MAXIMAL REPRESENTATIONS OF SURFACE GROUPS:

SYMPLECTIC ANOSOV STRUCTURES

MARC BURGER, ALESSANDRA IOZZI, FRANÇOIS LABOURIE, AND ANNA WIENHARD

Dedicated to the memory of Armand Borel, with affection and admiration

Abstract. Let G be a connected semisimple Lie group such that the associated symmetric space X is Hermitian and let Γ_{g} be the fundamental group of a compact orientable surface of genus g ≥ 2. We survey the study of maximal representations of Γ_{g} into G that is the subset of Hom(Γ_{g}, G) which is a union of components characterized by the maximality of the Toledo invariant ([16] and [14]). Then we concentrate on the particular case G = Sp(2n, ℝ), and we show that if ρ is any maximal representation then the image ρ(Γ_{g}) is a discrete, faithful realizations of Γ_{g} as a Kleinian group of complex motions in X with an associated Anosov system, and whose limit set in an appropriate compactification of X is a rectifiable circle.

Contents

1. Introduction 2 2. Hermitian Symmetric Spaces and Examples 5 3. The Toledo Invariant and Maximal Representations 10 4. Tube Type Subdomains and Maximal Representations 17 5. Tight Homomorphisms 21 6. Symplectic Anosov Structures 28 7. Bounded Cohomology at Use 30 8. Symplectic Anosov Structures: Proofs 34 References 46

Date: June 16, 2005. A.I. and A.W. were partially supported by FNS grant PP002-102765.

1. Introduction

Let Γ_{g} be the fundamental group of a compact orientable topological surface Σ_{g} of genus g ≥ 2. For a general real algebraic group G the representation variety Hom(Γ_{g}, G) is a natural geometric object which reflects properties both of the discrete group Γ_{g} and of the algebraic group G and enjoys an extremely rich structure. For example, Hom(Γ_{g}, G) is not only a topological space, but also a real algebraic variety, which in addition parametrizes flat principal G-bundles over Σ_{g}; furthermore, it admits an action of the group of automorphisms of Γ_{g} by precomposition which commutes with the action by postcomposition with inner automorphisms of G. It is natural to consider homomorphisms up to conjugation, thus we introduce

Rep(Γ_{g}, G) := Hom(Γ_{g}, G)/G;

although this is not necessarily a Hausdorff space, it contains a large part which is Hausdorff, namely the space Rep_{red}(Γ_{g}, G) of homomorphisms with reductive image modded out by G-conjugation. The general theme of this note is the study of certain connected components of Hom(Γ_{g}, G) or Rep(Γ_{g}, G) analogous to Teichmüller space, and their relation to geometric and dynamical structures on Σ_{g}.

Recall that if G = PU(1, 1), Rep(Γ_{g}, G) has 4g − 3 connected components ([33], [35]), two of which are homeomorphic to ℝ^{6g−6} and correspond to the two Teichmüller spaces T_{g} – one for each orientation of Σ_{g} – that is to the space of marked complex, alternatively hyperbolic, structures on the topological surface Σ_{g}.

If on the other hand G = SL(3, ℝ), Goldman and Choi proved [20] that Hom(Γ_{g}, G) has three connected components, one of which parametrizes convex projective structures on Σ_{g}, that is diffeomorphisms of Σ_{g} with Ω/Γ, where Γ < SL(3, ℝ) is a faithful discrete image of Γ_{g} and Ω ⊂ ℙ(ℝ^{3}) is a convex invariant domain.

If G = PSL(2, ℂ), there is an open subset of Rep(Γ_{g}, G) consisting of all quasi-Fuchsian deformations of Γ_{g}, which is diffeomorphic to the product T_{g} × T_{g} of two copies of Teichmüller space.

In each of these three cases, a representation belonging to such a “special component” in Rep(Γ_{g}, G) is faithful and with discrete image, and ρ(Γ_{g}) < G gives rise, as a Kleinian group, to many interesting dynamical and geometric structures.

When G is a simple split real Lie group, such as for instance G = PSL(n, ℝ), PSp(2n, ℝ), PO(n, n) or PO(n, n + 1), Hitchin, using Higgs bundle techniques, singled out a component Rep_{H}(Γ_{g}, G) of Rep(Γ_{g}, G) diffeomorphic to ℝ^{χ(Σ_{g}) dim G}, [41]. For example, if G = PSL(n, ℝ) for n ≥ 2, this component is the one containing the homomorphisms of Γ_{g} into SL(n, ℝ) obtained by composing a hyperbolization with the ndimensional irreducible representation of SL(2, ℝ). As Hitchin however points out, the analytic point of view does not shed any light on the geometric significance of this component Rep_{H}(Γ_{g}, G), from now on called Hitchin component.

Recently the concept of Anosov representation, which links the surface Σ_{g} to flag manifolds associated to G was introduced in [49] and used to show that, if G = PSL(n, ℝ), representations in the Hitchin component are discrete and faithful, and that they provide quasiisometric embeddings of Γ_{g} into G, [49], [50].

In parallel, Goncharov and Fock developed for surfaces with nonempty boundary and for the same class of Lie groups a tropical-algebro-geometric viewpoint of Rep(Γ_{g}, G), singling out positive real points in Rep(Γ_{g}, G) which correspond to discrete and faithful representations, [29], [28].

There is another natural extension of the case G = PU(1, 1) in a different direction, that is to connected semisimple Lie groups G such that the associated symmetric space X admits a G-invariant complex structure, just like in the case of the Poincaré disk. This includes notably groups like SU(p, q), Sp(2n, ℝ), SO^{∗}(2n), SO(2, n). Symmetric spaces with this property are called Hermitian.

In the same framework, the topology and the number of connected components of the space of reductive representations into SU(p, q) and Sp(2n, ℝ) have been studied in a series of papers by Bradlow, García-Prada, Gothen, Mundet i Riera and Xia ([7], [4], [6], [36], [31], [67]), extending the analytic approach introduced by Hitchin.

The additional feature for symmetric spaces which are Hermitian is the presence of a Kähler form ω_{X} on X which allows to associate to every representation ρ : Γ_{g} → G a characteristic number, called the Toledo invariant T_{ρ} (see § 3), which is constant on connected components of Hom(Γ_{g}, G) and which satisfies a Milnor–Wood type inequality

|T_{ρ}| ≤ |χ(Σ_{g})| rk_{X} , (1.1)

where rk_{X} is the real rank of X. A representation is maximal if equality holds in (1.1), and the set Hom_{max}(Γ_{g}, G) of such representations is then a union of components of Hom(Γ_{g}, G).

In the first part of this article we illustrate, mostly without proofs, results concerning the geometric significance of maximal representations. To fix the notation, let G := G(ℝ)^{∘}, where G is a semisimple real algebraic group and assume that the symmetric space X associated to G is Hermitian. In complete analogy with Goldman’s theorem, any maximal representation ρ : Γ_{g} → G is injective with discrete image (Theorem 4.6). This fact depends on a careful study of the Zariski closure L of the image ρ(Γ_{g}) and the fact that there is an essential restriction on L := L(ℝ)^{∘}, namely that it is reductive and it preserves a subHermitian symmetric space of X which is of tube type and maximal with respect to this property. On the constructive side, the study of maximal representations into G does not reduce to the study of classical Teichmüller space; in fact, if X is of tube type, any representation which is the composition of a hyperbolization Γ_{g} → SU(1, 1) with the homomorphism SU(1, 1) → G associated to the realization of the Poincaré disk diagonally in a maximal polydisk in X can be deformed into a representation with Zariski dense image in G (Theorem 4.7); such a representation is by construction maximal. For examples of discrete representations into SU(1, n) with prescribed Toledo invariant see [34].

These results are proven in greater generality in [14], where for the representation of the fundamental group of a surface with boundary, we define a Toledo invariant whose definition and properties however require some vigorous use of bounded cohomology. In the context of this paper, continuous bounded cohomology will appear as a tool in the proofs; in particular it allows to define the notion of tight homomorphism, more general and flexible than that of maximal representation, and which is an essential tool to study the geometric properties of the inclusion X_{L} ↪ X, where X_{L} is the subsymmetric space associated to L (see above). A systematic study of tight homomorphisms and the companion notion of tight embedding is the subject matter of [15] and a few highlights of this theory are presented in § 5.

While the first part of the paper is expository, in the second part we give an elementary treatment of a certain number of results on maximal representations into the symplectic group Sp(V) of a real symplectic vector space V. The results are stated in § 6 and their proofs in § 8 are independent of the rest of the paper (see § 8).

Observe at this point that Sp(V) is at the same time real split, and hence falls into the context of the Hitchin component, and is the group of automorphisms of the Siegel upper half space, a fundamental class of Hermitian symmetric spaces. We have the inclusion

Rep_{H} (Γ_{g}, Sp(V)) ⊂ Rep_{max} (Γ_{g}, Sp(V)) ,

but while the representations in the Hitchin component are all irreducible [49], there are (at least when dim V ≥ 4) components of maximal representations which contain reducible representations, so that the above inclusion is strict.

For a representation ρ : Γ_{g} → Sp(V) and a fixed hyperbolization Σ of Σ_{g}, we associate the flat symplectic bundle E^{ρ} over the unit tangent bundle T^{1}Σ of Σ with fiber V. The geodesic flow lifts canonically to a flow g_{t}^{ρ} on E^{ρ}; we adapt some of the ideas in [49] to our situation and, combining them with the results in § 5 and § 7, prove that if ρ is maximal then E^{ρ} is the sum of two continuous Lagrangian subbundles E_{+}^{ρ} ⊕ E_{−}^{ρ} on which g_{t}^{ρ} acts contracting and expanding respectively. Moreover, this bundle will also come with a field of complex structures in each fiber, exchanging E_{±}^{ρ} and positive for the symplectic structure (see § 5). As a consequence, one deduces that any maximal representation ρ : Γ_{g} → Sp(V) is a quasiisometry, where Sp(V) is equipped with a standard invariant metric. This implies that the action of the mapping class group Out(Γ_{g}) on Rep_{max} (Γ_{g}, Sp(V)) is properly discontinuous.

Acknowledgments: The authors thank Domingo Toledo for bringing to their attention [61] and Domingo Toledo and Nicolas Monod for useful comments on a preliminary version of the paper.

2. Hermitian Symmetric Spaces and Examples

Let X be a symmetric space and let G := Isom(X)^{∘} be the connected component of its group of isometries; in this paper we shall consider only symmetric spaces of noncompact type. Recall that X is Hermitian if it admits a G-invariant complex structure. An equivalent definition is that X is a Hermitian manifold such that every point x ∈ X is the isolated fixed point of an isometric involution s_{x}.

Let J : X → End(T X) be the G-invariant complex structure and let g : T X ×_{p} T X → ℝ be the Riemannian metric, where T X ×_{p} T X denotes the fibered product over the projection p : T X → X. Then

ω_{X} (X, Y) := g(JX, Y)

defines a G-invariant differential two-form on X which is nondegenerate.

Lemma 2.1. Let X be a symmetric space and G = Isom(X)^{∘}. Then any G-invariant differential form on X is closed.

Proof. Let α be a G-invariant differential k-form on X and let s ∈ Isom(X) be the geodesic symmetry at a basepoint 0 ∈ X. Since G is normal in Isom(X), then sgs^{−1} ∈ G and hence sα is also G-invariant, since

g(sα) = s^{2}g(sα) = s(sgs)α = sα .

Moreover, since s|_{T_{0}X} = −Id we have that (sα)_{0} = (−1)^{k}α_{0}, and since α and sα are both G-invariant, the equality (sα)_{x} = (−1)^{k}α_{x} holds for every x ∈ X. Hence since dα is also G-invariant, from

(−1)^{k}dα = d(sα) = s(dα) = (−1)^{k+1}dα

we deduce that dα = 0. □

The immediate consequence of the above lemma is that a Hermitian symmetric space X is a Kähler manifold with Kähler form ω_{X}. Furthermore, using the existence of a Kähler form on X, one can prove that for an irreducible symmetric space X being Hermitian is equivalent to the center of a maximal compact subgroup of Isom(X)^{∘} having positive dimension (and in fact being one-dimensional).

A fundamental result which makes the study of Hermitian symmetric spaces accessible to techniques from function theory à la Bergmann is the following theorem of Harish-Chandra which for classical domains is due to E. Cartan, [19].

Theorem 2.2 (Harish-Chandra, [39]). Any Hermitian symmetric space of noncompact type is biholomorphic to a bounded domain in a complex vector space.

The bounded realization D ⊂ ℂ^{N} of a Hermitian symmetric space X has a natural compactification, namely the topological closure \overline{D} in ℂ^{N}, on which G := Isom(X)^{∘} acts by restriction of birational isomorphism of ℂ^{N}. The Shilov boundary Š ⊂ ∂D of the bounded domain, which can be defined in function theoretical terms, is also the unique closed G-orbit in \overline{D}. It is a homogeneous space of the form G/Q, where Q is a (specific) maximal parabolic subgroup of G, and plays a prominent role in our study, for example as target of appropriate boundary maps. Notice that only if X is of real rank one, it coincides with the whole boundary ∂D.

Recall that the rank rk_{X} of a symmetric space X is the maximal dimension of a flat subspace, that is an isometric copy of Euclidean space.

Expositions of different aspects of the geometry of Hermitian symmetric spaces are [45], [22], [57], [56], and [66].

2.1. Examples of Hermitian Symmetric Spaces. We give here examples of two families of Hermitian symmetric spaces which are of fundamental nature and with which we shall illustrate our results.

2.1.1. SU(W). Let W be a complex vector space of dimension n, and h( · , ·) a nondegenerate Hermitian form of signature (p, q), p ≤ q, so that p is the maximal dimension of a subspace L ⊂ W on which the restriction h( · , ·)|_{L} is positive definite. A model for the symmetric space associated to

SU(W) := {g ∈ SL(V) : h(gx, gy) = h(x, y), ∀x, y ∈ V }

is

X_{SU(W)} := {L ∈ Gr_{p}(W) : h( · , ·)|_{L} is positive definite}

which, as an open subset of the Grassmannian Gr_{p}(W) of p-dimensional subspaces of W, is a complex manifold on which G acts by automorphisms. To realize X_{SU(W)} as a bounded domain, fix W_{+} ∈ X_{SU(W)}, and let W_{−} := W_{+}^{⊥} be its orthogonal complement with respect to the form h. Since the orthogonal projection p_{+} : W → W_{+} is an isomorphism when restricted to any L ∈ X_{SU(W)}, we can define

(2.1)  E : X_{SU(W)} → Lin(W_{+}, W_{−})

by

E(L) := pr_{−} ∘ (pr_{+}|_{L})^{−1} .

It is easy to see that this defines a biholomorphic map from X_{SU(W)} to the bounded domain

(2.2) D_{SU(W)} := {A ∈ Lin(W_{+}, W_{−}) : Id − A^{∗}A is positive definite}

where the adjoint is taken with respect to the structures of the unitary spaces (W_{+}, h( · , ·)|_{W_{+}}) and (W_{−}, −h( · , ·)|_{W_{−}}). Moreover the closure \overline{X}_{SU(W)} of X_{SU(W)} in Gr_{p}(W) is mapped by E to

\overline{D_{SU(W)}} = \{A ∈ Lin(W_{+}, W_{−}) : Id − A^{∗}A is positive semidefinite\} .

