}([α_{2}, β_{2}])^{−1},

obtaining a representation ρ_{1} : π → PU(2, 1) with

τ(ρ_{1}) = \frac{2}{3},

because the totally real part ρ_{1}|_{π_{1}(Σ_{2})} does not contribute to the Toledo invariant. The proof of the proposition is complete.

Note that no representation ρ : π → PU(2, 1) with non-integer Toledo invariant can preserve a complex geodesic. The reason is that the stabilizer of a complex geodesic in PU(2, 1) is conjugate to SU(1, 1) and therefore canonically lifts to SU(2, 1).

Proposition 5.2. The image of ρ_{2} is a discrete subgroup of PU(2, 1).

Proof. The group ϕ(F_{2}) is contained in a (2, 4, 12)-Schwarz triangle group, which (up to conjugation) is commensurable with the arithmetic Fuchsian group G_{1} of matrices preserving the Hermitian form

z_{1}z̄_{1} − \sqrt{3}z_{2}z̄_{2}

whose entries lie in the ring 𝔒 = ℤ[\sqrt{3}, \sqrt{−1}]. Thus we adjust ϕ by a conjugation so that it maps F_{2} onto a subgroup of an arithmetic group commensurable with G_{1}.

Consider the Hermitian form

H(z) = z_{1}z̄_{1} + z_{2}z̄_{2} − \sqrt{3}z_{3}z̄_{3}

on ℂ^{3} and let G_{2} be the unitary group of H with entries from 𝔒. Then the group generated by G_{1} and θG_{1}θ^{−1} lies in a group commensurable with the projection of G_{2} to PU(2, 1). Since G_{2} is an arithmetic lattice in PU(2, 1), it is discrete. It follows that ρ_{2}(π) ⊂ G_{2} is a discrete subgroup of PU(2, 1).

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 Σ be a closed surface of genus g ≥ 2 which is split along a separating simple loop c into two subsurfaces Σ_{1} and Σ_{2} of genera g_{1} = g − 1 and g_{2} = 1, respectively. We pick singular hyperbolic surfaces T_{j} (j = 1, 2) of genus g_{j} with one cone point k_{j} of angle σ; such surfaces can be constructed by suitably identifying the sides of a regular hyperbolic 4g_{j}-gon with angles σ/4g_{j}. Since π_{1}(T_{j} − {k_{j}}) is a free group, the holonomy representation

hol_{j} : π_{1}(T_{j} − {k_{j}}) → PU(1, 1)

for the singular hyperbolic structure on T_{j} lifts to a representation

h̃ol_{j} : π_{1}(T_{j} − {k_{j}}) → SU(1, 1) ⊂ PU(2, 1)

Let W be the complex geodesic preserved by SU(1, 1). An element γ representing a small simple loop around k_{j} is carried by hol_{j} to a rotation in W by angle σ around a point o ∈ W. The lifted holonomy h̃ol_{j}(γ) acts as a rotation by angle π + σ/2 in 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 σ/2 or π + σ/2. To exclude the first possibility, we note that the hyperbolic cone structure of T_{j} can be continuously deformed so that σ increases to 2π. 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 σ to the special value 2π/3 and proceed as in the proof of Proposition 5.1. By combining h̃ol_{1} and h̃ol_{2} we construct a representation with Toledo invariant −χ(Σ) − 2/3. By combining h̃ol_{1} with the appropriate conjugate of the representation ϕ′ we get a representation with Toledo invariant −χ(Σ) − 4/3. Thus we can realize the extremal positive non-integer values

−χ(Σ) − 2/3, −χ(Σ) − 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 Σ onto a surface Σ′ of genus k + 1. Take representations ρ′ : Σ′ → PU(2, 1) with Toledo’s invariant 2k − 2/3 and 2k − 4/3 as constructed above and compose them with the induced map π_{1}(Σ) → π_{1}(Σ′) 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 H_{ℂ}^{2}.

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Department of Mathematics, University of Maryland, College Park, MD 20742 USA E-mail address: wmg@math.umd.edu (Goldman)

Department of Mathematics, University of Utah, Salt Lake City, Utah 84112 USA E-mail address: kapovich@math.utah.edu (Kapovich)

Fachbereich Mathematik, Johannes-Gutenberg Universität, D-55099 Mainz, Germany

E-mail address: leeb@mathematik.uni-mainz.de (Leeb)