To determine the preimage of the Shilov boundary in the hyperboloid model X_{SU(W)}, observe that there are precisely (p + 1) orbits of SU(W) in \overline{X}_{SU(W)}, only one of which is closed, namely the Grassmannian of maximal isotropic subspaces

Is_{p}(W) := {L ∈ Gr_{p}(W) : h( · , ·)|_{L} = 0} ,

which is sent via E to the Shilov boundary

Š_{SU(W)} = \{A ∈ Lin(W_{+}, W_{−}) : Id − A^{∗}A = 0\} ⊂ \overline{D_{SU(W)}}

of the bounded domain D_{SU(W)}. The real rank of X_{SU(W)} is p.

Identifying W with ℂ^{p+q} in such a way that h is the standard Hermitian form of signature (p, q), we denote SU(p, q) := SU(W) and

D_{p,q} = \{Z ∈ M_{q,p}(ℂ) : Id −^{t}\overline{Z}Z is positive definite\}

the corresponding bounded domain with Shilov boundary

Š_{p,q} = \{Z ∈ M_{q,p}(ℂ) : Id −^{t}\overline{Z}Z ≡ 0\} ⊂ \overline{D_{p,q}} .

In particular D_{1,1} is the Poincaré disk.

2.1.2. The Symplectic Group. Let V be a real vector space equipped with a symplectic form ⟨ · , · ⟩, that is a nondegenerate antisymmetric bilinear form. In particular V must be even dimensional and we fix dim V = 2n. The group

Sp(V) := {g ∈ GL(V) : ⟨gx, gy⟩ = ⟨x, y⟩, ∀x, y ∈ V }

is the real symplectic group. The fact that on a complex vector space the imaginary part of a nondegenerate Hermitian form is a symplectic form for the underlying real structure suggests to introduce the space

X := {J ∈ GL(V) : J is a complex structure on V and

h_{J}(x, y) := ⟨x, Jy⟩ + i⟨x, y⟩ is a positive definite

Hermitian form on (V, J)} ,

so that, if J ∈ X, then ℜh_{J}(x, y) = ⟨x, Jy⟩ is a symmetric positive definite form. It is easy to see that, among complex structures on V, this property characterizes the elements of X. Furthermore, for the transitive action by conjugation of Sp(V) on X, the stabilizer of J is a maximal compact subgroup isomorphic to U(n) and hence X is the symmetric space X_{Sp(V)} associated to Sp(V); in particular, since the center of U(n) has positive dimension, X_{Sp(V)} is Hermitian symmetric and, as such, there is a Sp(V)-invariant complex structure on X_{Sp(V)} which one can explicit as follows.

Let V_{ℂ} be the complexification of V and let σ : V_{ℂ} → V_{ℂ} be the complex conjugation σ(x+iy) := x−iy for x, y ∈ V. Then there is a bijective correspondence between complex structures J on V and decompositions V_{ℂ} = W_{+} ⊕ W_{−} into complex subspaces satisfying σ(W_{±}) = W_{∓}, given by the eigenspace decomposition of J ⊗_{ℂ} 11 into ±i-eigenspaces. If now ⟨ · , · ⟩_{ℂ} is the complexification of ⟨ · , · ⟩, then

h(x, y) := i⟨x, σ(y)⟩_{ℂ}

is a nondegenerate Hermitian form of signature (n, n) on V_{ℂ}; if in particular J ∈ X_{Sp(V)}, then the restriction h|_{W_{+}×W_{+}} is positive definite, so that we obtain a map

\begin{matrix}X_{Sp(V)} & → X_{SU(V_{ℂ})} \\ J & ↦ W_{+} \end{matrix} (2.3)

which is equivariant with respect to the natural homomorphism

\begin{matrix}λ : Sp(V) & → SU(V_{ℂ}) \\ g & ↦ g ⊗_{ℂ} 11 . \end{matrix}

Since X_{SU(V_{ℂ})} inherits a natural complex structure as an open subset of the Grassmannian Gr_{n}(V_{ℂ}) of n-planes in V_{ℂ} and

M := {L ∈ Gr_{n}(V_{ℂ}) : ⟨ · , · ⟩_{ℂ}|_{L×L} ≡ 0}

is an algebraic subvariety of Gr_{n}(V_{ℂ}), then X_{SU(V_{ℂ})} ∩ M acquires a natural complex structure as an open subset of M. But

⟨ · , · ⟩_{ℂ}|_{W_{±}×W_{±}} ≡ 0

for all J ∈ X_{Sp(V)}, so that the map in (2.3) is a λ-equivariant bijection between X_{Sp(V)} and X_{SU(V_{ℂ})} ∩M by the use of which the complex structure on X_{SU(V_{ℂ})} ∩ M defines the Sp(V)-invariant complex structure on X_{Sp(V)}.

Let us denote by S_{Sp(V)} := X_{SU(V_{ℂ})} ∩ M the Siegel space associated to Sp(V) with its Sp(V)-action via the homomorphism λ.

To determine the bounded domain realization of X_{Sp(V)}, it is enough to observe that the Siegel space S_{Sp(V)} is mapped by the map E defined in (2.1) to the subdomain

D_{Sp(V)} := {A ∈ D_{SU(V_{ℂ})} : ⟨ · , A · ⟩_{ℂ}|_{W_{+}×W_{+}} is symmetric} ,

and, accordingly, \overline{S}_{Sp(V)} is mapped to

\overline{D_{Sp(V)}} := \{A ∈ \overline{D_{SU(V_{ℂ})}} : ⟨ · , A · ⟩_{ℂ}|_{W_{+}×W_{+}} is symmetric\} .

One can verify again that the closure \overline{S_{Sp(V)}} in Gr_{n}(V_{ℂ}) decomposes into (n + 1) orbits under the symplectic group, only one of which is closed, namely

Is_{n}(V_{ℂ}) ∩ \overline{S_{Sp(V)}} = \{L ∈ Gr_{n}(V_{ℂ}) : ⟨ · , · ⟩_{ℂ}|_{L×L} = 0 and h|_{L×L} ≡ 0\}

and hence is the preimage, under the map E in (2.1), of the Shilov boundary in the bounded domain realization of S_{Sp(V)}. To give an intrinsic description of the Shilov boundary observe that, since we have the alternative description

Is_{n}(V_{ℂ}) ∩ \overline{S_{Sp(V)}} = \{L ∈ Gr_{n}(V_{ℂ}) : ⟨ · , · ⟩_{ℂ}|_{L×L} = 0 and σ(L) = L\} ,

we conclude that the map

\begin{matrix}L(V) & → Is_{n}(V_{ℂ}) ∩ \overline{S_{Sp(V)}} \\ L & ↦ L ⊗ ℂ , \end{matrix}

where L(V) is the space of Lagrangians in V, is an Sp(V)-equivariant bijection. Thus the space of Lagrangian subspaces is as a Sp(V)- homogeneous space isomorphic to the Shilov boundary of the bounded domain D_{Sp(V)}.

If we identify V with the direct sum of n symplectic planes, that is copies of ℝ^{2} with the standard symplectic form, then accordingly we denote the symplectic group by Sp(2n, ℝ) and the Siegel space by S_{n}.

3. The Toledo Invariant and Maximal Representations

Let Σ_{g} be a compact oriented surface of genus g ≥ 2, G a connected semisimple Lie group with finite center and associated symmetric space X which we assume to be Hermitian, and ρ : Γ_{g} → G a homomorphism. Then there is a smooth Γ_{g}-equivariant map f̃: Σ̃_{g} → X, where Σ̃_{g} denotes the universal covering of Σ_{g}, which can be obtained by lifting a smooth section of the flat bundle Γ_{g}\(Σ̃_{g} × X ) → Σ_{g} with contractible fiber X. The pullback f̃^{∗}ω_{X} is then a Γ-invariant differential form on Σ̃_{g}, which hence descends to a form on Σ_{g}. Since any two such sections are homotopic and hence the map f̃ is unique up to ρ-equivariant homotopy, the result of the integration over Σ_{g} of any two forms obtained in this way does not depend on the particular choice of a section: we can hence define the Toledo invariant of ρ

T_{ρ} := \frac{1}{2π} ∫_{Σ_{g}} f̃^{∗}(ω_{X}) .

Normalizing the metric on X once and for all so that the minimal holomorphic sectional curvature is −1, we can summarize the properties of the Toledo invariant in the following

Proposition 3.1. There exists a nonzero rational number ℓ_{X} ∈ ℚ^{∗} such that the Toledo invariant T_{ρ} of a representation ρ : Γ_{g} → G has the following properties:

(1) T_{ρ} ∈ ℓ_{X} ℤ;

(2) the map T : Hom(Γ_{g}, G) → ℓ_{X} ℤ is constant on connected components of the representation variety, and

(3) |T_{ρ}| ≤ |χ(Σ_{g})| rk_{X}, where rk_{X} is the real rank of X.

Remark 3.2. (1) The constant ℓ_{X} can be explicitly computed from the restricted root system of the real Lie group G (see [14]). In fact, the metric of minimal holomorphic sectional curvature −1 on X is the ℓ_{X} -multiple of the Bergmann metric given by the bounded domain realization of X.

(2) The inequality in Proposition 3.1(3) is due to J. Milnor in the case G = SL(2, ℝ) [53] and to V. Turaev in the case G = Sp(2n, ℝ) [63].

We want to give now a very concrete interpretation of the Toledo invariant of a representation Γ_{g} → G in the case in which G = Isom(X)^{∘}, in terms of generators of Γ_{g}; this is very much in the spirit of Milnor’s formula for the Euler number (see [53]).

Let 0 ∈ X be a basepoint and K its stabilizer in G. We already alluded to the fact that the center of K is of positive dimension (see § 2). In fact, the ℂ-vector space structure on the tangent space T_{0}X gives an action of U(1) := {z ∈ ℂ : |z| = 1} which can be “integrated”, in the sense that there is a continuous homomorphism

u_{0} : U(1) → K (3.1)

such that for z ∈ U(1) the differential of the isometry of X defined by u_{0}(z) at 0 is the multiplication v ↦ zv for v ∈ T_{0}X. In particular, since K acts on T_{0}X faithfully by ℂ-linear maps, the image of u_{0} is in the center Z(K) of K and coincides with the center of K when X is irreducible.

Assume hence for the following discussion that X is irreducible. The homomorphism in (3.1) induces a homomorphism on the level of fundamental groups ℤ → π_{1}(K) = π_{1}(G) and hence an isomorphism

ℤ → π_{1}(G)/π_{1}(G)_{tor} .

Denoting by Ĝ the covering of G associated to π_{1}(G)_{tor}, we obtain a topological central extension

\xymatrix{0 \ar[r] & ℤ \ar[r] & Ĝ \ar[r] & G \ar[r] & (e) .}

The commutator map Ĝ×Ĝ → Ĝ factors via ℤ to give rise to a smooth map

\begin{matrix}G × G & → Ĝ \\ (a, b) & ↦ ˆ[a, b] . \end{matrix}

Given a standard presentation of Γ_{g},

Γ_{g} = ⟨a_{1}, b_{1}, …, a_{g}, b_{g} : ∏_{i=1}^{g} [a_{i}, b_{i}] = e⟩ ,

to any homomorphism ρ : Γ_{g} → G we can thus associate

∏_{i=1}^{g} { \atop [ρ(a_{i}), ρ(b_{i})]} ∈ ℤ

and the same argument as in [53], shows that

T_{ρ} = ℓ_{X} ∏_{i=1}^{g} { \atop [ρ(a_{i}), ρ(b_{i})]} .

As an immediate consequence we have that the Toledo invariant depends continuously on the representation and hence (1) and (2) of Proposition 3.1 follow at once. The proof of part (3) of the same proposition is more delicate; one efficient way to prove it uses the value of the simplicial area of Σ_{g} and the norm of the bounded Kähler class in bounded cohomology established by Domic and Toledo [23] and by Clerc and Ørsted [21], and will be explained in § 5.

If G = PU(1, 1) and X is the Poincaré disk, the constant ℓ_{X} in Proposition 3.1 equals 1 and hence for the Toledo invariant of a homomorphism

ρ : Γ_{g} → PU(1, 1)

we have that T_{ρ} ∈ ℤ and |T_{ρ}| ≤ 2g − 2. Thus T_{ρ} can achieve at most 4g − 3 values. In fact:

Theorem 3.3 (Goldman, [33], [35]). The fibers of

\begin{matrix}Hom (Γ_{g}, PU(1, 1)) & → \{−(2g − 2), …, 0, …, 2g − 2\} \\ ρ & ↦ T_{ρ} \end{matrix}

are exactly the connected components of Hom (Γ_{g}, PU(1, 1)). Moreover T_{ρ} = ±(2g −2) if and only if ρ is a hyperbolization, that is it is faithful with discrete image.

Let now G be any connected semisimple Lie group with finite center such that the associated symmetric space X is Hermitian. Inspired by Proposition 3.1(3) and by Goldman’s result, we give the following

Definition 3.4. We say that a representation ρ : Γ_{g} → G is maximal if T_{ρ} = ±χ(Σ_{g}) rk_{X}.

In the sequel we shall show that, in this degree of generality, maximal representations contain a remarkable amount of structure. Historically, the following result of H. Kneser seems to be the birth certificate of this theme.

Theorem 3.5 (Kneser, [44]). Let f : Σ_{g} → Σ_{h} be a continuous map of compact orientable surfaces, where h ≥ 2. If d_{f} is the degree of f we have

|d_{f}(h − 1)| ≤ |g − 1| ,

with equality if and only if f is homotopic to a covering map, necessarily of degree d_{f}.

G. Lusztig observed that this inequality is a consequence of Milnor’s inequality (see [25] for this remark as well as a proof using harmonic maps), by taking a hyperbolization

ρ : π_{1}(Σ_{h}) → PU(1, 1)

of Σ_{h} and considering the Toledo invariant of the composed homomorphism

ρ ∘ f_{∗} : π_{1}(Σ_{g}) → PU(1, 1) .

Actually, continuing along these lines and applying Goldman’s theorem characterizing maximal representations into PU(1, 1) leads to a proof of the equality case in Kneser’s theorem. The reader might find instructive to fill in the details.

3.1. Examples of Maximal Representations. It is a nontrivial and remarkable geometric fact that any maximal flat in a Hermitian symmetric space can be “complexified” hence leading to the existence of maximal polydisks.

Definition 3.6. A maximal polydisk in X is an isomorphic copy of (D_{1,1})^{r}, where r = rk_{X}.

The fact that maximal polydisks exist (and are all conjugate under Isom(X)^{∘}) [66, p. 280] is a crucial ingredient in some of the examples to follow and their generalization (see Theorem 4.7).

Example 3.7. The embedding

\begin{matrix}(D_{1,1})^{p} & → D_{p,q} \\ (z_{1}, …, z_{p}) & ↦ (\begin{matrix}z_{1} \\ . & . & . \\ & & z_{p} \\ 0 & … & 0 \\ ⋮ & & ⋮ \\ 0 & … & 0 \end{matrix} ) \end{matrix}

is isometric and holomorphic and hence defines a maximal polydisk P ⊂ D_{p,q} associated to the obvious homomorphism

τ_{P} : SU(1, 1)^{p} → SU(p, q) .

Example 3.8. Let ℝ^{2n} = ℝ^{2}⊕⋯⊕ℝ^{2} be the direct sum of n symplectic planes. Then the embedding

\begin{matrix}(X_{Sp(2,ℝ)})^{n} & → X_{Sp(2n,ℝ)} \\ (J_{1}, …, J_{n}) & ↦ (\begin{matrix}J_{1} \\ & . & . & . \\ & & & & J_{n} \end{matrix}) \end{matrix}

defines a maximal polydisk P ⊂ X_{Sp(2n,ℝ)} associated to the obvious homomorphism

τ_{P} : (Sp(2, ℝ))^{n} → Sp(2n, ℝ) .

Now we present some examples of maximal representations, though the proof of their maximality is not necessarily immediate at this point.

Example 3.9. Let X be any Hermitian symmetric space of rank r, and P ⊂ X a maximal polydisk with associated homomorphism

τ_{P} : SU(1, 1)^{r} → G := Isom(X)^{∘} .

Given now r orientation preserving hyperbolizations h_{1}, …, h_{r} : Γ_{g} → SU(1, 1), the representation

\begin{matrix}h : Γ_{g} & ⟶ SU(1, 1)^{r} \\ γ & ↦ (h_{1}(γ), …, h_{r}(γ)) \end{matrix}

as well as the composition

τ_{P} ∘ h : Γ_{g} → G

are maximal.

Example 3.10. Let h : Γ_{g} → SL(2, ℝ) be an orientation preserving hyperbolization and let ρ_{2n} : SL(2, ℝ) → SL(2n, ℝ) be the 2n-dimensional irreducible representation of SL(2, ℝ). Since ρ_{2n}(SL(2, ℝ)) preserves the standard symplectic form on ℝ^{2n}, we obtain a representation

ρ_{2n} ∘ h : Γ_{g} → Sp(2n, ℝ)

which is maximal. Observe that such a representation ρ_{2n} ∘ h is in the Hitchin component Rep_{H} (Γ_{g}, Sp(2n, ℝ)) (see § 1).

The fact that τ_{P} ∘h and ρ_{2n} ∘h are maximal depends on the property of τ_{P} and ρ_{2n} being a “tight homomorphism”, a concept to which we shall return in § 5.

Example 3.11. Because of Proposition 3.1(2), any deformation of any of the above representations is maximal. In particular, all representations in Rep_{H} (Γ_{g}, Sp(2n, ℝ)) are maximal.

Example 3.12. Let Σ_{g} be a compact oriented surface of genus g ≥ 2. Our objective is to give an example of maximal representation with Zariski dense image in Sp(4, ℝ) constructed via an explicit deformation of the representation in Example 3.9. This was triggered by a comment of Toledo, who pointed out that a disk diagonally embedded into a polydisk does not determine uniquely the polydisk.

To this end, write Σ_{g} as the sum of two surfaces Σ_{A}, Σ_{B} identified along a separating simple closed loop γ on which we choose a basepoint p and realize Γ_{g} as an amalgamated product Γ_{A} ∗_{⟨γ⟩} Γ_{B} of Γ_{A} := π_{1}(Σ_{A}, p) and Γ_{B} := π_{1}(Σ_{B}, p) over the infinite cyclic subgroup ⟨γ⟩.

PSfrag replacements Σ_{A} Σ_{A} γ

Figure 1.

Let h : Γ_{g} → PSp(2, ℝ)^{2} be defined by h(γ) := (h_{1}(γ), h_{2}(γ)), where h_{1}, h_{2} : Γ_{g} → PSp(2, ℝ) are two inequivalent hyperbolizations, and let us choose some lift to Sp(2, ℝ)^{2}, denoted again by h with a small abuse of notation,

\begin{matrix}h : Γ_{g} & → Sp(2, ℝ)^{2} \\ γ & ↦ (h_{1}(γ), h_{2}(γ)) . \end{matrix}

We shall assume that:

(∗) h(γ) ∈ Δ, where Δ is the diagonal of Sp(2, ℝ)^{2}, and (∗∗) the restriction of the two hyperbolic structures to Σ_{A} and to

Σ_{B} are inequivalent, and we denote by

\begin{matrix}h_{A} : Γ_{A} & → Sp(2, ℝ)^{2} \\ α & ↦ (h_{1,A}(α), h_{2,A}(α)) \end{matrix}

and

\begin{matrix}h_{B} : Γ_{B} & → Sp(2, ℝ)^{2} \\ β & ↦ (h_{1,B}(β), h_{2,B}(β)) \end{matrix}

the restrictions of h to Γ_{A} and Γ_{B} respectively.

Let now ℝ^{4} = ℝ^{2} ⊕ ℝ^{2} be the sum of two standard symplectic real planes (as in 2.1.2), so that

Sp(4, ℝ) = \{g = (\begin{matrix}A & B \\ C & D \end{matrix}) ∈ GL(4, ℝ) : ^{t}g (\begin{matrix}J & 0 \\ 0 & J \end{matrix}) g = (\begin{matrix}J & 0 \\ 0 & J \end{matrix})\}

where J = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}). Consider the homomorphism

\begin{matrix}τ_{P} : Sp(2, ℝ)^{2} & → Sp(4, ℝ) \\ (A, D) & ↦ (\begin{matrix}A & 0 \\ 0 & D \end{matrix}) . \end{matrix}

associated to the maximal polydisk

P = \{(\begin{matrix}J_{1} & 0 \\ 0 & J_{2} \end{matrix}) ∈ X_{Sp(4,ℝ)} : , J_{1}, J_{2} are complex structures on ℝ^{2}\} .

Then the centralizer Z in Sp(4, ℝ) of the image τ_{P}(Δ) of the diagonal is

Z = \{(\begin{matrix}aId_{2} & bId_{2} \\ cId_{2} & dId_{2} \end{matrix}) : (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ O(2)\} .

Denoting by Int(z) the conjugation by z ∈ Z, the homomorphisms τ_{P} ∘ h_{A} and Int(z) ∘ τ_{P} ∘ h_{B} coincide on ⟨γ⟩ (see (∗)). Thus by the universal property of amalgams, there is a unique homomorphism

ρ_{z} : Γ_{g} → Sp(4, ℝ)

whose restriction to Γ_{A} is τ_{P} ∘ h_{A} and to Γ_{B} is Int(z) ∘ τ_{P} ∘ h_{B}.

Proposition 3.13. With the above notations:

(1) For every z ∈ Z^{∘} the representation ρ_{z} : Γ_{g} → Sp(4, ℝ) is maximal, and

(2) if z = (\begin{matrix}aId_{2} & bId_{2} \\ cId_{2} & {I d \atop .}_{2} \end{matrix}) ∈ Z satisfies ac ≠ 0, then ρ_{z}(Γ_{g}) is Zariski

dense in Sp(4, ℝ).

Proof. (1) If z = Id, ρ_{Id} is the composition of h : Γ_{g} → Sp(2, ℝ)^{2} with τ_{P} : Sp(2, ℝ)^{2} → Sp(4, ℝ). The latter homomorphism is associated to an embedding realizing (X_{Sp(2,ℝ)})^{2} as a maximal polydisk in the Siegel space X_{Sp(4,ℝ)} and hence ρ_{Id} is maximal (see Example 3.9). Every z ∈ Z^{∘} can be connected to Id by a continuous path, thus ρ_{z} is in the same component of Hom (Γ_{g}, Sp(4, ℝ)) as ρ_{Id} and thus maximal.

(2) It follows from (∗∗) that the image ρ_{z}(Γ_{g}) is Zariski dense in the algebraic group L < Sp(4, ℝ) generated by P := τ_{P}(Sp(2, ℝ)^{2}) and zP z^{−1}. Now the condition on z guarantees that the Lie algebra 𝔭 of P is strictly contained in the Lie algebra 𝔩 of L. But it is easily verified that the representation of Sp(2, ℝ)^{2} on the Lie algebra 𝔰𝔭(4, ℝ) of Sp(4, ℝ) obtained by composing τ_{P} with the adjoint representation is a sum of 𝔭 and the irreducible representation in dimension 4, tensor product of the standard 2-dimensional representation of Sp(2, ℝ) with itself. This implies that 𝔩 = 𝔰𝔭(4, ℝ) and hence proves the proposition. □

Example 3.14. A smooth fiber bundle over a surface Σ_{g} with typical fiber Σ_{n} leads, via the monodromy representation on the first homology group of the fiber, to a representation

ρ : Γ_{g} → Sp(2n, ℤ) .

For such representations, D. Kotschick [46] showed that

|T_{ρ}| ≤ \frac{1}{2}|χ(Σ_{g})|(n − 1) ,

and in particular ρ is far from being maximal. On the other hand D. Toledo has given examples of maximal representations into Sp(4n, ℤ) for all n ≥ 1, [61].

4. Tube Type Subdomains and Maximal Representations

Let G be a semisimple real algebraic group with associated symmetric space X of Hermitian type. In view of Goldman’s theorem, a basic question concerning a maximal representation ρ : Γ_{g} → G is whether it is faithful with discrete image. In addition, when X is not the Poincaré disk, ρ(Γ_{g}) cannot be a lattice in G and thus in this general setting there is the question of determining how “large” the image of ρ can be. Concerning the latter question, it is natural to turn one’s attention to the Zariski closure L of ρ(Γ_{g}). While in the preceding examples we have seen that L(ℝ) can be a product Sp(2, ℝ)^{r}, or more interestingly Sp(4, ℝ), it turns out however that there are restrictions on L and that moreover the determination of these restrictions is an essential step in order to answer the question about faithfulness and discreteness of ρ.

To this purpose, an instructive special case is the family of Hermitian symmetric spaces of real rank one, that is the complex hyperbolic spaces D_{1,q}. This case, beyond q = 1, was examined by Toledo and in order to state his result we recall that a complex geodesic is an isomorphic copy of D_{1,1} in D_{1,q}; equivalently, complex geodesics are obtained by taking the exponential of a complex line in T_{x}D_{1,q}, for x ∈ D_{1,q}; they constitute the maximal polydisks in D_{1,q}. Theorem 4.1 (Toledo, [62]). Any maximal representation ρ : Γ_{g} → PU(1, q) stabilizes a complex geodesic.

Since the stabilizer in PU(1, q) of a complex geodesic is, modulo a compact kernel, isomorphic to PU(1, 1), Goldman’s theorem [33] applies, and thus ρ is basically obtained via a hyperbolization of Σ_{g}.

The proof of Toledo’s theorem is very much in the spirit of the Gromov–Thurston proof of Mostow rigidity theorem and uses notably ℓ^{1}-homology and smearing. (Incidentally, ℓ^{1}-homology will play a role also in our treatment of the Milnor–Wood type inequality in Proposition 3.1(3) as described in § 5). A special case of Theorem 4.1 was already proven by Toledo in [60] using harmonic mappings techniques. In the same spirit, taking up Hitchin’s approach via Higgs bundles, Bradlow, García-Prada and Gothen made a comprehensive study of the topology of the connected components of Hom_{red} (Γ_{g}, PU(p, q)) and obtained in particular for the maximal representations the following Theorem 4.2 (Bradlow–García-Prada–Gothen, [4]). Assume that 1 ≤ p ≤ q. If ρ : Γ_{g} → PU(p, q) is maximal and reductive, then its image is contained in P(U(p, p) × U(q − p)) up to conjugation.

This result had been previously obtained by L. Hernández in the case p = 2, [40]. Moreover, an analogous result has been proven by Bradlow, García-Prada and Gothen for SO^{∗}(2n), if n is odd [5].

An equivalent way of stating the theorem asserts that ρ(Γ_{g}) preserves a Hermitian symmetric subspace of D_{p,q} conjugate to D_{p,p}. In order to understand the situation in general, the relevant concept here is the one of tube type domain. For instance, the Hermitian symmetric space associated to PU(1, 1) has a realization as upper half plane but, unlike the bounded domain realization, this type of realization is not available for all Hermitian symmetric spaces. Definition 4.3. A Hermitian symmetric space X is of tube type if it is biholomorphic to a domain

{u + iv : u ∈ V, v ∈ Ω} ⊂ V ⊕ iV

where V is a real vector space and Ω ⊂ V is a proper open cone.

Example 4.4. The space D_{p,q} associated to SU(p, q) is of tube type if and only if p = q, in which case it is biholomorphic to

Herm_{p}(ℂ) + i Herm_{p}^{+}(ℂ) ,

where Herm_{p}(ℂ) is the real vector space of Hermitian matrices and Herm_{p}^{+}(ℂ) is the open cone of positive definite ones. The biholomorphism is given explicitly by restricting to D_{p,p} the Cayley transformation

\begin{matrix}C : M_{p,p}(ℂ) & → M_{p,p}(ℂ) \\ Z & ↦ \frac{i(Id + Z)}{(Id − Z)} . \end{matrix}

The Cayley transformation C sends the real Zariski open subset of the Shilov boundary Š_{p,p} of D_{p,p} consisting of matrices Z ∈ M_{p,p}(ℂ) such that det(I − Z) ≠ 0 bijectively into Herm_{p}(ℂ). Considering Herm_{p}^{+}(ℂ) as an open cone in the tangent space of Herm_{p}(ℂ) at 0, and taking its image under the differential at 0 of C^{−1}, one obtains a cone Ω in the tangent space T_{−Id} Š_{p,p} which is invariant under the stabilizer of −Id in SU(p, p). Translating this cone by the SU(p, p)-action, one obtains a smooth family of open cones Ω_{x} ⊂ T_{x}Š_{p,p}, that is a causal structure on the Shilov boundary, which is SU(p, p)-invariant.

Example 4.5. Under the same Cayley transformation, the bounded domain realization of the Siegel space is sent biholomorphically to

Sym_{n}(ℝ) + i Sym_{n}^{+}(ℝ) ,

where Sym_{n}(ℝ) denotes the vector space consisting of real symmetric n×n matrices, and realizes the symmetric space associated to Sp(2n, ℝ) as a tube type domain.

The open cone Sym^{+}(ℝ) defines in the same way as above an Sp(V)- invariant causal structure on L(V) which will be used in the proof of Corollary 6.3.

The examples above serve to illustrate the general fact that the Shilov boundary of a bounded symmetric domain of tube type admits an invariant causal structure. Among bounded symmetric domains, this property characterizes those of tube type. The general classification of Hermitian symmetric spaces relative to the notion of tube type is as follows:

Tube Type Non-Tube Type
SU(p, p)  SU(p, q)
p < q
Sp(2n, ℝ)
SO^{∗}(2n)  SO^{∗}(2n)
n even  n odd
SO(2, n)
E_{7}(−25)  E_{6}(−14)

where E_{7}(−25) and E_{6}(−14) correspond to the exceptional Hermitian symmetric spaces of complex dimension 27 and 16 respectively.

An essential feature of a Hermitian symmetric space of rank rk_{X} is that (holomorphically embedded) maximal Hermitian subdomains of tube type always exist, are of rank rk_{X} and are all conjugate.

Notice that in the rank one case, that is for complex hyperbolic n-space, the notion of maximal tube type subdomain and maximal polydisk coincide. This ambiguity left open the correct generalization of Toledo’s theorem until the construction of a maximal representation with Zariski dense image in a tube type domain [16], of which Example 3.12 is a particular case.

We can finally state the structure theorem for maximal representations.

Theorem 4.6 ([16], [14]). Let G be a connected real algebraic group and assume that the symmetric space X_{G} associated to G := G(ℝ)^{∘} is Hermitian symmetric. Let ρ : Γ_{g} → G be a maximal representation. Then:

(1) Γ_{g} acts properly discontinuously, via ρ, on X_{G};

(2) the Zariski closure L of ρ(Γ_{g}) is reductive;

(3) the connected component L := L(ℝ)^{∘} stabilizes a maximal tube type subdomain T ⊂ X_{G};

(4) the symmetric space X_{L} associated to L is Hermitian of tube type and the isometric embedding X_{L} ↪ T is tight.

We illustrate the above theorem in the examples of § 3. For the notion of tight embedding see § 5.

Example. 3.9 The orientation preserving hyperbolizations h_{1}, …, h_{r} fall into s equivalence classes modulo SU(1, 1)-equivalence, with 1 ≤ s ≤ r; the group L is then isomorphic to SU(1, 1)^{s} and X_{L} is a product of s-subdiagonals in the maximal polydisk P. Any diagonal disk in P determines the same maximal tube type domain T and X_{L} ⊂ P ⊂ T. The first statement of the theorem is obvious in this example.

Example. 3.10 In this case L ≅ Sp(2, ℝ), T = X_{Sp(2n,ℝ)}, and X_{L} ⊂ X_{Sp(2n,ℝ)} is a totally geodesically embedded disk, holomorphic if and only if n = 1. Item (1) of the theorem is obvious also in this case.

Example. 3.12 Under the conditions of Proposition 3.13(2), the maximal representation

ρ_{z} : Γ_{g} → Sp(4, ℝ)

has Zariski dense image, so L = Sp(4, ℝ) and X_{L} = T = X_{Sp(4,ℝ)} by construction. On the other hand, Theorem 4.6(1) implies that ρ_{z} is injective with discrete image, a fact that in this case is not at all obvious from the construction.

As alluded to earlier, Example 3.12 is a particular case of a general fact which we now state. Let r = rk_{X}, let P ⊂ X be a maximal polydisk and

τ_{P} : SU(1, 1)^{r} → Isom(X)^{∘} .

the associated homomorphism.

Theorem 4.7 ([16], [14]). Assume that X is of tube type and let ρ_{0} : Γ_{g} → Isom(X)^{∘} be the maximal representation obtained by composing a hyperbolization of Γ_{g} → SU(1, 1) with the diagonal embedding of SU(1, 1) in SU(1, 1)^{r} followed by τ_{P}. Then ρ_{0} admits a continuous deformation ρ_{t} : Γ_{g} → Isom(X)^{∘}, for t ≥ 0, such that ρ_{t}(Γ_{g}) is Zariski dense in Isom(X)^{∘} for t > 0.

Observe that ρ_{t}, being a continuous deformation of a maximal representation, is maximal as well.

5. Tight Homomorphisms

A fundamental role in the study of maximal representations of surface groups is played by tight homomorphisms, which generalize maximal representations of surface groups, in that it is a notion defined for any continuous homomorphism of a locally compact group into the group of isometries of a Hermitian symmetric space.

The definition of tight homomorphism rests on basic concepts in bounded continuous cohomology which we briefly recall; for a comprehensive treatment see [55] and [18]. We start with the more familiar concept of continuous group cohomology. For a locally compact group G, its continuous cohomology H_{c}^{•}(G, ℝ) is the cohomology of the complex (C(G^{•}, ℝ)^{G}, d^{•}) of G-invariant real valued continuous cochains, where d^{•} is the usual homogeneous coboundary. The bounded continuous cohomology H_{cb}^{•}(G, ℝ) is then the cohomology of the subcomplex (C_{b}(G^{•}, ℝ)^{G}, d^{•}) of G-invariant bounded continuous cochains. The complex (C_{b}(G^{•}, ℝ)^{G}, d^{•}) equipped with the supremum norm is a complex of Banach spaces with continuous coboundary operators, and hence H_{cb}^{•}(G, ℝ) is endowed with a quotient seminorm. Also, the inclusion of the complex of bounded continuous functions into the one of continuous functions gives rise to a comparison map

c_{G}^{•} : H_{cb}^{•}(G, ℝ) → H_{c}^{•}(G, ℝ) (5.1)

which encodes subtle properties of G of geometric and algebraic nature. See [1], [32], [54], [17], [18, § V.13], [13], and also [8], [9], [37], [59], [2], [27], [30], [26], [3], [48], [47] in relation with the existence of quasimorphisms.

If now G is a connected semisimple Lie group with finite center and associated symmetric space X, we have seen that the complex (Ω^{•}(X)^{G}, d^{•}) of G-invariant differential forms on X coincides with its cohomology (see Lemma 2.1) and, in fact, there is a canonical isomorphism [64]

H_{c}^{•}(G, ℝ) ≅ Ω^{•}(X)^{G} .

Let us now specialize to the case of interest to us, namely when X is Hermitian symmetric and ω_{X} ∈ Ω^{2}(X)^{G} is its Kähler form. A continuous cocycle defining the class κ_{X} ∈ H_{c}^{2}(G, ℝ) corresponding to ω_{X} is

c_{X} (g_{1}, g_{2}, g_{3}) = ∫_{Δ(g_{1}x,g_{2}x,g_{3}x)} ω_{X} ,

where x ∈ X is a basepoint and Δ(g_{1}x, g_{2}x, g_{3}x) is any smooth twosimplex with geodesic sides and vertices g_{1}x, g_{2}x, g_{3}x.

There is a general conjecture of Dupont to the extent that cocycles obtained by integrating G-invariant differential forms (of any degree) should be bounded, [24]. In terms of the comparison map, this suggests the following

Question. Let G be a connected semisimple Lie group with finite center. Is the comparison map (5.1) surjective in all degrees?

This turns out to be true for forms representing specific classes (see [38], [24], [58], [10], [51]) and in particular for the Kähler form was first shown by Dupont [24]. In fact, with the assumed normalization, one has the sharp bound

∥c_{X} ∥_{∞} = π rk_{X} (5.2) due to Domic and Toledo for classical domains [23] and to Clerc and Ørsted in the general case [21]. Thus c_{X} defines a continuous bounded class κ_{X}^{b} ∈ H_{cb}^{2}(G, ℝ) to which we shall refer to the bounded Kähler class; and for which one has the following theorem (see also Proposition 7.4):

Theorem 5.1 (Domic–Toledo [23], Clerc–Ørsted [21]). If the metric on X is normalized to have minimal holomorphic sectional curvature −1, then

∥κ_{X}^{b} ∥ = π rk_{X} .

Given now locally compact groups H and G, any continuous homomorphism ρ : H → G induces canonical pullbacks ρ^{•} and ρ_{b}^{•} respectively in continuous and bounded continuous cohomology, by precomposition of continuous (bounded) cochains with ρ; the resulting linear maps have the property that the diagram

\xymatrix{H_{cb}^{•}(G, ℝ) \ar[r]^{ρ_{b}^{•}} \ar[d]_{c_{G}^{•}} & H_{cb}^{•}(H, ℝ) \ar[d]^{c_{H}^{•}} \\ H_{c}^{•}(G, ℝ) \ar[r]^{ρ^{•}} & H_{c}^{•}(H, ℝ)}

(5.3)

commutes, and moreover the pullback in bounded continuous cohomology is norm decreasing, namely for all α ∈ H_{cb}^{n}(G, ℝ),

∥ρ_{b}^{(n)}(α)∥ ≤ ∥α∥ .

Definition 5.2 ([15], [65]). Let G be a connected semisimple group with finite center and such that the associated symmetric space X is Hermitian, and let H be any locally compact group. A continuous homomorphism ρ : H → G is tight if it preserves the norm of the bounded Kähler class, that is if

∥ρ_{b}^{(2)}(κ_{X}^{b} )∥ = ∥κ_{X}^{b} ∥ .

To motivate this definition, we sketch a proof of the inequality in Proposition 3.1(3). Since Σ_{g} is a K(Γ_{g}, 1), we have in particular a canonical isomorphism

H^{2}(Γ_{g}, ℝ) → H^{2}(Σ_{g}, ℝ)

which, if κ_{X} ∈ H_{c}^{2}(G, ℝ) allows us to see ρ^{(2)}(κ_{X}) ∈ H^{2}(Γ_{g}, ℝ) as a singular class in H^{2}(Σ_{g}, ℝ) and evaluate it on the fundamental class [Σ_{g}] ∈ H_{2}(Σ_{g}, ℝ) of Σ_{g}; recall that Σ_{g} is oriented once and for all. Then if

⟨ · , · ⟩ : H^{2}(Σ_{g}, ℝ) × H_{2}(Σ_{g}, ℝ) → ℝ

denotes the pairing, analogously to the classical case of the Euler number, we have

T_{ρ} = \frac{1}{2π} ⟨ρ^{(2)}(κ_{X}), [Σ_{g}]⟩ .

The proof of the Milnor–Wood type inequality in Proposition 3.1(3) will follow from the interpretation of this invariant in bounded cohomology. To this purpose, following Gromov [38], recall that the ℓ^{1}-homology of Σ_{g} is the homology H_{•,ℓ^{1}}(Σ_{g}, ℝ) of the complex of singular ℓ^{1}-chains, while the bounded cohomology H_{b}^{•}(Σ_{g}, ℝ) is the cohomology of the dual Banach space complex; consequently, ℓ^{1}-homology and bounded cohomology acquire quotient seminorms and there is the canonical pairing

⟨ · , · ⟩_{b} : H_{b}^{n}(Σ_{g}, ℝ) × H_{n,ℓ^{1}}(Σ_{g}, ℝ) → ℝ

which satisfies the property that for all α ∈ H_{b}^{n}(Σ_{g}, ℝ) and all a ∈ H_{n,ℓ^{1}}(Σ_{g}, ℝ)

|⟨α, a⟩_{b}| ≤ ∥α∥ ∥a∥_{ℓ^{1}} .

These notions have been introduced by Gromov for any topological space X and one has the Gromov–Brooks canonical isometric isomorphism (see [38] and [8])

H_{b}^{•}(π_{1}(X), ℝ) ≅ H_{b}^{•}(X, ℝ) ,

a rather deep fact depending on higher homotopy groups being Abelian and hence amenable. In our situation one can explicitly write an isometric isomorphism

H_{b}^{2}(Γ_{g}, ℝ) ≅ H_{b}^{2}(Σ_{g}, ℝ) (5.4)

compatible with the isomorphism in ordinary cohomology, by choosing a hyperbolic metric on Σ_{g} and using the technique of straightening simplices.

Starting now with the bounded Kähler class κ_{X}^{b} ∈ H_{cb}^{2}(G, ℝ), and applying the pullback in bounded cohomology and the isomorphism (5.4), we obtain the class ρ_{b}^{(2)}(κ_{X}^{b} ) ∈ H_{b}^{2}(Σ_{g}, ℝ) which corresponds, using (5.3), to ρ^{(2)}(κ_{X}) ∈ H^{2}(Σ_{g}, ℝ) under the comparison map in singular cohomology

H_{b}^{2}(Σ_{g}, ℝ) → H^{2}(Σ_{g}, ℝ) .

This latter being the dual of the natural map

H_{2}(Σ_{g}, ℝ) → H_{2,ℓ^{1}}(Σ_{g}, ℝ), (5.5) we have that

⟨ρ^{(2)}(κ_{X}), [Σ_{g}]⟩ = ⟨ρ_{b}^{(2)}(κ_{X}^{b} ), [Σ_{g}]_{ℓ^{1}}⟩_{b} ,

where [Σ_{g}]_{ℓ^{1}} denotes the image of [Σ_{g}] under (5.5). Thus

|T_{ρ}| ≤ \frac{1}{2π} ∥ρ_{b}^{(2)}(κ_{X}^{b} )∥ ∥[Σ_{g}]∥_{ℓ^{1}} .

Recall now that the ℓ^{1}-norm ∥[Σ_{g}]∥_{ℓ^{1}} of the fundamental class is the simplicial area of Σ_{g} and, by [38],

∥[Σ_{g}]∥_{ℓ^{1}} = 4g − 4 .

This, together with the norm decreasing property of the pullback in bounded cohomology and the value of the norm of the Kähler class in Theorem 5.1, implies on the one hand the inequality in Proposition 3.1(3) and on the other the following

Proposition 5.3. Any maximal representation is a tight homomorphism.

The following general result about tight homomorphisms, together with the above proposition, implies part of Theorem 4.6.

Theorem 5.4 ([15], [65]). Let G be a semisimple real algebraic group, H a locally compact group, and assume that the symmetric space X_{G} associated to G := G(ℝ)^{∘} is Hermitian. Then for a tight homomorphism ρ : H → G the following holds:

(1) The Zariski closure L of the image ρ(H) is reductive;

(2) the real reductive group L := L(ℝ)^{∘} has compact centralizer in G; and

(3) the symmetric space X_{L} ⊂ X_{G} associated to L is Hermitian.

Notice that the totally geodesic embedding X_{L} ⊂ X_{G} in (3) is not necessarily holomorphic. However, there is a notion of tight embedding for Hermitian symmetric spaces which parallels the one for homomorphisms.

Definition 5.5 ([15], [65]). Given a totally geodesic embedding

f : Y → X

of Hermitian symmetric spaces, we say that f is tight if

sup_{Δ⊂Y} ∫_{Δ} f^{∗}ω_{X} = sup_{Δ⊂X} ∫_{Δ} ω_{X}

where the supremum is taken over all smooth triangles with geodesic sides.

This corresponds for the associated homomorphism

ρ : H_{Y} → Isom(X)^{∘} , (5.6)

where H_{Y} is an appropriate finite covering of Isom(Y)^{∘}, to be a tight homomorphism. With this terminology, the inclusion X_{L} ⊂ X_{G} in Theorem 5.4(3) is a tight embedding.

Here are some examples of tight embeddings:

Example 5.6. The homomorphism τ_{P} : SU(1, 1)^{r} → X associated to a maximal polydisc P ⊂ X is tight; evidently, the embedding P ⊂ X is both holomorphic and tight.

Example 5.7. The irreducible representation ρ_{2n} : SL(2, ℝ) → Sp(2n, ℝ) is tight, and the associated totally geodesic embedding of H_{ℝ}^{2} → X_{Sp(2n,ℝ)} is a tight embedding which is holomorphic only if n = 1.

Example 5.8. The embedding T ⊂ X of a maximal tube type subdomain in X is tight and holomorphic.

Example 5.9. If Y is an irreducible Hermitian symmetric space and f : Y → X is a totally geodesic embedding, then f is tight if and only if

f^{∗}ω_{X} = ±\begin{matrix}rk & X \\ rk & Y \end{matrix} ω_{Y} .

Example 5.10. The embedding X_{Sp(V)} → X_{SU(V_{ℂ})} in Example 2.1.2 is tight and holomorphic.

Observe now the following simple

Proposition 5.11. Let H, G be connected semisimple Lie groups with finite center and associated symmetric spaces of Hermitian type. If ρ : Γ_{g} → H is maximal and ρ′ : H → G is tight, then ρ′ ∘ ρ is maximal.

This, together with Examples 5.6 and 5.7 above justifies the maximality of the representations in Examples 3.9 and 3.10.

Notice that in general totally geodesic embeddings between bounded symmetric domains do not induce maps between the corresponding Shilov boundaries even if they are holomorphic. This is however something else that tight homomorphism can provide for us, namely

Theorem 5.12 ([15], [65]). Let X , Y be Hermitian symmetric spaces and f : Y → X a tight embedding with associated homomorphism ρ : H_{Y} → Isom(X)^{∘} (see (5.6)). Then there exists a ρ-equivariant map

f̌: Š_{Y} → Š_{X} .

Remark that, since the Shilov boundary of a Hermitian symmetric space is a homogeneous space, if such ρ-equivariant map exists, it is unique (up to translations).

The above theorem allows us also to deduce in great generality the existence of boundary maps for tight homomorphisms. Let Λ be a countable discrete group and let θ be a probability measure on Λ. Recall that a Poisson boundary of the pair (Λ, θ) is a measurable Λspace B with a probability measure ν such that there exists an isometric isomorphism between the space of bounded θ-harmonic functions

H^{∞}(Λ, θ) := {f : Λ → ℝ : f(g) = ∫_{Λ} f(gh)dθ(h), ∀g ∈ Λ}

and the space L^{∞}(B, ν), given by the Poisson formula

f(g) = ∫_{B} ψ(gx)dν(x) . (5.7)

Although we shall not need it here, we recall that, under natural assumptions on the measure θ, a Poisson boundary in fact exists even for locally compact second countable groups, [43].

An immediate consequence of the Poisson formula (5.7) is that the measure ν is θ-stationary, that is θ∗ν = ν. Moreover, it will be essential for our purposes that the action of Λ on the Poisson boundary B is amenable with respect to the measure ν, [68].

Theorem 5.13. Let Λ be a countable discrete group with probability measure θ and let G be a semisimple real algebraic group such that the symmetric space X associated to G := G(ℝ)^{∘} is Hermitian. If (B, ν) is a Poisson boundary for (Λ, θ) and ρ : Λ → G is a tight homomorphism, then there exists a ρ-equivariant measurable map

φ : B → Š_{X} .

Proof. Let L be the Zariski closure of ρ(Λ). By Theorem 5.4 the symmetric space Y associated to L := L(ℝ)^{∘} is Hermitian symmetric and the embedding Y → X is tight, so that Theorem 5.12 implies the existence of a ρ-equivariant map f̌ between the corresponding Shilov boundaries

f̌: Š_{Y} → Š_{X} . (5.8)

Let Q < L be a maximal parabolic subgroup defined over ℝ such that Š_{Y} ≅ L(ℝ)/Q(ℝ), and let P < Q be a minimal parabolic subgroup defined over ℝ contained in Q, so that we have an equivariant map

L(ℝ)/P(ℝ) ↠ L(ℝ)/Q(ℝ) ≅ Š_{Y} . (5.9)

Since the action of Λ on (B, ν) is amenable, there exists a ρ-equivariant measurable map

φ_{0} : B → M^{1}(L(ℝ)/P(ℝ))

where M^{1}(L(ℝ)/P(ℝ)) denotes the space of probability measure on L(ℝ)/P(ℝ). Since ρ : Λ → L has Zariski dense image, the Λ-action on L(ℝ)/P(ℝ) is mean proximal (see [13, Theorem 7.3]): this, together with the fact that ν is θ-stationary, implies that for ν-a. e. b ∈ B, φ_{0}(b) is a Dirac measure, thus providing a map

φ_{0} : B → L(ℝ)/P(ℝ)

which composed with the maps in (5.8) and (5.9) provides the required ρ-equivariant map. □

As an application, given a compact surface group Γ_{g}, choose a hyperbolization of Σ_{g} and let Γ be the realization of Γ_{g} as a cocompact lattice in PU(1, 1). Then Γ acts naturally on 𝕊^{1} = ∂D_{1,1} and, in fact, a theorem of Furstenberg asserts that there exists a probability measure θ on Γ such that 𝕊^{1} with the Lebesgue measure λ is a Poisson boundary of (Γ, θ).

Corollary 5.14. Let G be a semisimple real algebraic group such that the symmetric space X associated to G := G(ℝ)^{∘} is Hermitian and let ρ : Γ → G be a tight embedding of a cocompact lattice Γ < PU(1, 1). Then there exists a ρ-equivariant measurable map

φ : 𝕊^{1} → Š_{X} .

6. Symplectic Anosov Structures

We focus in this section on maximal representations into a symplectic group Sp(V). Let thus ρ : Γ_{g} → Sp(V) be any representation. We choose a hyperbolization Σ of Σ_{g}, and let Γ < PU(1, 1) = Aut(D_{1,1})^{∘} be the resulting realization of Γ_{g}. From now on we consider ρ as a representation of Γ. The geodesic flow g̃_{t} on the unit tangent bundle T^{1}D_{1,1} gives rise to a flow g̃_{t}^{ρ} on the total space of the flat symplectic bundle Ẽ^{ρ} := T^{1}D_{1,1} × V over T^{1}D_{1,1} commuting with the diagonal Γaction given by γ(u, x) := (γu, ρ(γ)x) which hence descends to a flow g_{t}^{ρ} on the quotient E^{ρ} := Γ\(T^{1}D_{1,1} × V) which is a flat symplectic bundle over the unit tangent bundle T^{1}Σ. The projection

p : E^{ρ} → T^{1}Σ (6.1) is then equivariant with respect to the g_{t}^{ρ}-action on E^{ρ} and to the action of the geodesic flow g_{t} on T^{1}Σ.

Let ⟨ · , · ⟩ : E^{ρ} ×_{p} E^{ρ} → ℝ be the symplectic form on E^{ρ}. A positive complex structure of the symplectic bundle is a continuous section

J : T^{1}Σ → End(E^{ρ})

such that (1) J_{u} is a complex structure on the fiber E^{ρ}(u), and

(2) the form ⟨ · , J· ⟩ is symmetric and positive definite in each fiber. We denote by ∥ · ∥ : E^{ρ} → ℝ_{+} the resulting Euclidean norm, and by ∥ · ∥_{u} its value on the fiber E^{ρ}(u) above the point u ∈ T^{1}Σ.

Observe that any symplectic bundle over a paracompact base admits a positive complex structure. A Lagrangian subbundle of a symplectic bundle is a subbundle such that each fiber is a Lagrangian subspace. With this terminology we have then the following

Theorem 6.1. Assume that ρ : Γ → Sp(V) is a maximal representation. Then there is a g_{t}^{ρ}-invariant splitting

E^{ρ} = E_{−}^{ρ} ⊕ E_{+}^{ρ}

into continuous Lagrangian subbundles, and there exist a positive complex structure J and a constant A > 0 such that

(1) J interchanges E_{−}^{ρ} and E_{+}^{ρ}, and

(2) for all t ≥ 0,

∥g_{t}^{ρ}ξ∥ ≤ e^{−At}∥ξ∥ for all ξ ∈ E_{+}^{ρ}

and

∥g_{−t}^{ρ}ξ∥ ≤ e^{−At}∥ξ∥ for all ξ ∈ E_{−}^{ρ} .

This result has interesting consequences on the metric properties of a maximal representation. To describe them, as well as for convenience in the proofs in § 8, we specify a left invariant metric on the symmetric space X_{Sp(V)} associated to Sp(V). Recall that X_{Sp(V)} is the set of complex structures J on V such that ⟨ · , J· ⟩ is symmetric and positive definite. Denoting by q_{J} the corresponding Euclidean norm on V, and by ∥Id∥_{J_{1},J_{2}} the norm of the identity map between (V, q_{J_{1}}) and (V, q_{J_{2}}), we set

d(J_{1}, J_{2}) := | ln ∥Id∥_{J_{1},J_{2}} | + | ln ∥Id∥_{J_{2},J_{1}} | J_{1}, J_{2} ∈ X_{Sp(V)} .

Of course, this distance is equivalent to the G-invariant Riemannian distance on X_{Sp(V)}, but it is more convenient for our purposes.

The statement of next corollary does not depend on the choice of a hyperbolization.

Corollary 6.2. Let ρ : Γ_{g} → Sp(V) a maximal representation, J ∈ X_{Sp(V)} a basepoint and ℓ the word length on Γ_{g}. Then the orbit map

\begin{matrix}ρ_{J} : Γ_{g} & → X_{Sp(V)} \\ γ & ↦ ρ(γ)J \end{matrix}

is a quasiisometry, that is there are constants A, B > 0 such that for every γ ∈ Γ

A^{−1}ℓ(γ) − B ≤ d(ρ(γ)J, J) ≤ Aℓ(γ) + B .

Essential in the proof of Theorem 6.1 is the existence of the boundary map obtained in Corollary 5.14 from the boundary 𝕊^{1} = ∂D_{1,1} of the Poincaré disk into the space of Lagrangians L(V) which relates the Maslov cocycle (see § 7) to the orientation cocycle on 𝕊^{1}. A priori this map is only measurable, but as a consequence of the continuity of the splitting in Theorem 6.1, it turns out to be continuous. In fact, this map plays a role analogous to the one of hyperconvex curves in the study of the Hitchin component of Hom (Γ_{g}, SL(n, ℝ)) in [49].

Corollary 6.3. Let ρ : Γ → Sp(V) be a maximal representation. Then there is a ρ-equivariant continuous injective map

φ : 𝕊^{1} → L(V)

with rectifiable image.

7. Bounded Cohomology at Use

The definition of continuous bounded cohomology in § 5 is not very useful from a practical point of view, as many natural cocycles of geometric origin are not continuous. The homological algebra approach developed in [18], [55], [12] and [11] allows us to overcome this obstacles in the usual way: as in the homological algebra approach to continuous cohomology, there are appropriate notions of coefficients modules, of relatively injective modules and of strong resolutions, that is resolutions with an appropriate homotopy operator. The underlying philosophy is that we need not restrict to the standard resolution in § 5, but any resolution satisfying certain conditions will suffice to compute the bounded cohomology in a completely canonical way. More specifically, the prominent role played by proper actions in the case of continuous cohomology is played by amenable actions in the case of bounded continuous cohomology.

Theorem 7.1. Let G be a locally compact second countable group and (S, ν) a regular amenable G-space. Then the continuous bounded cohomology of G is isometrically isomorphic to the cohomology of the complex

\xymatrix{0 \ar[r] & L_{alt}^{∞}(S, ℝ)^{G} \ar[r]^{d} & L_{alt}^{∞}(S^{2}, ℝ)^{G} \ar[r]^{d} & …}

with the usual homogeneous coboundary operator.

Here L_{alt}^{∞}(S^{n}, ℝ) denotes the subspace of L^{∞}(S^{n}, ℝ) consisting of functions such that f(s) = sign(σ)f(σ(s)) for all s ∈ S^{n} and σ any permutation of the coordinates.

Without getting into the details of the amenability of an action (for which we refer the reader to [69]), let us mention that the action of a group Λ on the Poisson boundary (B, ν) relative to a probability measure θ is amenable, as well as the action of a connected semisimple Lie group G on the quotient G/P by a minimal parabolic subgroup P < G. So, for example, the action of a surface group Γ_{g} on 𝕊^{1} via a hyperbolization is amenable, but if X is a Hermitian symmetric space the action of Isom(X)^{∘} on the Shilov boundary Š_{X} is not, unless the symmetric space has real rank one.

If in addition to being amenable the action of G on (S, ν) is mixing, that is the diagonal action on (S × S, ν × ν) is ergodic, then any Ginvariant measurable function on S × S must be essentially constant, and hence L_{alt}^{∞}(S^{2}, ℝ)^{G} = 0.

Corollary 7.2. Let G be a locally compact second countable group and (S, ν) a regular amenable mixing G-space. If ZL_{alt}^{∞}(S^{3}, ℝ) denotes the subspace of cocycles in L_{alt}^{∞}(S^{3}, ℝ), then we have a canonical isometric isomorphism

H_{cb}^{2}(G, ℝ) ≅ ZL_{alt}^{∞}(S^{3}, ℝ)^{G} .

Example 7.3. Since the Γ_{g}-action on 𝕊^{1} is amenable and mixing, then

H_{b}^{2}(Γ_{g}, ℝ) ≅ ZL_{alt}^{∞}((𝕊^{1})^{3}, ℝ)^{Γ_{g}} .

Likewise if G is a connected semisimple Lie group and P < G is a minimal parabolic, then

H_{cb}^{2}(G, ℝ) ≅ ZL_{alt}^{∞}((G/P)^{3}, ℝ)^{G} .

On the one hand this shows immediately that in degree two continuous bounded cohomology is a Banach space, on the other it allows us to represent bounded cohomology classes via meaningful cocycles defined on boundaries.

From now on we shall apply these considerations to the symplectic group G = Sp(V); for ease of notation, set dim V = 2n. Following Kashiwara [52, § 1.5], we recall that the Maslov index β_{n} of three Lagrangians L_{1}, L_{2}, L_{3} ∈ L(V) is defined as the index β_{n}(L_{1}, L_{2}, L_{3}) ∈ ℕ of the quadratic form

\begin{matrix}L_{1} ⊕ L_{2} ⊕ L_{3} & ⟶ ℝ \\ (x_{1}, x_{2}, x_{3}) & ↦ ⟨x_{1}, x_{2}⟩ + ⟨x_{2}, x_{3}⟩ + ⟨x_{3}, x_{1}⟩ . \end{matrix}

The function β_{n} : L(V)^{3} → ℕ is a cocycle which takes integer values in the interval [−n, n]; more specifically, on the space L(V)^{(3)} of triples of Lagrangians which are pairwise transverse, its set of values is {−n, −n+ 2, …, n − 2, n}, and each fiber of β_{n} is precisely an open Sp(V)-orbit. Remark also that β_{1} is nothing but the orientation cocycle on 𝕊^{1}.

The space F(V) of complete isotropic flags is a homogeneous space of Sp(V) with a minimal parabolic subgroup as stabilizer, and therefore the Sp(V)-action on F(V) is amenable. Let

pr : F(V) → L(V)

be the projection

pr ({0} ⊊ V_{1} ⊊ ⋯ ⊊ V_{p}) := V_{p} .

With these notations we have

Proposition 7.4. The map

β_{n} ∘ pr^{3} : F(V)^{3} → ℕ (7.1)

is a bounded Sp(V)-invariant alternating cocycle such that π(β_{n} ∘ pr^{3}) corresponds to the bounded Kähler class κ_{Sp(V)}^{b} ∈ H_{cb}^{2}(Sp(V, ℝ) under the isometric isomorphism in Corollary 7.2. In particular

∥κ_{Sp(V)}^{b}∥ = ∥π(β_{n} ∘ pr^{3})∥_{∞} = π n .

Of course the drawback of the acquired freedom in going from continuous functions to L^{∞} functions – or, more specifically, function classes – is that now the implementation of the pullback of a bounded cohomology class cannot be done mindlessly as before, since pullbacks even via continuous maps do not define, in general, a well defined equivalence class of measurable functions. However, the situation is much simpler in our case, given that our class admits as a representative the Borel function in (7.1) for which the cocycle identity holds everywhere. The following important result is a particular case of a general phenomenon for which we refer the reader to [12].

Theorem 7.5. Let Γ_{g} → Sp(V) be a homomorphism, and assume that there exists a ρ-equivariant measurable map

φ : 𝕊^{1} → L(V) ,

where Γ_{g} acts on 𝕊^{1} via a hyperbolization. Then the pullback

ρ_{b}^{(2)}(κ_{Sp(V)}^{b}) ∈ H_{cb}^{2}(Γ_{g}, ℝ)

is represented by the cocycle π(β_{n} ∘ φ^{3}) : (𝕊^{1})^{3} → ℝ defined by

(x, y, z) ↦ πβ_{n}(φ(x), φ(y), φ(z)) .

Now we succeeded in implementing the pullback in a rather effective way, but we find ourselves in the infinite dimensional Banach space H_{b}^{2}(Γ_{g}, ℝ). To size things down again, we shall need to make use of the transfer map.

Choose a hyperbolization of Σ_{g} and let as before Γ be the realization of Γ_{g} as a cocompact lattice in PU(1, 1). Inspired by Example 7.3 and by the fact that

H_{cb}^{2}(PU(1, 1), ℝ) ≅ ZL_{alt}^{∞}((𝕊^{1})^{3}, ℝ)^{PU(1,1)} ,

define a transfer map

t : L^{∞}((𝕊^{1})^{3}, ℝ)^{Γ} → L^{∞}((𝕊^{1})^{3}, ℝ)^{PU(1,1)}

by

tf(x, y, z) := ∫_{Γ\PU(1,1)} f(gx, gy, gz) dμ(g) ,

where μ is the PU(1, 1)-invariant probability measure on Γ\PU(1, 1). Since by Proposition 7.4

H_{cb}^{2}(Sp(V), ℝ) ≅ ℝ · (β_{n} ∘ pr^{3})

and

H_{cb}^{2}(PU(1, 1), ℝ) ≅ ℝ · β_{1} ,

composition of the pullback implemented as in Theorem 7.5 followed by the transfer map in cohomology

\xymatrix{H_{cb}^{2}(Sp(V), ℝ) \ar[r]^{ρ_{b}^{(2)}} & H_{b}^{2}(Γ, ℝ) \ar[r]^{t^{(2)}} & H_{cb}^{2}(PU(1, 1), ℝ)}

(7.2)

implies that there exists a constant c ≥ 0 such that for almost all x, y, z ∈ 𝕊^{1}

∫_{Γ\PU(1,1)} β_{n}(φ(gx), φ(gy), φ(gz)) dμ(g) = cβ_{1}(x, y, z) . (7.3)

An analogous composition of maps as in (7.2) in ordinary cohomology and their interplay via the comparison map which for Sp(V) and PU(1, 1) are isomorphisms [18], allow us to explicit the constant c in (7.3) as explained in [42, § 3] in the context of Matsumoto’s theorem.

Theorem 7.6. Let ρ : Γ_{g} → Sp(V) be a homomorphism, Γ < PU(1, 1) a hyperbolization of Γ_{g}, and assume that there exists a ρ-equivariant measurable map φ : 𝕊^{1} → L(V). Then for almost every x, y, z ∈ 𝕊^{1}

(7.4) ∫_{Γ\PU(1,1)} β_{n}(φ(gx), φ(gy), φ(gz)) dμ(g) = \frac{T_{ρ}}{|χ(Σ_{g})|}β_{1}(x, y, z) .

Observe that if either ρ(Γ) is Zariski dense or ρ is tight, such a measurable Γ-equivariant map exists. The following corollary is then immediate from Theorem 5.13 and Theorem 7.6.

Corollary 7.7. Let ρ : Γ → Sp(V) be a maximal representation. Then there exists a ρ-equivariant measurable map φ : 𝕊^{1} → L(V) and it satisfies

β_{n}(φ(x), φ(y), φ(z)) = nβ_{1}(x, y, z)

for almost every x, y, z ∈ 𝕊^{1}.

8. Symplectic Anosov Structures: Proofs

In this section we prove the results stated in § 6. These proofs rest entirely on Corollary 7.7 and are otherwise independent of the machinery used to establish Corollary 7.7.

8.1. The Geometry of Triples of Lagrangians. Here we collect a few basic facts about the Maslov cocycle. Our reference is [52, § 1.5].

The space L(V)^{(3)} of triples of pairwise transverse Lagrangians decomposes as a union ⊔_{j=0}^{n}O_{n−2j} of (n + 1) open Sp(V)-orbits such that O_{n−2j} is the level set of β_{n} where β_{n} takes the value n − 2j.

The maximal value n is special in that, if L_{1}, L_{2}, L_{3} are not pairwise transverse, then |β_{n}(L_{1}, L_{2}, L_{3})| < n, [52, Proposition 1.5.10]. Thus we observe that

(8.1)

if β_{n}(L_{1}, L_{2}, L_{3}) = ±n,
then L_{1},L_{2}, L_{3} are pairwise transverse.

Given L_{1}, L and L_{3} with L_{1} and L transverse to L_{3}, consider the linear map T_{13} : L_{1} → L_{3} defined by

L = {ℓ_{1} + T_{13}(ℓ_{1}) : ℓ_{1} ∈ L_{1}}

and the quadratic form Q_{L}^{L_{1},L_{3}} : L_{1} → ℝ defined by

Q_{L}^{L_{1},L_{3}}(x) := ⟨x, T_{13}x⟩ .

PSfrag replacements ℓ_{1} ℓ_{3}

ℓ_{1} + ℓ_{3} L_{1} L_{2} L_{3} 0

Figure 2. Let L_{1} and L_{2} be transverse to L_{3}. Then the vector ℓ_{3} ∈ L_{3} is the image of the vector ℓ_{1} ∈ L_{1} under the isomorphism T_{13} : L_{1} → L_{3} defined by L_{2}. The value at ℓ_{1} of the quadratic form Q_{L}^{L_{1},L_{3}} on L_{1} based at L_{3} and induced by L_{2} measures the signed area of the parallelogram with vertices 0, ℓ_{1}, ℓ_{1} + ℓ_{3}, ℓ_{3}. Moreover, if also L_{1} and L_{2} are transverse, then Q_{L}^{L_{1},L_{3}}(ℓ_{1}) = −Q_{L}^{L_{3},L_{1}}(ℓ_{3}), where ℓ_{1} and ℓ_{3} are related as above.

Let now

t(L_{3}) := {L ∈ L(V) : L ∩ L_{3} = {0}}

and let Q(L_{1}) be the space of quadratic forms on L_{1}. Then we have a diffeomorphism

\begin{matrix}t(L_{3}) & → Q(L_{1}) \\ L & ↦ Q_{L}^{L_{1},L_{3}} \end{matrix} (8.2)

and moreover (see [52, Lemma 1.5.4])

β_{n}(L_{1}, L, L_{3}) = sign (Q_{L}^{L_{1},L_{3}}) . (8.3)

If τ := (L_{1}, L_{2}, L_{3}) is a triple of pairwise transverse Lagrangians, we have an endomorphism J(τ) of V = L_{1} ⊕ L_{3} given in block form by

J(τ) := (\begin{matrix}0 & −T_{31} \\ T_{13} & 0 \end{matrix}) (8.4)

which, since J(τ)^{2} = −Id, defines a complex structure on V; moreover ⟨ · , J(τ)· ⟩ is symmetric and the associated quadratic form q_{J(τ)} has signature 2β_{n}(L_{1}, L_{2}, L_{3}) = sign (Q_{L_{2}}^{L_{1},L_{3}}) − sign (Q_{L_{2}}^{L_{3},L_{1}}).

In particular if L(V)_{max}^{3} denotes the set of triples τ for which β_{n}(τ) = n, we obtain as in (8.4) an Sp(V)-equivariant map

\begin{matrix}L(V)_{max}^{3} → & X_{Sp(V)} \\ τ ⟼ & J(τ) \end{matrix} (8.5)

into the symmetric space X_{Sp(V)} associated to Sp(V). This has the property that the positive quadratic form q_{J(τ)} on V is the orthogonal direct sum of Q_{L_{2}}^{L_{1},L_{3}} on L_{1} and Q_{L_{2}}^{L_{3},L_{1}} on L_{3} (see Figure 2).

Definition 8.1. We say that a quadruple τ′ of Lagrangians is maximal if β(τ) = n for any subtriple of Lagrangians τ taken in the same cyclic order as in τ′.

In particular (8.1) implies that a maximal quadruple consists of pairwise transverse Lagrangians. Finally we have the following monotonicity property:

Lemma 8.2. Assume that L_{0}, L_{1}, L_{2}, L_{∞} is maximal. Then

0 < Q_{L_{1}}^{L_{0},L_{∞}} < Q_{L_{2}}^{L_{0},L_{∞}}

and

Q_{L_{1}}^{L_{∞},L_{0}} < Q_{L_{2}}^{L_{∞},L_{0}} < 0 .

Proof. For ℓ_{0} ∈ L_{0}, let ℓ_{∞}, ℓ′_{∞} ∈ L_{∞} with ℓ_{0}+ℓ_{∞} ∈ L_{1} and ℓ_{0}+ℓ′_{∞} ∈ L_{2}. Then

Q_{L_{2}}^{L_{0},L_{∞}}(ℓ_{0}) − Q_{L_{1}}^{L_{0},L_{∞}}(ℓ_{0}) = ⟨ℓ_{0}, ℓ′_{∞} − ℓ_{∞}⟩
= ⟨ℓ_{0} + ℓ_{∞}, ℓ′_{∞} − ℓ_{∞}⟩
= Q_{L_{2}}^{L_{1},L_{∞}}(ℓ_{0} + ℓ_{∞}) ,

where the last equality follows from the fact that (ℓ_{0} + ℓ_{∞}) ∈ L_{1}, ℓ′_{∞} −ℓ_{∞} ∈ L_{∞}, and their sum ℓ_{0} +ℓ′_{∞} ∈ L_{2}. Maximality of (L_{1}, L_{2}, L_{∞}) implies that Q_{L_{2}}^{L_{1},L_{∞}} > 0, and maximality of (L_{0}, L_{1}, L_{∞}) implies that Q_{L_{1}}^{L_{0},L_{∞}} > 0. Hence the assertion. □

Notice that in the proof of Lemma 8.2 what was used is exactly the fact that the Lagrangians L_{0}, L_{1}, L_{2}, L_{∞} are pairwise transverse and that the triples (L_{0}, L_{1}, L_{∞}) and (L_{1}, L_{2}, L_{∞}) are maximal, which however, via the cocycle identity for β_{n}, is equivalent to the maximality of the quadruple (L_{0}, L_{1}, L_{2}, L_{∞}) (see the proof of Lemma 8.4).

8.2. Proofs of the Results in § 6. Let ρ : Γ → Sp(V) be a maximal representation and let φ : 𝕊^{1} → L(V) be the ρ-equivariant measurable map given by Corollary 7.7. Paramount in the study of regularity properties of the map φ is the closer analysis of its essential graph which we now introduce. Let λ be the Lebesgue measure on 𝕊^{1}. The essential graph E_{φ} of φ is the closed subset E_{φ} ⊂ 𝕊^{1} × L(V) which is the support of the pushforward of the measure λ under the map

\begin{matrix}𝕊^{1} & → 𝕊^{1} × L(V) \\ x & ↦ (x, φ(x)) . \end{matrix}

Here and in the sequel we shall often use the observation that

(8.6) for almost every x ∈ 𝕊^{1}, (x, φ(x)) ∈ E_{φ} .

Lemma 8.3. Let (x_{1}, L_{1}), (x_{2}, L_{2}), (x_{3}, L_{3}) ∈ E_{φ}, and assume that (1) x_{1}, x_{2}, x_{3} are pairwise distinct, and

(2) L_{1}, L_{2}, L_{3} are pairwise transverse. Then β_{n}(L_{1}, L_{2}, L_{3}) = nβ_{1}(x_{1}, x_{2}, x_{3}) .

Proof. We may assume that β_{1}(x_{1}, x_{2}, x_{3}) = 1. Using that (x_{i}, L_{i}) ∈ E_{φ}, Corollary 7.7 and the definition of essential graph imply that we may find sequences L_{i}^{(k)}, i = 1, 2, 3 such that β_{n}(L_{1}^{(k)}, L_{2}^{(k)}, L_{3}^{(k)}) = n and (L_{1}^{(k)}, L_{2}^{(k)}, L_{3}^{(k)}) converges to (L_{1}, L_{2}, L_{3}). In particular, (L_{1}, L_{2}, L_{3}) is in the closure \overline{O_{n}} in L(V)^{3} of O_{n}. Since on the other hand this triple belongs to ⊔_{j=0}^{n}O_{n−2j}, observing that O_{k} ∩ \overline{O_{n}} = ∅ for k ≠ n, we conclude that (L_{1}, L_{2}, L_{3}) ∈ O_{n}. □

Notice now that any two (distinct) points x_{1}, x_{2} ∈ 𝕊^{1} determine an interval in 𝕊^{1}, by defining

((x_{1}, x_{2})) := {t ∈ 𝕊^{1} : β_{1}(x_{1}, t, x_{2}) = 1} .

Lemma 8.4. Let (x_{1}, L_{1}) and (x_{2}, L_{2}) ∈ E_{φ} with x_{1} ≠ x_{2}. Then L_{1} and L_{2} are transverse.

Proof. Using Corollary 7.7 and (8.6) twice, we may choose a ∈ ((x_{1}, x_{2})) such that (a, φ(a)) ∈ E_{φ} and φ(a) is transverse to L_{1}, L_{2}, and choose b ∈ ((x_{2}, x_{1})) so that (b, φ(b)) ∈ E_{φ} and φ(b) is transverse to φ(a), L_{1}, L_{2}.

Applying the cocycle property of β_{n} to the quadruple φ(a), L_{2}, φ(b), L_{1}, we have that

\begin{matrix}β_{n}(L_{2}, φ(b), L_{1}) − β_{n}(φ(a), φ(b), L_{1}) \\ + & β_{n}(φ(a), L_{2}, L_{1}) − β_{n}(φ(a), L_{2}, φ(b)) = 0 ; \end{matrix}

since it follows from Lemma 8.3 that

β_{n}(φ(a), φ(b), L_{1}) = n = β_{n}(φ(a), L_{2}, φ(b)) ,

we obtain that

β_{n}(L_{2}, φ(b), L_{1}) + β_{n}(φ(a), L_{2}, L_{1}) = 2n ,

which implies in turn that

β_{n}(L_{2}, φ(b), L_{1}) = β_{n}(φ(a), L_{2}, L_{1}) = n .

It follows hence from (8.1) that L_{1} and L_{2} are transverse. □

From Lemmas 8.3 and 8.4 we deduce the following

Corollary 8.5. For (x_{1}, L_{1}), (x_{2}, L_{2}), (x_{3}, L_{3}) ∈ E_{φ} with (x_{1}, x_{2}, x_{3}) pairwise distinct, we have

β_{n}(L_{1}, L_{2}, L_{3}) = nβ_{1}(x_{1}, x_{2}, x_{3}) .

For the following, it will be convenient to define for A ⊂ 𝕊^{1} the “image of A” by E_{φ}

F_{A} := {L ∈ L(V) : there exists a ∈ A such that (a, L) ∈ E_{φ}}

which is closed if A ⊂ 𝕊^{1} is so. Now let us fix any two distinct points x, y ∈ 𝕊^{1}.

Lemma 8.6. The sets \overline{F}_{((y,x))} ∩ F_{\{x\}} and \overline{F}_{((x,y))} ∩ F_{\{x\}} both consist of one point.

Proof. Assume that there are L_{0}, L′_{0} ∈ \overline{F}_{((x,y))} ∩F_{\{x\}} and fix L_{∞} ∈ F_{\{y\}}. By hypothesis, there are sequences (x_{n}, L_{n}) and (x′_{n}, L′_{n}) in E_{φ} with

(1) x_{n}, x′_{n} ∈ ((x, y)), and lim x_{n} = lim x′_{n} = x;

(2) lim L_{n} = L_{0} and lim L′_{n} = L′_{0} . By Lemma 8.4 all L_{n} and L′_{n} are transverse to L_{∞} and we may thus use the diffeomorphism

\begin{matrix}t(L_{∞}) & → Q(L_{0}) \\ L & ↦ Q_{L}^{L_{0},L_{∞}} \end{matrix}

and study the situation in the model Q(L_{0}). Dropping the superscript L_{0}, L_{∞}, we have that lim Q_{L_{n}} = Q_{L_{0}} = 0. For every k ≥ 1, there is N(k) such that x′_{n} ∈ ((x, x_{k})) for all n ≥ N(k), and consequently L_{0}, L′_{n}, L_{k}, L_{∞} is maximal; using Lemma 8.2, this implies that

Q_{L_{0}} = 0 ≤ Q_{L′_{n}} ≤ Q_{L_{k}}

and hence lim_{n} Q_{L′_{n}} = 0. This shows that lim_{n} L′_{n} = L_{0} and hence L′_{0} = L_{0}. □

According to Lemma 8.6, for every x ∈ 𝕊^{1} define

φ_{+}(x) ∈\overline{F}_{((y,x))} ∩ F_{\{x\}} and φ_{−}(x) ∈ \overline{F}_{((x,y))} ∩ F_{\{x\}} .

From the definitions one deduces immediately the following Corollary 8.7. The maps

φ_{+}, φ_{−} : 𝕊^{1} → L(V)

defined above are respectively left and right continuous and strictly Γ-equivariant.

Now we turn to our symplectic bundle E^{ρ} introduced in § 6 and the study of the properties of the flow g_{t}^{ρ}. To define the Lagrangian splitting of E^{ρ} we parametrize T^{1}D_{1,1} by the set (𝕊^{1})^{(3)} of distinct triples of points on 𝕊^{1}, as follows: to a unit vector u ∈ T^{1}D_{1,1} based at x associate the triple (u_{−}, u_{0}, u_{+}) ∈ 𝕊^{1}, where u_{−} ∈ 𝕊^{1} and u_{+} ∈ 𝕊^{1} are respectively the initial and ending point of the geodesic [u_{−}, u_{+}] determined by u, and u_{0} ∈ 𝕊^{1} is the endpoint of the geodesic perpendicular to [u_{−}, u_{+}] at x ∈ D_{1,1} and oriented in such a way that u_{0} ∈ ((u_{−}, u_{+})). Notice that as u moves along the geodesic [u_{−}, u_{+}] in the positive direction, the point u_{0} approaches u_{+} but the points u_{−}, u_{+} stay unchanged, so that the vector g_{t}u corresponds to the triple (u_{−}, u_{t}, u_{+}) (see Figure 3).

PSfrag replacements u_{−}

u_{0}
u_{t}
u_{+}
u
g_{t}u
Figure 3. The identification of T^{1}D_{1,1} with (𝕊^{1})^{(3)}.

Let φ_{−}, φ_{+} : 𝕊^{1} → L(V) be respectively the right and left continuous Γ-equivariant map in Corollary 8.7. For every u ∈ T^{1}D_{1,1}, since u_{−} ≠ u_{+}, Lemma 8.4 implies that φ_{−}(u_{−}) and φ_{+}(u_{+}) define transverse and hence complementary Lagrangians

V = φ_{−}(u_{−}) ⊕ φ_{+}(u_{+}) .

In this way we obtain a splitting of Ẽ^{ρ} into (g̃_{t}^{ρ})-invariant Borel subbundles Ẽ^{ρ} = Ẽ_{−}^{ρ} ⊕ Ẽ_{+}^{ρ} which descends to a (g_{t}^{ρ})-invariant splitting

E^{ρ} = E_{−}^{ρ} ⊕ E_{+}^{ρ} .

Using Corollary 8.5 we deduce that the triples (φ_{−}(u_{−}), φ_{±}(u_{t}), φ_{+}(u_{+})) are maximal for every t, so that we can associate to each of them complex structures J(g_{t}u, +) and J(g_{t}u, −) on V as in (8.5), and hence positive quadratic forms q_{J(g_{t}u,+)} and q_{J(g_{t}u,−)}, which thus give rise to two families ∥ · ∥_{g_{t}u}^{+} and ∥ · ∥_{g_{t}u}^{−} of Euclidean metrics on E^{ρ}(g_{t}u), for t ∈ ℝ.

Lemma 8.8. Let p : E^{ρ} → T^{1}Σ be the projection defined in (6.1) and, if ξ ∈ E^{ρ}, let u := p(ξ) ∈ T^{1}Σ. Then

(1) For every ξ ∈ E_{+}^{ρ}

lim_{t→+∞} ∥g_{t}^{ρ}ξ∥_{g_{t}u}^{+} = 0 monotonically, and ∥g_{−t}^{ρ}ξ∥_{g_{−t}u}^{±} ≥ ∥ξ∥_{u}^{±} for all t ≥ 0 .

(2) For every ξ ∈ E_{−}^{ρ}

lim_{t→+∞} ∥g_{−t}^{ρ}ξ∥_{g_{−t}u}^{−} = 0 monotonically, and ∥g_{t}^{ρ}ξ∥_{g_{t}u}^{±} ≥ ∥ξ∥_{u}^{±}, for all t ≥ 0 .

PSfrag replacements φ_{−}(u_{−}) φ_{+}(u_{0}) φ_{+}(u_{t}) φ_{+}(u_{+})

Figure 4.

Proof. We prove (1), as the proof of (2) is analogous. Working in Ẽ^{ρ} as we may, let ξ ∈ Ẽ^{ρ}, ξ = (u, v), v ∈ V. Let v ∈ φ_{+}(u_{+}). We use the Euclidean metrics ∥ · ∥_{g_{t}u}^{+} defined by the triple

(φ_{−}(u_{−}), φ_{+}(u_{t}), φ_{+}(u_{+})) ,

that is

∥g̃_{t}^{ρ}ξ∥_{g_{t}u}^{+} = |Q_{φ_{+}(u_{t})}^{φ_{+}(u_{+}),φ_{−}(u_{−})}(v)| ,

which, since φ_{+} is left continuous and hence

lim_{t→+∞} φ_{+}(u_{t}) = φ_{+}(u_{+}) ,

implies immediately that

lim_{t→+∞} ∥g̃_{t}^{ρ}ξ∥_{g_{t}u}^{+} = 0 .

Monotonicity follows from Lemma 8.2. In fact, for every 0 ≤ t_{1} ≤ t_{2}, the quadruple

(φ_{−}(u_{−}), φ_{+}(u_{t_{1}}), φ_{+}(u_{t_{2}}), φ_{+}(u_{+}))

is maximal and hence Lemma 8.2 implies that

∥g̃_{t_{2}}^{ρ}ξ∥_{g_{t_{2}}u}^{+} ≤ ∥g̃_{t_{1}}^{ρ}ξ∥_{g_{t_{1}}u}^{+} .

To prove the second statement in (1), observe that for t ≥ 0 the quadruple

(φ_{−}(u_{−}), φ_{+}(u_{−t}), φ_{+}(u_{0}), φ_{+}(u_{+}))

is maximal and hence Lemma 8.2 implies that

∥g̃_{−t}^{ρ}ξ∥_{g_{−t}u}^{+} = |Q_{φ_{+}(u_{−t})}^{φ_{+}(u_{+}),φ_{−}(u_{−})}(v)| ≥ |Q_{φ_{+}(u_{0})}^{φ_{+}(u_{+}),φ_{−}(u_{−})}(v)| = ∥ξ∥_{u}^{+} .

The statement for the metrics ∥ · ∥_{g_{t}u}^{−} follows analogously. □

The metrics ∥ · ∥_{u}^{+} and ∥ · ∥_{u}^{−} are Borel metrics on the bundle E^{ρ}. Since the basis T^{1}Σ is compact, any two continuous Euclidean metrics on E^{ρ} are equivalent: we have then

Lemma 8.9. The metrics ∥·∥_{u}^{+} and ∥·∥_{u}^{−} are equivalent to a continuous metric.

This follows easily from the following two facts:

- The proper action of Γ on (𝕊^{1})^{(3)} has compact quotient.

- For any compact subset C ⊂ (𝕊^{1})^{(3)}, the set of metrics

{∥ · ∥_{u}^{±} : (u_{−}, u_{0}, u_{+}) ∈ C}

is bounded.

Proof of Theorem 6.1. Fix a continuous Euclidean metric ∥ · ∥ on E^{ρ}. Then it follows from Lemmas 8.9 and 8.8 that

E_{±}^{ρ} := {ξ ∈ E^{ρ} : lim_{t→±∞} ∥g_{t}^{ρ}ξ∥ = 0} .

This implies by the following classical argument that the subbundles E_{+}^{ρ} and E_{−}^{ρ} are continuous. Let u_{m} be a converging sequence in T^{1}Σ with limit u, and let F ⊂ E^{ρ}(u) be any accumulation point of

{E_{+}^{ρ}(u_{m}) : m ≥ 1}

in the Grassmann n-bundle of E^{ρ}. Let {m_{k}} be a subsequence with lim_{k→∞} E_{+}^{ρ}(u_{m_{k}}) = F. For every ξ ∈ F take ξ_{k} ∈ E_{+}^{ρ}(u_{m_{k}}) with lim_{k→∞} ξ_{k} = ξ. Then the function

\begin{matrix}ℝ^{+} & → ℝ^{+} \\ t & ↦ ∥g_{t}^{ρ}ξ∥ \end{matrix}

being a uniform limit on compacts of the sequence of functions t ↦ ∥g_{t}^{ρ}ξ_{k}∥ which vanish at infinity, vanishes at infinity as well, which implies that ξ ∈ E_{+}^{ρ}(u) and hence F ⊆ E_{+}^{ρ}(u); since both spaces have the same dimension, we conclude that F = E_{+}^{ρ}(u). This shows continuity of the splitting.

This implies by the definition of Ẽ_{±}^{ρ} that both maps φ_{+} and φ_{−} from 𝕊^{1} to L(V) are continuous. But this implies easily that φ_{−} = φ_{+}; we shall denote from now on by φ this continuous Γ-equivariant map. This implies now the first assertion of Corollary 6.3.

We are thus in the following situation: for every u ∈ T^{1}D_{1,1}, we have the splitting

V = φ(u_{−}) ⊕ φ(u_{+}), u = (u_{−}, u_{0}, u_{+})

which gives rise to the splittings

Ẽ^{ρ} = Ẽ_{−}^{ρ} ⊕ Ẽ_{+}^{ρ}

and

E^{ρ} = E_{−}^{ρ} ⊕ E_{+}^{ρ}

into continuous g̃_{t}^{ρ} and g_{t}^{ρ} invariant subbundles. We denote by J(u) ∈ X_{Sp(V)} the complex structure associated to the triple

(φ(u_{−}), φ(u_{0}), φ(u_{+}))

as in (8.5). It is now immediate that the map

\begin{matrix}T^{1}D_{1,1} → & X_{Sp(V)} \\ u ⟼ & J(u) \end{matrix} (8.7)

gives a positive complex structure J of E^{ρ} with the required properties (see (8.4)). Let ∥ · ∥_{u} be the Euclidean metric on E^{ρ} induced by the quadratic form q_{J(u)}.

In the notation of Lemma 8.8, we have ∥ · ∥_{u}^{+} = ∥ · ∥_{u}^{−} = ∥ · ∥_{u} and hence for every ξ ∈ E_{±}^{ρ} with p(ξ) = u

lim_{t→∞} ∥g_{±t}ξ∥_{g_{±t}u} = 0 monotonically.

We claim now that there exists T > 0 such that for every ξ ∈ E_{+}^{ρ},

∥g_{t}^{ρ}ξ∥_{g_{t}u} ≤ \frac{1}{2}∥ξ∥_{u} for t ≥ T .

Indeed, if this were not the case, by Lemma 8.8 there would exist a sequence ξ_{n} ∈ E_{+}^{ρ} and T_{n} → +∞ with ∥ξ_{n}∥ = 1 and ∥g_{T_{n}}^{ρ}ξ_{n}∥_{g_{Tn}u_{n}} = \frac{1}{2}.

We may assume that ξ_{n} converges to a point ξ ∈ E_{+}^{ρ}. Then the sequence of functions

\begin{matrix}ℝ^{+} & ⟶ ℝ^{+} \\ t & ↦ ∥g_{t}^{ρ}ξ_{n}∥_{g_{t}u_{n}} \end{matrix}

converges uniformly on compact sets to

t ↦ ∥g_{t}^{ρ}ξ∥_{g_{t}u} .

But, by monotonicity, we have that

∥g_{t}^{ρ}ξ_{n}∥_{g_{t}u_{n}} ≥ \frac{1}{2}, for t ∈ [0, T_{n}],

and since T_{n} → +∞, we deduce that

∥g_{t}^{ρ}ξ∥_{g_{t}u} ≥ \frac{1}{2} for all t ≥ 0 ,

which contradicts the fact that ξ ∈ E_{+}^{ρ}. Applying the inequality

∥g_{T}^{ρ} ξ∥ ≤ \frac{1}{2}∥ξ∥

to nT, for n ∈ ℕ, we obtain the exponential decay. □

Proof of Corollary 6.2. The proof will rely on the metric properties of the map defined in (8.7).

Fix a unit tangent vector v ∈ T^{1}D_{1,1} based at 0 ∈ D_{1,1} and let J_{0} := J(v) ∈ X_{Sp(V)} be the corresponding complex structure on V. Observe first of all that d(J_{0}, ρ(γ)J_{0}) is bounded above linearly by the word length ℓ(γ) of γ, as an argument by recurrence on ℓ(γ) easily shows. In order to show the lower bound, we shall use the contraction– dilation property of the Anosov flow in Theorem 6.1(2).

The essential step is estimating the distance in X_{Sp(V)} between J(u) and J(g_{t}u), given by

d(J(u), J(g_{t}u)) = | ln ∥Id∥_{J(u),J(g_{t}u)}| + | ln ∥Id∥_{J(g_{t}u),J(u)}| ,

for any u ∈ T^{1}D_{1,1} and any t ≥ 0 (see § 6).

For x ∈ φ(u_{−}), applying Theorem 6.1, we have that

q_{J(g_{t}u)}(x) ≥ e^{2At}q_{J(u)}(x)

and likewise for x ∈ φ(u_{+})

q_{J(g_{t}u)}(x) ≤ e^{−2At}q_{J(u)}(x) .

These inequalities, together with the fact that φ(u_{−}) ⊕ φ(u_{+}) is an orthogonal decomposition for both q_{J(u)} and q_{J(g_{t}u)}, imply that

∥Id∥_{J(u),J(g_{t}u)} ≥ e^{At}

and

∥Id∥_{J(g_{t}u),J(u)} ≥ e^{At} ,

from which we deduce that

d(J(u), J(g_{t}u)) ≥ 2At . (8.8)

Let now γ ∈ Γ and let us choose u ∈ T^{1}D_{1,1} to be the tangent vector at 0 ∈ D_{1,1} to the geodesic segment connecting 0 to ρ(γ) and let t = d(0, ρ(γ)0). Applying (8.8) to this situation and observing that g_{t}u = γu, we get that

d(J(u), ρ(γ)J(u)) ≥ 2Ad(0, γ0)

and hence

d(J_{0}, ρ(γ)J_{0}) ≥ 2Ad(0, γ0) − 2C ,

where

C := sup {d(J(w_{1}), J(w_{2})) : w_{1}, w_{2} are based at 0} .

Finally, d(0, γ0) is bounded linearly below in terms of ℓ(γ), as follows from the Milnor–Svarc lemma. □

Proof of Corollary 6.3. The injectivity of the Γ-equivariant continuous map

φ : 𝕊^{1} → L(V)

obtained in the proof of Theorem 6.1, follows for instance from Corollary 7.7 because of continuity. So we finally turn to the proof of the rectifiability of the image of φ. For this we shall put to use the Sp(V)-invariant causal structure on L(V).

Let us fix a ≠ b ∈ 𝕊^{1}, let L_{0} := φ(a) and L_{∞} := φ(b), so that on 𝕊^{1} \ {b}, φ takes values in t(L_{∞}). Composing the restriction of φ to 𝕊^{1} \ {b} with the usual diffeomorphism

\begin{matrix}t(L_{∞}) & → Q(L_{0}) \\ L & ↦ Q_{L}^{L_{0},L_{∞}} , \end{matrix}

gives rise to a continuous map

c : 𝕊^{1} \ {b} → Q(L_{0})

whose restriction to the interval ((a, b)) has the following properties:

(1) it takes values in the cone Q^{+}(L_{0}) of positive definite quadratic forms, and

(2) for every t_{1}, t_{2} ∈ ((a, b)) such that a, t_{1}, t_{2}, b are in positive cyclic order, c(t_{2}) − c(t_{1}) ∈ Q^{+}(L_{0}).

Fixing a scalar product on L_{0}, we can identify Q(L_{0}) with the space Sym(L_{0}) of symmetric endomorphisms of L_{0} and Q^{+}(L_{0}) with the cone Sym^{+}(L_{0}) of positive definite ones. On Sym(L_{0}) we have a natural scalar product

⟨⟨A, B⟩⟩ := tr AB

and we have that for every A, B ∈ Sym^{+}(L_{0})

⟨⟨A, B⟩⟩ > 0 ,

that is Sym^{+}(L_{0}) is an open convex acute cone. The assertion then follows from the following general fact

Lemma 8.10. Let C ⊂ E be an open convex acute cone in an Euclidean space and let f : [0, 1] → C be a continuous map such that for every t_{1} < t_{2},

f(t_{1}) − f(t_{2}) ∈ C .

Then f is of finite length.

Proof. Fix e ∈ C. We claim that since C is acute

k := inf_{x∈C} \frac{⟨⟨x, e⟩⟩}{∥x∥} > 0 .

Indeed, otherwise there is a nonzero x ∈ \overline{C} such that ⟨⟨x, e⟩⟩ = 0. On the other hand, since C is open, for s < 0 and |s| small enough we have that

e′ := sx + (1 − s)e ∈ C ,

which implies that ⟨⟨e′, x⟩⟩ < 0 and contradicts the fact that ⟨⟨u, v⟩⟩ ≥ 0 for all u, v ∈ \overline{C}.

Let 0 ≤ s < t ≤ 1; then f(t) − f(s) ∈ C and applying the claim, we obtain:

∥f(t) − f(s)∥ ≤ \frac{1}{k}⟨f(t) − f(s), e⟩ .

Given any subdivision 0 = t_{0} < t_{1} < ⋯ < t_{n−1} < t_{n} = 1 of the interval [0, 1], we deduce that

∑_{i=1}^{n} ∥f(t_{i}) − f(t_{i−1})∥ ≤ \frac{1}{k} ∑_{i=1}^{n}⟨f(t_{i}) − f(t_{i−1}), e⟩ = \frac{⟨f(1) − f(0), e⟩}{k}

which proves that f is rectifiable. □

References

1. Ch. Bavard, Longueur stable des commutateurs, Enseign. Math. (2) 37 (1991), no. 1-2, 109–150.

2. M. Bestvina and K. Fujiwara, Bounded cohomology of subgroups of mapping class groups, Geom. Topol. 6 (2002), 69–89 (electronic).

3. P. Biran, M. Entov, and L. Polterovich, Calabi quasimorphisms for the symplectic ball, Commun. Contemp. Math. 6 (2004), no. 5, 793–802.

4. S. B. Bradlow, O. García Prada, and P. B. Gothen, Surface group representations in PU(p, q) and Higgs bundles, J. Diff. Geom. 64 (2003), no. 1, 111–170.

5. , oral communication, 2005.

6. , Representations of surface groups in the general linear group, preprint, arXiv:math.AG/0401064, January 2004.

7. , Surface group representations, Higgs bundles, and holomorphic triples, preprint, arXiv:math.AG/0206012, June 2002.

8. R. Brooks, Some remarks on bounded cohomology, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) (Princeton, N.J.), Ann. of Math. Stud., vol. 97, Princeton Univ. Press, 1981, pp. 53–63.

9. R. Brooks and C. Series, Bounded cohomology for surface groups, Topology 23 (1984), no. 1, 29–36.

10. M. Bucher, Boundedness of characteristic classes for flat bundles, Ph.D. Thesis, ETH, 2004.

11. M. Burger and A. Iozzi, Bounded cohomology and deformation rigidity in hyperbolic geometry, preprint, arXiv:math.MG/0505069.

12. , Boundary maps in bounded cohomology, Geom. Funct. Anal. 12 (2002), 281–292.

13. , Bounded Kähler class rigidity of actions on Hermitian symmetric spaces, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 1, 77–103.

14. M. Burger, A. Iozzi, and A. Wienhard, Surface group representations with maximal Toledo invariant, preprint.

15. , Tight embeddings, preprint.

16. , Surface group representations with maximal Toledo invariant, C. R. Acad. Sci. Paris, Sér. I 336 (2003), 387–390.

17. M. Burger and N. Monod, Bounded cohomology of lattices in higher rank Lie groups, J. Eur. Math. Soc. 1 (1999), no. 2, 199–235.

18. , Continuous bounded cohomology and applications to rigidity theory, Geom. Funct. Anal. 12 (2002), 219–280.

19. E. Cartan, Sur le domains bornés homogènes de l’espace de n variables complexes, Abh. Math. Sem. Hamburg 11 (1935), 116–162.

20. S. Choi and W. M. Goldman, Convex real projective structures on closed surfaces are closed, Proc. Amer. Math. Soc. 118 (1993), no. 2, 657–661.

21. J. L. Clerc and B. Ørsted, The Gromov norm of the Kaehler class and the Maslov index, Asian J. Math. 7 (2003), no. 2, 269–295.

22. P. Deligne, Espaces hermitiens symétriques, transcripted by J. Milne from Deligne’s handwritten notes, Spring 1973. Available at http://www.jmilne.org/math/ under the title ”Deligne’s notes on Hermitian symmetric spaces”.

23. A. Domic and D. Toledo, The Gromov norm of the Kähler class of symmetric domains, Math. Ann. 276 (1987), no. 3, 425–432.

24. J. L. Dupont, Bounds for characteristic numbers of flat bundles, Algebraic topology, Aarus 1978, Lecture Notes in Mathematics, vol. 763, Springer Verlag, 1979.

25. J. Eells and J. C. Wood, Restriction on harmonic maps of surfaces, Topology 15 (1976), 263–266.

26. M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not. (2003), no. 30, 1635–1676.

27. D. B. A. Epstein and K. Fujiwara, The second bounded cohomology of wordhyperbolic groups, Topology 36 (1997), no. 6, 1275–1289.

28. V. V. Fock and A. B. Goncharov, Moduli spaces of convex projective structures on surfaces, preprint, arXiv:math.DG/0405348.

29. , Moduli spaces of local systems and higher Teichmüller theory, preprint, arXiv:math.AG/0311149.

30. J.-M. Gambaudo and É. Ghys, Commutators and diffeomorphisms of surfaces, Ergodic Theory Dynam. Systems 24 (2004), no. 5, 1591–1617.

31. O. García-Prada and I. Mundet i Riera, Representations of the fundamental group of a closed oriented surface in Sp(4, ℝ), Topology 43 (2004), no. 4, 831– 855.

32. E. Ghys, Groupes d’homéomorphismes du cercle et cohomologie bornée, The Lefschetz centennial conference, Part III, (Mexico City 1984), Contemp. Math., vol. 58, American Mathematical Society, RI, 1987, pp. 81–106.

33. W. M. Goldman, Discontinuous groups and the Euler class, Thesis, University of California at Berkeley, 1980.

34. W. M. Goldman, M. Kapovich, and B. Leeb, Complex hyperbolic manifolds homotpy equivalent to a riemann surface, Comm. Anal. Geom. 9 (2001), no. 1, 61–95.

35. W.M. Goldman, Topological components of spaces of representations, Invent. Math. 93 (1988), no. 3, 557–607.

36. P. B. Gothen, Components of spaces of representations and stable triples, Topology 40 (2001), no. 4, 823–850.

37. R. I. Grigorchuk, Some results on bounded cohomology, Combinatorial and geometric group theory (Edinburgh, 1993), London Math. Soc. Lecture Note Ser., vol. 204, Cambridge Univ. Press, Cambridge, 1995, pp. 111–163.

38. M. Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99.

39. Harish-Chandra, Representations of semisimple Lie groups. VI. Integrable and square-integrable representations, Amer. J. Math. 78 (1956), 564–628.

40. L. Hernàndez Lamoneda, Maximal representations of surface groups in bounded symmetric domains, Trans. Amer. Math. Soc. 324 (1991), 405–420.

41. N. J. Hitchin, Lie groups and Teichmüller space, Topology 31 (1992), no. 3, 449–473.

42. A. Iozzi, Bounded cohomology, boundary maps, and representations into Homeo_{+}(S^{1}) and SU(1, n), Rigidity in Dynamics and Geometry, Cambridge, UK, 2000, Springer Verlag, 2002, pp. 237–260.

43. V. A. Kaimanovich, The Poisson boundary of a discrete group, preprint available at http://name.math.univ-rennes1.fr/vadim.kaimanovich/poifrm1w.ps.

44. H. Kneser, Die kleinste Bedeckungszahl innerhalb einer Klasse von Flächenabbildungen, Math. Ann. 103 (1930), 317–358.

45. A. Korányi, Function spaces on bounded symmetric domains, Analysis and Geometry on Complex Homogeneous Domains (J. Faraut et al., ed.), Progress in Mathematics, Birkhäuser, Boston, 2000, pp. 183–277.

46. D. Kotschick, Signatures,monopoles and mapping class groups, Math. Res. Lett. 5 (1998), no. 1-2, 227–234.

47. , Quasi-homomorphisms and stable lengths in mapping class groups, Proc. Amer. Math. Soc. 132 (2004), no. 11, 3167–3175 (electronic). MR MR2073290

48. , What is… a quasi-morphism?, Notices Amer. Math. Soc. 51 (2004), no. 2, 208–209.

49. F. Labourie, Anosov flows, surface groups and curves in projective space, preprint, arXiv:math.DG/0401230.

50. , Crossratios, surface groups, SL(n, ℝ) and C^{1,h}(S^{1}) ⋊ Diff^{h}(S^{1}), preprint, arXiv:math.DG/0502441.

51. J. F. Lafont and B. Schmidt, Simplicial volume of closed locally symmetric spaces of non-compact type, preprint, arXiv:math.DG/0504338.

52. G. Lion and M. Vergne, The Weil representation, Maslov index and theta series, Progress in Mathematics, vol. 6, Birkhäuser Boston, Mass., 1980.

53. J. Milnor, On the existence of a connection with curvature zero, Comment. Math. Helv. 32 (1958), 215–223.

54. I. Mineyev, Straightening and bounded cohomology of hyperbolic groups, Geom. Funct. Anal. 11 (2001), no. 4, 807–839.

55. N. Monod, Continuous bounded cohomology of locally compact groups, Lecture Notes in Math., no. 1758, Springer-Verlag, 2001.

56. I. I. Piateskii-Shapiro, Geometry of classical domains and theory of automorphic functions, Gordon and Breach, New York, 1969.

57. I. Satake, Algebraic structures of symmetric domains, Kanô Memorial Lectures, vol. 4, Iwanami Shoten, Tokyo, 1980.

58. R. Savage, The space of positive definite matrices and Gromov’s invariant, Trans. Amer. Math. Soc. 274 (1982), no. 1, 241–261.

59. T. Soma, The third bounded cohomology and Kleinian groups, Topology and Teichmüller spaces (Katinkulta, 1995), World Sci. Publishing, River Edge, NJ, 1996, pp. 265–277.

60. D. Toledo, Harmonic maps from surfaces to certain Kähler manifolds, Math. Scand. 45 (1979), 13–26.

61. , Non-existence of certains closed complex geodesics in the moduli space of curves, Pacific. J. Math. 129 (1987), no. 1, 187–192.

62. , Representations of surface groups in complex hyperbolic space, J. Diff. Geom. 29 (1989), no. 1, 125–133.

63. V. Turaev, A cocycle for the symplectic first Chern class and the Maslov index, Funct. Anal. Appl. 18 (1984), 35–39.

64. W. T. van Est, Group cohomology and Lie algebra cohomology in Lie groups, I, II, Nederl. Akad. Wetensch. Proc. Series A. {56}=Indag. Math. 15 (1953), 484–504.

65. A. Wienhard, Bounded cohomology and geometry, Ph. D. thesis, Universität Bonn, 2004.

66. J.A. Wolf, Fine structure of Hermitian symmetric spaces, Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970), Dekker, New York, 1972, pp. 271–357. Pure and App. Math., Vol. 8.

67. E. Z. Xia, The moduli of flat U(p, 1) structures on Riemann surfaces, Geom. Dedicata 97 (2003), 33–43, Special volume dedicated to the memory of Hanna Miriam Sandler.

68. R. J. Zimmer, Amenable ergodic group actions and an application to Poisson boundaries of random walks, J. Funct. Anal. 27 (1978), 350–372.

69. , Ergodic theory and semisimple groups, Birkhäuser, Boston, 1984.

E-mail address: burger@math.ethz.ch

FIM, ETH Zentrum, Rämistrasse 101, CH-8092 Zürich, Switzerland

E-mail address: Alessandra.Iozzi@unibas.ch

Institut für Mathematik, Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland

Départment de Mathématiques, Université de Strasbourg, 7, rue René Descartes, F-67084 Strasbourg Cedex, France

E-mail address: labourie@math.u-psud.fr

Topologie et Dynamique, Bâtiment 425, Université de Paris Sud, F-91405 Orsay Cedex, France

E-mail address: Anna.Wienhard@unibas.ch

Institut für Mathematik, Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland