Higher Teichmüller Spaces: from SL(2,R) to other Lie groups
aa r X i v : . [ m a t h . G T ] D ec Higher Teichm¨uller Spaces:from SL(2 , R ) to other Lie groups M. Burger and A. Iozzi ∗ and A. Wienhard ∗∗ Department MathematikETH ZentrumR¨amistrasse 101, CH-8092 Z¨urich, Switzerlandemail: [email protected]
Department MathematikETH ZentrumR¨amistrasse 101, CH-8092 Z¨urich, Switzerlandemail: [email protected]
Department of MathematicsPrinceton UniversityFine Hall, Washington Road, Princeton, NJ 08540, USAemail: [email protected]
Contents
I Teichm¨uller Space and Hyperbolic Structures 5 G -bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Central extensions . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Description of H (cid:0) π ( S ) , Z (cid:1) , a digression . . . . . . . . . . . . . 14 ∗ Work partially supported by the Swiss National Science Foundation project 2000021-127016/2 ∗∗ Work partially supported by NSF Grant No.DMS-0803216 and NSF CAREER GrantNo. DMS-0846408
M. Burger and A. Iozzi and A. Wienhard
II Higher Teichm¨uller Spaces 41 igher Teichm¨uller Spaces: from SL(2 , R ) to other Lie groups Let S be a connected surface of finite topological type. The Teichm¨uller space T ( S ) is the moduli space of marked complex structures on S . It is isomorphicto the moduli space of marked complete hyperbolic structures on S , sometimescalled the Fricke space F ( S ). Associating to a hyperbolic structure its holon-omy representation naturally embeds the Fricke space F ( S ) into the varietyof representations Hom (cid:0) π ( S ) , PSU(1 , (cid:1) / PSU(1 , § S ) of hyperbolic structures on S and constructing insome details the map δ : Hyp( S ) → Hom (cid:0) π ( S ) , PSU(1 , (cid:1) , as well as the embedding δ ′ : F ( S ) = Diff +0 ( S ) \ Hyp( S ) → Hom (cid:0) π ( S ) , PSU(1 , (cid:1) / PSU(1 , , where Diff +0 ( S ) is the group of orientation preserving diffeomorphisms whichare homotopic to the identity. Then §§ δ (cid:0) Hyp( S ) (cid:1) ⊂ Hom (cid:0) π ( S ) , PSU(1 , (cid:1) . When S is acompact surface, δ (Hyp( S )) is:(1) the set of injective orientation preserving homomorphisms with discreteimage (see Theorem 2.5 and Corollary 2.13);(2) identified with one connected component of Hom (cid:0) π ( S ) , PSU(1 , (cid:1) (see § § S is a noncompact surface of finite type, the description of δ (cid:0) Hyp( S ) (cid:1) is more involved, and the characterizations (1) and (2) do not hold in this case.In § §
3, which allow us to give in § δ (cid:0) Hyp( S ) (cid:1) for noncompact surfaces S , generalizing (3), (4) and (5) above.In the second part we ask how much of this “PSU(1 ,
1) picture” generalizesto an arbitrary Lie group G . We discuss two classes of (semi)simple Lie groupsfor which one can make this question precise by defining, in very different ways,components (or specific subsets when S is not compact) of Hom (cid:0) π ( S ) , G (cid:1) which play the role of Teichm¨uller space.The terminology Higher Teichm¨uller spaces, coined by Fock and Gon-charov, has now come to mean subsets of Hom (cid:0) π ( S ) , G (cid:1) , where G is a simple M. Burger and A. Iozzi and A. Wienhard
Lie group, which share essential geometric and algebraic properties with clas-sical Teichm¨uller space considered as a subset of Hom (cid:0) π ( S ) , PSU(1 , (cid:1) . Upto now higher Teichm¨uller spaces are defined for two classes of Lie groups,namely for split real simple Lie groups, e.g. SL( n, R ), Sp(2 n, R ), SO( n, n + 1)or SO( n, n ) and for Lie groups of Hermitian type, e.g. Sp(2 n, R ), SO(2 , n ),SU( p, q ) or SO ∗ (2 n ).The invariants defined in § § π ( S ) with values in any Lie group G but, when G is a Lie group ofHermitian type these invariants are particularly meaningful. We describe howthe basic objects available in the case of PSU(1 ,
1) can be generalized to higherrank in §
5. Considering the maximal value level set of the numerical invariantthus constructed leads us to consider the space of maximal representationsHom max (cid:0) π ( S ) , G (cid:1) ⊂ Hom (cid:0) π ( S ) , G (cid:1) , some of whose properties are discussed in §
5. In particular we state a result(”structure theorem”) which describes the Zariski closure in G of the image ofa maximal representation; a major part of §§ G is a split real Liegroup. We review the definition of these spaces shortly ( § § § § § § § Acknowledgments:
We thank A. Papadopoulos for undertaking thisproject. We thank also O. Guichard and T. Hartnick for carefully reading apreliminary version of this paper providing many helpful comments, W. Gold-man for many bibliographical comments and F. Labourie for positive feedback.Our thanks go also to D. Toledo and N. A’Campo for useful discussions on thisgeneral topic over the years and to T. Delzant for helpful comments concerningcentral extensions of surface groups. Finally, we thank the referee for helpfulcomments.
Part I
Teichm¨uller Space andHyperbolic Structures
In this section we review briefly how one associates to a hyperbolic struc-ture on a surface a homomorphism of its fundamental group into the group oforientation preserving isometries of the Poincar´e disk, and how an appropri-ate quotient of the set of hyperbolic structures injects into the representationvariety.Let D = { z ∈ C : | z | < } be the unit disk endowed with the Poincar´emetric | dz | (cid:0) −| z | (cid:1) , and let G := PSU(1 ,
1) = SU(1 , / {± Id } denote the quotientof SU(1 ,
1) by its center. The group G acts on b C = C ∪{∞} by linear fractionaltransformations preserving D and hence can be identified with the group oforientation preserving isometries of D .Given a surface S , that is a two-dimensional smooth manifold a hyper-bolic metric on S is a Riemannian metric with sectional curvature −
1. A( G, D )-structure on S is an atlas on S consisting of charts taking values in D , whose change of charts are locally restrictions of elements of G , [94]. As-suming from now on that S is orientable, we observe that by the local versionof Cartan’s theorem, an orientation together with a hyperbolic metric on S determines a ( G, D )-structure on S (the converse is also true and straightfor-ward). Also, the hyperbolic metric is complete if and only if the same is true Note that all manifolds here are without boundary. In particular a compact surface isnecessarily closed. for the corresponding ( G, D )-structure, i.e. if the developing map e S → D is adiffeomorphism.The group Diff( S ), and hence its subgroup Diff + ( S ) consisting of orien-tation preserving diffeomorphisms of S , act on the set Hyp( S ) of completehyperbolic metrics on S in a contravariant way. In the sequel let e S = D be asmooth oriented disk with a basepoint ∗ ∈ D and let us fix once and for all abase tangent vector v ∈ T ∗ D , v = 0. By the correspondence between hyper-bolic metrics and ( G, D )-structures, let us also consider, for every h ∈ Hyp( D ),the unique orientation preserving isometry f h : ( D, ∗ ) → ( D , df h ( v ) ∈ R + e , where e = 1 ∈ C . If ϕ ∈ Diff + ( D ), then for any h ∈ Hyp( D ), ϕ is by definition an orientation preserving isometry between thehyperbolic metrics ϕ ∗ ( h ) and h . Therefore c ( ϕ, h ) := f h ◦ ϕ ◦ f − ϕ ∗ ( h ) is an element of G . In this way we obtain a map c : Diff + ( D ) × Hyp( D ) → G which verifies the cocycle relation c ( ϕ ϕ , h ) = c ( ϕ , h ) c (cid:0) ϕ , ϕ ∗ ( h ) (cid:1) . Let now ( S, ∗ ) be a connected oriented surface with base point ∗ and assumethat Hyp( S ) = ∅ . Let ( e S, ∗ ) = ( D, ∗ ), let p : D → S be the canonicalprojection and Γ = { T γ : γ ∈ π ( S, ∗ ) } < Diff + ( D )the group of covering transformations. Then the pullback via p gives a bijectionbetween Hyp( S ) and the set Hyp( D ) Γ of Γ-invariant elements in Hyp( D ).Furthermore, it follows from the cocycle identity that, for every h ∈ Hyp( D ) Γ ,the map ρ h : π ( S, ∗ ) −→ Gγ c ( T γ , h )is a homomorphism with respect to which the isometry f h is equivariant. Thuswe obtain the map δ , assigning to a hyperbolic structure its holonomy homo-morphism δ : Hyp( S ) → Hom (cid:0) π ( S, ∗ ) , G (cid:1) h ρ p ∗ ( h ) , which has certain important equivariance properties which we now explain.To this end, let N + be the normalizer of Γ in Diff + ( D ). Then we have thediagram with exact line { e } / / Γ / / N + π / / a (cid:15) (cid:15) Diff + ( S ) / / { e } Aut (cid:0) π ( S, ∗ ) (cid:1) , where π associates to every ϕ ∈ N + the corresponding diffeomorphism of S obtained by observing that ϕ permutes the fibers of p ; the fact that π is surjective follows from covering theory. The homomorphism a is the oneassociating to ϕ the automorphism a ϕ of Γ, or rather of π ( S, ∗ ), obtained byconjugation. With these definitions, a computation gives ρ ϕ ∗ ( h ) ( γ ) = c ( ϕ, h ) − ρ h (cid:0) a ϕ ( γ ) (cid:1) c ( ϕ, h ) , (2.1)for every ϕ ∈ N + , h ∈ Hyp( D ) Γ and γ ∈ Γ.In view of (2.1), it is important to determine those ϕ ∈ N + such that a ϕ is an inner automorphism of Γ. Let N +i be the subgroup consisting of all suchdiffeomorphisms. Then we have: Lemma 2.1.
The map π induces an isomorphism Γ \N +i ∼ = Diff +0 ( S ) , where Diff +0 ( S ) is the subgroup of Diff + ( S ) consisting of those diffeomorphismswhich are homotopic to the identity.Proof. If f : S → S is homotopic to the identity, then by covering theory theconjugation of Γ by any lift ˜ f of f gives an inner automorphism of Γ.Conversely, if ϕT γ ϕ − = T ηγη − for some η and all γ , then the diffeomor-phism T − η ϕ : D → D commutes with the Γ-action on D ; if we fix h ∈ Hyp( D ) Γ then the geodesic homotopy from T − η ϕ to Id D is Γ-equivariant and hence de-scends to a homotopy between π ( T − η ϕ ) = π ( ϕ ) and Id S .Thus combining the inverse of π with a we obtain an injective homomor-phism Map( S ) := Diff +0 ( S ) \ Diff + ( S ) α / / Out (cid:0) π ( S, ∗ ) (cid:1) of the mapping class group Map( S ) of S into the group of outer automorphismsof π ( S ). It follows then from (2.1) that the map which to h ∈ Hyp( S )associates the class of the homomorphism [ ρ p ∗ ( h ) ] and which takes values inthe quotient Hom (cid:0) π ( S, ∗ ) , G (cid:1) /G by the G -conjugation action on the target,is invariant under the Diff +0 ( S )-action so that finally we obtain a map fromthe Fricke space F ( S ) = Diff +0 ( S ) \ Hyp( S ) to the representation variety δ ′ : Diff +0 ( S ) \ Hyp( S ) → Hom (cid:0) π ( S, ∗ ) , G (cid:1) /G [ h ] [ ρ p ∗ ( h ) ]which is α -equivariant. Proposition 2.2. If S is a connected, oriented surface admitting a completehyperbolic structure, then δ ′ is injective.Proof. If h , h ∈ Hyp( D ) Γ are such that ρ h and ρ h are conjugated by g ∈ G ,then it follows from the definitions that f − h gf h is an orientation preservingdiffeomorphism sending h to h , which furthermore is Γ-equivariant; by theargument used in Lemma 2.1, we get that π ( f − h gf h ) ∈ Diff +0 ( S ).We now describe the image of the homomorphism α and of the map δ ′ inthe case in which S is a compact oriented surface of genus g ≥
2. This lattercondition guarantees that the classifying map S → Bπ ( S, ∗ ) (2.2)is a homotopy equivalence; we use this fact to equip H (cid:0) π ( S, ∗ ) , Z (cid:1) with thecanonical generator, image of the fundamental class [ S ] via the isomorphismH ( S, Z ) → H (cid:0) π ( S, ∗ ) , Z (cid:1) induced by (2.2). An isomorphism between thefundamental groups of two compact oriented surfaces S and S is said to beorientation preserving if the generator of H (cid:0) π ( S , ∗ ) , Z (cid:1) is mapped to thegenerator of H (cid:0) π ( S , ∗ ) , Z (cid:1) . Theorem 2.3.
Let S and S be compact oriented surfaces of genus g ≥ .Then any orientation preserving isomorphism π ( S , ∗ ) → π ( S , ∗ ) is inducedby an orientation preserving diffeomorphism S → S . Let us denote by Aut + (cid:0) π ( S, ∗ ) (cid:1) the group of the orientation preservingautomorphisms of a compact orientable surface S of genus g ≥
1, and byOut + (cid:0) π ( S, ∗ ) (cid:1) its quotient by the group of inner automorphisms. From The-orem 2.3 we conclude: Corollary 2.4 (Dehn-Nielsen–Baer Theorem, see [33] for a proof). If S is acompact orientable surface of genus g ≥ , the map α : Map( S ) = Diff + ( S ) / Diff +0 ( S ) → Out + (cid:0) π ( S, ∗ ) (cid:1) is an isomorphism. Let now Hom d,i denote the subset of Hom consisting of all injective homo-morphisms with discrete image. The following classical identification of theimage of δ uses the Nielsen realization: Theorem 2.5. If S is a compact orientable surface of genus g ≥ , then theimage of δ consists precisely of (cid:8) ρ ∈ Hom d,i (cid:0) π ( S, ∗ ) , G ) (cid:1) : im ρ \ D is compact and ρ is orientation preserving (cid:9) . Remark 2.6.
Since π ( S ) is the fundamental group of a compact surface and ρ is a discrete embedding, the quotient im ρ \ D is automatically compact. Here,we include this property explicitly in order to stress the similarity with thedefinition of Hom (Γ , G ) in § ρ is orientation preserving if the induced map ρ ∗ maps the generator of H (cid:0) π ( S, ∗ ) , Z (cid:1) to the generator of H (cid:0) π (im ρ \ D , ∗ ) , Z (cid:1) . Proof.
Given h ∈ Hyp( S ), it is immediate, by using f h , that ρ h belongs to theabove set.Conversely, apply Nielsen realization to the orientation preserving isomor-phism ρ to get an orientation preserving diffeomorphism f : S → im ρ \ D ; if h = f ∗ ( h P ), where h P is the Poincar´e metric on im ρ \ D , then one verifies that[ ρ ] = [ ρ h ]. Remark 2.7.
Contrary to what happens in the compact case, if S is a non-compact orientable surface of negative Euler characteristic, the inclusion δ (cid:0) Hyp( S ) (cid:1) ∪ δ (cid:0) Hyp( S ) (cid:1) ⊂ Hom d,i (cid:0) π ( S, ∗ ) , PSU(1 , (cid:1) , where S denotes the surface S with the opposite orientation is always proper.In fact, if ρ : π ( S, ∗ ) → G is just discrete and injective, the surfaces im ρ \ D and S need not be diffeomorphic, although they have the ”same” fundamen-tal group. For example, the once punctured torus and the thrice puncturedsphere have isomorphic fundamental groups F and admit complete hyperbolicstructures. We will see in § In this section we review some basic properties of the set of discrete and faithfulrepresentations in Hom(Γ , G ) in the general context of a finitely generatedgroup Γ and a connected reductive Lie group G .One way to approach the problem of determining the image of Hyp( S )under the map δ is to equip Hom(Γ , G ) with a topology. Quite generally ifΓ is a discrete group and G is a topological group, Hom(Γ , G ) inherits thetopology of the product space G Γ . In case Γ is finitely generated with finitegenerating set F ⊂ Γ, let p : F | F | → Γ be the corresponding presentation and R a set of generators of the relators ker p . Since every r ∈ R is a word in F | F | ,0it determines a product map m r : G F → G by evaluation on G . The mapHom(Γ , G ) −→ G F π (cid:0) π ( s ) (cid:1) s ∈ F identifies the topological space Hom(Γ , G ) with the closed subset ∩ r ∈ R m − r ( e ) ⊂ G F . In particular, Hom(Γ , G ) is locally compact if G is so, and a real algebraicset if G is a real algebraic group. We record the following Proposition 2.8 ([3, 98]). If Γ is finitely generated and G is a real algebraicgroup, then Hom(Γ , G ) has finitely many connected components and each ofthem is a real semialgebraic set, that is, it is defined by a finite number ofpolynomial equations and inequalities. Remark 2.9.
Proposition 2.8 fails if G is not algebraic. An example for this,given in [48], is the quotient of the three dimensional Heisenberg group bya cyclic central subgroup, where a simple obstruction class detects infinitelymany connected components in the representation variety.In order to proceed further we assume that Γ is finitely generated, G is aLie group and introduce (see [49, 97]) the following subset of Hom(Γ , G )Hom (Γ , G ) = { ρ ∈ Hom d,i (Γ , G ) such that ρ (Γ) \ G is compact } , where, as in § d,i refers to the set of injective homomorphisms withdiscrete image, so that Hom (Γ , G ) ⊂ Hom d,i (Γ , G ) ⊂ Hom(Γ , G ).The first result on the topology of Hom d,i (Γ , G ) requires a hypothesis onΓ: Definition 2.10.
We say that Γ has property (H) if every normal nilpotentsubgroup of Γ is finite.Observe that this condition is fulfilled by every nonabelian free group andevery fundamental group of a compact surface of genus g ≥
2. With this wecan now state the following
Theorem 2.11 ([49]).
Let Γ be a finitely generated group with property (H) and G a connected Lie group. Then Hom d,i (Γ , G ) is closed in Hom(Γ , G ) .Proof. The essential ingredient is the theorem of Kazhdan–Margulis–Zassenhaus[89, Theorem 8.16] saying that there exists an open neighborhood
U ⊂ G of e such that whenever Λ < G is a discrete subgroup, then U ∩
Λ is containedin a connected nilpotent group. We fix now such an open neighborhood andassume in addition that [it does not contain any nontrivial subgroup of G ; letalso ℓ be an upper bound on the degree of nilpotency of connected nilpotentLie subgroups of G .1Let now { ρ n } n ≥ be a sequence in Hom d,i (Γ , G ) with limit ρ . We showthat ρ is injective. For every finite set E ⊂ ker ρ , we have that ρ n ( E ) ⊂ U for n large, which, by [89, Theorem 8.16], implies that for all k ≥ ℓ , the k -thiterated commutator of ρ n ( E ) is trivial, and the same holds therefore for E since ρ n is injective.As a result, ker ρ is nilpotent and hence, by property (H), finite; thus ρ n (ker ρ ) ⊂ U for large n which, by the choice of U implies that ρ n (ker ρ ) = e and hence that ρ is injective.We prove now that ρ is discrete. To this end, let L = ρ (Γ) be the closureof ρ (Γ); then L is a Lie subgroup of G and L is open in L . Let V be anopen neighborhood of the identity on L with V ⊂ U ; then ρ (Γ) ∩ V is densein V and V generates L , from which we conclude that ρ (Γ) ∩ V generates adense subgroup of L . For every finite set F ′ ⊂ Γ with ρ ( S ) ⊂ V ⊂ U we havethat ρ n ( S ) ⊂ U for n large, which implies as before that for all k ≥ l the k -thiterated commutator of F ′ , and hence of ρ ( F ′ ) is trivial; thus L is nilpotentand so is ρ − ( L ) since ρ is injective. But then ρ − ( L ) is finite and hence L = { e } , which shows that ρ (Γ) is discrete and concludes the proof.Next we turn to the set Hom (Γ , G ) of faithful, discrete and cocompactrealizations of Γ in G ; this set was considered by A. Weil as a tool in hiscelebrated local rigidity theorem in which the following general result playedan important role. Theorem 2.12 ([89]).
Assume that Γ is finitely generated and that G is aconnected Lie group. Then Hom (Γ , G ) is an open subset of G . There are by now several approaches available: we refer to the paper byBergeron and Gelander [4], where the geometric approach due essentially toEhresmann and Thurston [94] is explained (see also [45, 23, 77]); this approach,based on a reformulation of the problem in terms of variations of (
G, X )-structures leads to a more general stability result also valid for manifolds withboundary.We content ourselves with noticing the following consequence:
Corollary 2.13.
Assume that Γ is finitely generated, torsion-free and hasproperty (H) . Assume that G is a connected reductive Lie group and that thereexists a discrete, injective and cocompact realization of Γ in G . Then Hom (Γ , G ) = Hom d,i (Γ , G ) and both sets are therefore open and closed, in particular a union of connectedcomponents of Hom(Γ , G ) .Proof. Let ρ ∈ Hom (Γ , G ) and let K < G be a maximal compact subgroup;since by the Iwasawa decomposition X := G/K is contractible and since ρ (Γ)2acts on X as a group of covering transformations with compact quotient, wehave that, for n = dim X , H n (Γ , R ) = H n (cid:0) ρ (Γ) , R (cid:1) = 0. Therefore, if ρ : Γ → G is any discrete injective embedding, we have that H n (cid:0) ρ (Γ) \ X, R (cid:1) does notvanish and hence ρ (Γ) \ X is compact, thus implying that ρ ∈ Hom (Γ , G ).Applying the preceding discussion to our compact surface S of genus g ≥ δ (cid:0) Hyp( S ) (cid:1) ⊂ Hom (cid:0) π ( S, ∗ ) , PSU(1 , (cid:1) is a union of components of the representation variety Hom (cid:0) π ( S, ∗ ) , PSU(1 , (cid:1) . In this section we will discuss various aspects of a fundamental invariant at-tached to a representation ρ : π ( S, ∗ ) → G , where G = PSU(1 , Euler number of ρ . This leads to a quite different way of characterizing theimage of δ : Hyp( S ) → Hom (cid:0) π ( S, ∗ ) , G (cid:1) in the case in which S is compact. This invariant can also be defined fortargets belonging to a large class of Lie groups G , and this leads to naturalgeneralizations of Teichm¨uller space (see the discussion in § § S denotes a compact surface of genus g ≥ π ( S, ∗ ) and we set D = e S . G -bundles Given a connected Lie group G and a homomorphism ρ : π ( S ) → G , weobtain, in the notation of § D × G by γ ∗ ( x, g ) = (cid:0) T γ x, ρ ( γ ) g (cid:1) whose quotient π ( S ) \ ( D × G ) is the total space G ( ρ ) of a flat principal (right) G -bundle over S , where the projection map comes from the projection D × G → D on the first factor.Given a G -bundle E over S the first obstruction to find a continuous sectionof E → S lies in H (cid:0) S, π ( G ) (cid:1) . Namely, let K be a triangulation of S ; choosepreimages in E for the vertices of K and extend this section over the 0-skeletonof K to the 1-skeleton by using that G is connected; for each 2-simplex σ we3have thus a section over its boundary ∂σ . Using the flat connection, thissection of E over ∂σ can be deformed into a loop lying in a single fixed fiber;identifying this fiber with G we get for every σ a free homotopy class of loopsin G and hence a well defined element c ( σ ) ∈ π ( G ), since the latter is Abelian.The map c is a simplicial 2-cocycle on K with values in π ( G ) and hence definesan element in H (cid:0) S, π ( G ) (cid:1) which depends only on ρ . In this way we obtain amap o : Hom (cid:0) π ( S ) , G (cid:1) → H (cid:0) S, π ( G ) (cid:1) which assigns to ρ the obstruction o ( ρ ) ∈ H (cid:0) S, π ( G ) (cid:1) of the flat G -bundle G ( ρ ). An important observation is that if ρ and ρ lie in the same compo-nent of Hom (cid:0) π ( S ) , G (cid:1) , then the associated G -bundles G ( ρ ) and G ( ρ ) areisomorphic. As a result, the invariant o is constant on connected componentsof Hom (cid:0) π ( S ) , G (cid:1) . (See also [44] for a discussion of characteristic classed andrepresentations.) An invariant closely related to the one defined above is obtained by consideringthe central extension of G given by the universal covering { e } / / π ( G ) / / e G p / / G / / { e } , where the neutral element is taken as basepoint.A homomorphism ρ : π ( S ) → G then gives a central extension Γ ρ of π ( S )by π ( G ) in the familiar wayΓ ρ = (cid:8) ( γ, g ) ∈ π ( S ) × e G : ρ ( γ ) = p ( g ) (cid:9) . Observing now that the isomorphism classes of central extensions of π ( S ) by π ( G ) are classified by H (cid:0) π ( S ) , π ( G ) (cid:1) , we get a map c : Hom (cid:0) π ( S ) , G (cid:1) → H (cid:0) π ( S ) , π ( G ) (cid:1) . So far the discussion in §§ G . In case G = PSU(1 , π ( G ) from theorientation of D ⊂ C ; by considering the loop[0 , → PSU(1 , s (cid:18) e iπs e − iπs (cid:19) , we identify π (cid:0) PSU(1 , (cid:1) with Z ; we will denote by t ∈ π (cid:0) PSU(1 , (cid:1) theimage of 1 ∈ Z .4 (cid:0) π ( S ) , Z (cid:1) , a digression Let g ≥ S . Then π ( S ) admits as presentation * a , b , . . . , a g , b g : g Y i =1 [ a i , b i ] = e + . The orientation of S is built in, in that, when drawing the lifts to e S of theloop a b a − b − a b . . . a − g b − g , one gets a 4 g -gone whose boundary is traveledthrough in the positive sense.Now defineΓ g := * A , B , . . . , A g , B g , z : g Y i =1 [ A i , B i ] = z and [ z, A i ] = [ z, B i ] = e + . This group Γ g surjects onto π ( S ) with kernel the cyclic subgroup generatedby z , which incidentally is central. In order to see that z has infinite order,observe first that Γ is isomorphic to the integer Heisenberg group x z y : x, y, z ∈ Z by A B z . Then conclude by considering the surjectionΓ g → Γ obtained by sending A i and B i to e for i ≥
2. Thus Γ g gives a central extensionof π ( S ) by Z ; denoting by (cid:2) Γ g (cid:3) its image in H (cid:0) π ( S g ) , Z (cid:1) , we have thefollowing Proposition 3.1. H (cid:0) π ( S g ) , Z (cid:1) = Z (cid:2) Γ g (cid:3) . In fact if 0 / / Z i / / Λ / / π ( S g ) / / { e } is any central extension by Z , take lifts α j , β j ∈ Λ of a j , b j : then Q gj =1 [ α j , β j ]is independent of all choices and the image under i of a well defined n ∈ Z .Using the Baer product of extensions [1] one shows, by recurrence on n , that[Λ] = n [Γ g ] . § ρ ∈ Hom (cid:0) π ( S ) , G (cid:1) and use the identification π ( G ) → Z t c ( ρ ) ∈ H (cid:0) π ( S ) , Z (cid:1) . In terms of central extension we have then that c ( ρ ) = z ( ρ )[Γ g ] , where z ( ρ ) ∈ Z is defined by the formula g Y i =1 [ α i , β i ] = t z ( ρ ) , where α , β , . . . , α g , β g ∈ e G are lifts of ρ ( a ) , ρ ( b ) , . . . , ρ ( a g ) , ρ ( b g ). The following discussion is specific to the case where G = PSU(1 , G by homographieson D gives an action on the circle ∂ D bounding D , by orientation preservinghomeomorphisms. It will become apparent that considering homomorphismswith values in G as homomorphisms with target the group Homeo + ( S ) oforientation preserving homeomorphisms of the circle gives additional flexibility.Let us take the quotient Z \ R of R by the group generated by the integertranslations T ( x ) = x + 1 of the real line, and consider, as model of S , H + Z ( R ) = { f : R → R : increasing homeomorphisms commuting with T } . We obtain then the central extension0 / / Z / / H + Z ( R ) p / / Homeo + ( S ) / / H + Z ( R ) as universal covering of the group Homeo + ( S ), thelatter being endowed with the compact open topology. One obtains a sectionof p by associating to every f ∈ Homeo + ( S ) the unique lift f : R → R with0 ≤ f (0) <
1. The extent to which f f is not a homomorphism is measuredby an integral 2-cocycle ǫ given by f ◦ g = f ◦ g ◦ T ǫ ( f,g ) , where T is the image in H + Z ( R ) of the generator 1 ∈ Z . The Euler class is thenthe cohomology class e ∈ H (cid:0) Homeo + ( S ) , Z (cid:1) defined by ǫ . Definition 3.2.
The
Euler number e ( ρ ) of a representation ρ : π ( S ) → Homeo + ( S )6is the integer h ρ ∗ ( e ) , [ S ] i obtained by evaluation of the pullback ρ ∗ ( e ) ∈ H (cid:0) π ( S ) , Z (cid:1) of e on the fundamental class [ S ], or rather on its image under the isomorphismH ( S, Z ) → H (cid:0) π ( S ) , Z (cid:1) considered in (2.2). Contrary to § D is an instance of a Hermitian symmetric space with G -invariantK¨ahler form ω D := dz ∧ dz (1 − | z | ) , where G = PSU(1 , ρ : π ( S ) → G , consider thenthe bundle with total space the quotient D ( ρ ) := π ( S ) \ ( D × D ) of D × D bythe properly discontinuous and fixed point free action γ ( x, z ) := (cid:0) T γ x, ρ ( γ ) z (cid:1) ,and with basis S = π ( S ) \ D . Since the typical fiber D is contractible, one canconstruct, adapting the procedure described in § F : D → D .As a result, the pullback F ∗ ( ω D ) is a π ( S )-invariant 2-form on D which givesa 2-form on S denoted again, with a slight abuse of notation, by F ∗ ( ω D ). The Toledo number T( ρ ) of the representation ρ is thenT( ρ ) := 12 π Z S F ∗ ( ω D ) . Recall that we have fixed an orientation on S once and for all. Remark 3.3.
One verifies, using again geodesic homotopy, that any two ρ -equivariant smooth maps D → D are homotopic and hence, by Stokes’ theo-rem, one concludes that the de Rham cohomology class (cid:2) F ∗ ( ω D ) (cid:3) ∈ H ( S, R )is independent of F . This shows that T( ρ ) is independent of the choice of F . Let L → D be a Hermitian complex line bundle over the Poincar´e disk D and G ′ a finite covering of PSU(1 ,
1) acting by bundle isomorphisms on L ; thenthe curvature form Ω L is a G ′ -invariant 2-form on D . Given a representation ρ : π ( S ) → G ′ and a smooth equivariant map F : D → D , π ( S ) acts bybundle automorphisms on F ∗ L → D and, by passing to the quotient, we get acomplex line bundle L ( ρ ) over S . Then πı F ∗ Ω L descends to a 2-form ω L ( ρ ) S which, by Chern-Weil theory, represents the first Chern class of L ( ρ ), i. e. c (cid:0) L ( ρ ) (cid:1) = Z S ω L ( ρ ) ∈ Z . (3.1)Applying this to specific line bundles we obtain integrality properties for theToledo number. Namely, let θ → D ⊂ CP be the restriction of the tautologicalbundle over CP and θ be its square. Then Ω θ = i ω D and Ω θ = i ω D . Thegroup PSU(1 ,
1) acts by isomorphisms on θ , and (3.1) implies T ( ρ ) = 12 π Z S F ∗ ω D = − Z S ω θ ρ = − c ( θ ρ ) ∈ Z . The group SU(1 ,
1) acts naturally on θ , so for representations ρ : π ( S ) → SU(1 ,
1) the relation in (3.1) gives T ( ρ ) = 12 π Z S F ∗ ω D = − Z S ω θ ρ = − c ( θ ρ ) ∈ · Z . In particular, a representation ρ : π ( S ) → PSU(1 ,
1) lifts to SU(1 ,
1) if andonly if its Toledo number is divisible by 2.
For G = PSU(1 ,
1) we identify in the sequel π ( G ) with Z as described in § ρ ∈ Hom (cid:0) π ( S ) , G (cid:1) the obstruction class o ( ρ ) ∈ H ( S, Z ) and the class c ( ρ ) ∈ H (cid:0) π ( S ) , Z (cid:1) ; using the specific descrip-tion of the latter in terms of central extensions as Z [Γ g ], we get the invariant z ( ρ ) ∈ Z by setting c ( ρ ) = z ( ρ )[Γ g ]. Then h o ( ρ ) , [ S ] i = − z ( ρ ) , (3.2)(see [82, Lemma 2] and [101]). Turning to the Euler class, we observe thatthe injection PSU(1 , ֒ → Homeo + (cid:0) S (cid:1) is a homotopy equivalence as bothgroups retract on the (common) group of rotations. Therefore the restriction e | PSU(1 , ∈ H (cid:0) PSU(1 , , Z (cid:1) classifies the universal covering of PSU(1 ,
1) andhence for ρ : π ( S ) → PSU(1 ,
1) we have ρ ∗ ( e ) = z ( ρ )[Γ g ] , which implies that e ( ρ ) = h ρ ∗ ( e ) , [ S ] i = − z ( ρ ) . (3.3)To relate the previous invariant to the Toledo number we will recall the verygeneral principle that invariant forms on a symmetric space form a complex,with 0 as derivative, which equals the continuous cohomology of the connectedgroup of isometries. In our special case of the Poincar´e disk, this takes the8form Ω ( D ) G ∼ = H ( G, R ) , where, given ω D , we get a continuous cocycle c ( g , g ) := 12 π Z ∆ (cid:0) ,g (0) ,g g (0) (cid:1) ω D , where ∆ (cid:0) , g (0) , g g (0) (cid:1) denotes the oriented geodesic triangle having ver-tices at the points 0 , g (0) , g g (0) ∈ D . We call the resulting class κ G the K¨ahler class . In fact, it is not difficult to show that under the change ofcoefficients H ( G, Z ) → H ( G, R )the Euler class e goes to the K¨ahler class κ G . If ρ ∈ Hom (cid:0) π ( S ) , G (cid:1) , thisimplies that e ( ρ ) = h ρ ∗ ( e ) , [ S ] i = 12 π Z S F ∗ ( ω D ) = T( ρ ) . (3.4) In his seminal paper [82], J. Milnor treated the problem of characterizingthose classes in H ( S, Z ) which are Euler classes of flat principal GL +2 -bundles.The fact that, in general there are restrictions on the characteristic classes offlat principal G -bundles and in particular on o ( ρ ) ∈ H (cid:0) S, π ( G ) (cid:1) comesfrom the following observation: o is constant on connected components ofHom (cid:0) π ( S ) , G (cid:1) and the latter is a real algebraic set when G is a real algebraicgroup, thus possesses only finitely many connected components (see Proposi-tion 2.8). To get explicit restrictions, however, is not a trivial matter. In thecase of G = PSU(1 , Milnor–Woodinequality , is the following:
Theorem 3.4 ([82, 101]).
Let ρ ∈ Hom (cid:0) π ( S ) , G (cid:1) and let g be the genus of S . Then (cid:12)(cid:12)(cid:10) o ( ρ ) , [ S ] (cid:11)(cid:12)(cid:12) ≤ g − . In light of subsequent generalizations of this inequality it is instructive togive an outline of the original arguments. Consider the retraction r : PSU(1 , → K = (cid:26) ± (cid:18) e ısπ e − ısπ (cid:19) : s ∈ R / Z (cid:27) g = r ( g ) h ( g ) as a product of a rotation r ( g ) with aHermitian matrix h ( g ). Now lift r to the universal covering˜ r : ^ PSU(1 , → R in such a way that ˜ r ( e ) = 0. Then:(1) ˜ r ( t n g ) = n + ˜ r ( g ), where t is the generator of π (cid:0) PSU(1 , (cid:1) ;(2) ˜ r ( g − ) = − ˜ r ( g );(3) (cid:12)(cid:12) ˜ r ( ab ) − ˜ r ( a ) − ˜ r ( b ) (cid:12)(cid:12) < for all a, b ∈ PSU(1 , +2 ( R ) (see also [101]).Given ρ ∈ Hom (cid:0) π ( S ) , G (cid:1) let now α i , β i be lifts to ^ PSU(1 ,
1) of ρ ( a i ) and ρ ( b i ) (see § t z ( ρ ) = g Y i =1 [ α i , β i ] . On applying the above properties several times we obtain (cid:12)(cid:12) z ( ρ ) (cid:12)(cid:12) = | ˜ r ( t z ( ρ ) ) | = | ˜ r ( g Y i =1 [ α i , β i ]) | ≤ (4 g −
1) 12 = 2 g − z ( ρ ) is an integer, implies that (cid:12)(cid:12) z ( ρ ) (cid:12)(cid:12) ≤ g − . This is not quite the announced result. An additional argument is needed andcan be found in [101]; we present instead an argument in the spirit of Gromov’strick to compute the simplicial area of a surface. Namely, let p : S ′ → S bea covering of degree n ≥ p ∗ : π ( S ′ ) → π ( S ) be the resultingmorphism. The inequality above, applied to ρ ◦ p ∗ gives (cid:12)(cid:12) h o ( ρ ◦ p ∗ ) , [ S ′ ] i (cid:12)(cid:12) ≤ g ′ − , where g ′ is the genus of S ′ . Since o is a characteristic class, we have o ( ρ ◦ p ∗ ) = p ∗ (cid:0) o ( ρ ) (cid:1) , where p ∗ : H ( S, Z ) → H ( S ′ , Z ) and thus (cid:10) o ( ρ ◦ p ∗ ) , [ S ] (cid:11) = (cid:10) p ∗ (cid:0) o ( ρ ) (cid:1) , [ S ′ ] (cid:11) = (cid:10) o ( ρ ) , p ∗ [ S ′ ] (cid:11) = n (cid:10) o ( ρ ) , [ S ] (cid:11) , since p is of degree n . Using the relation g ′ − n ( g − (cid:12)(cid:12) h o ( ρ ) , [ S ] (cid:11)(cid:12)(cid:12) < n ( g −
1) + 32 n , which gives the desired inequality as soon as n ≥
2, since the left hand side isan integer.0 In Milnor’s paper [82] the construction and the property (3) of ˜ r come asa complete surprise. With hindsight, it is an instance of a quasimorphism andit is in the context of bounded cohomology that its relation to the Euler classand the specific constant in (3) are explained.Concerning the optimality of the inequality, it is shown also in [101] thatevery integer between − (2 g −
2) and 2 g − ρ h : π ( S ) → G corresponding to a hyperbolic structure h on S . For this we have at our disposal the orientation preserving isometry f h : D → D and hence the form f ∗ h ( ω D ) on S coincides with the area 2-form ω h given by the hyperbolic structure. ThusT( ρ h ) = 12 π Z S ω h = (cid:12)(cid:12) χ ( S ) (cid:12)(cid:12) = 2 g − . It should be observed that the value of the area of S can be obtained directlyfrom the formula of the area of a geodesic triangle in D applied to the triangu-lation of a ”standard” fundamental polygon, taking into account that the sumof the internal angles if 2 π . In light of this computation it is a very naturalquestion what is the nature of the homomorphisms ρ for which T( ρ ) = 2 g − Theorem 3.5 ([43]).
A representation ρ : π ( S ) → PSU(1 , corresponds toa hyperbolic structure on S if and only if T( ρ ) = 2 g − . A reformulation of Theorem 3.5 is given in (4.1). In particular, the imageof Hyp( S ) in Hom (cid:0) π ( S ) , G (cid:1) being the preimage of 2 g − − ( n ), for n ∈ Z ∩ (cid:2) − (2 g − , g − (cid:3) are exactly the componentsof Hom (cid:0) π ( S ) , G (cid:1) [46]. The component where T = 2 − g corresponds tohyperbolic structures on S with the reversed orientation. This theorem takes its motivation in the question of what are the possibledegrees of continuous maps from a compact oriented surface S to itself. If S iseither the sphere or the torus, then maps of arbitrarily high degree exist. Thisis not the case anymore if the genus of S is at least two, and more generallywe have the following: Theorem 3.6 ([64]).
Let f : S → S be a continuous map between compactoriented surfaces S i of genus at least . Then | deg f | ≤ | χ ( S ) || χ ( S ) | , with equality if and only if f is homotopic to a covering map, necessarily ofdegree χ ( S ) χ ( S ) .Proof. Let f ∗ : π ( S ) → π ( S ) be the homomorphism induced on the levelof fundamental groups, and pick ρ ∈ Hom (cid:0) π ( S ) , G (cid:1) . Then (cid:10) o ( ρ ◦ f ∗ ) , [ S ] (cid:11) = (cid:10) f ∗ (cid:0) o ( ρ ) (cid:1) , [ S ] (cid:11) = (cid:10) o ( ρ ) , f ∗ (cid:0) [ S ] (cid:1)(cid:11) = deg f (cid:10) o ( ρ ) , [ S ] (cid:11) . (3.5)Specializing now to ρ = ρ h , for h ∈ Hyp( S ), we get (cid:10) o ( ρ h ) , [ S ] (cid:11) = (cid:12)(cid:12) χ ( S ) (cid:12)(cid:12) while the Milnor-Wood inequality gives (cid:12)(cid:12) h o ( ρ h ◦ f ∗ ) , [ S ] i (cid:12)(cid:12) ≤ (cid:12)(cid:12) χ ( S ) (cid:12)(cid:12) which, together with (3.5), gives the inequality on | deg f | .Assume now that we have equality and, without loss of generality, thatT( ρ h ◦ f ∗ ) = (cid:12)(cid:12) χ ( S ) (cid:12)(cid:12) . Then Goldman’s theorem implies that ρ h ◦ f ∗ corresponds to a hyperbolicstructure on S and, in particular, f ∗ is injective. Letting p : T → S denotethe covering of S corresponding to the image of f ∗ , we have that f ∗ : π ( S ) → π ( T )is an isomorphism, which implies that T is compact and (by Nielsen’s theorem)that f ∗ is induced by a homeomorphism F : S → T .We have then that the homomorphisms f ∗ and ( p ◦ F ) ∗ coincide; let now˜ f : ˜ S → ˜ S and ] p ◦ F : ˜ S → ˜ S be lifts of f . These are continuous maps whichare equivariant with respect to the same homomorphism π ( S ) → π ( S ).Upon choosing a hyperbolic metric on S , we conclude by using a geodesichomotopy that ] p ◦ F and ˜ f are equivariantly homotopic and hence p ◦ F and f are homotopic. Let S be a compact (connected, oriented) surface. Then the image of Hyp( S ) inHom (cid:0) π ( S ) , G (cid:1) , G = PSU(1 , t be the generator of π ( G ), a , b , . . . , a g , b g π ( S ) defined in § G × G −→ e G ( g, h ) [ g, h ] e where [ g, h ] e is the commutator of any two lifts of g and h , Goldman’s theorem(Theorem 3.5) can be restated as δ (cid:0) Hyp( S ) (cid:1) = ( ρ ∈ Hom (cid:0) π ( S ) , G (cid:1) : g Y i =1 (cid:2) ρ ( a i ) , ρ ( b i ) (cid:3) e = t − g ) . (4.1)The aim of this section is to present a circle of ideas, rooted in the theory ofbounded cohomology, which will, among other things, lead to an analogousexplicit description of δ (cid:0) Hyp( S ) (cid:1) in the case in which S is not compact. Wewill however always assume that π ( S ) is finitely generated; equivalently S isdiffeomorphic to the interior of a compact surface with boundary. The genus g of this surface and the number n of boundary components together determine S up to diffeomorphism. We say that S is of finite topological type .The first observation is that the invariants introduced in § S is not compact. In fact, for a connected surface S the following areequivalent:(1) H ( S, Z ) = H (cid:0) π ( S ) , Z (cid:1) = 0;(2) π ( S ) is a free group;(3) S is not compact.Elaborating a little on (2), if r is the rank of π ( S ) as a free group, we haveclearly that Hom (cid:0) π ( S ) , G (cid:1) ∼ = G r and, as a result, this space of homomor-phisms is always connected. Thus δ (cid:0) Hyp( S ) (cid:1) will not be a connected compo-nent.The second observation, and this will lead us in the right direction, is toconsider more closely the inclusions δ (cid:0) Hyp( S ) (cid:1) ⊂ Hom (cid:0) π ( S ) , G (cid:1) ⊂ Hom (cid:0) π ( S ) , Homeo + (cid:0) S (cid:1) (cid:1) in the case in which S is compact. For h , h ∈ Hyp( S ), the diffeomorphism f h ◦ f − h : D → D clearly conjugates ρ h to ρ h within Diff + ( D ). Since S is com-pact, f h ◦ f − h is a quasi isometry; it is then a fundamental fact in hyperbolicgeometry that f h ◦ f − h extends to an (orientation preserving) homeomor-phism of S = ∂ D . Thus any two representations in δ (cid:0) Hyp( S ) (cid:1) are conjugatein Homeo + (cid:0) S (cid:1) and it is an easy exercise to see that any ρ ∈ Hom (cid:0) π ( S ) , G (cid:1) that is conjugate to an element in δ (cid:0) Hyp( S ) (cid:1) in Homeo + (cid:0) S (cid:1) in fact belongsto δ (cid:0) Hyp( S ) (cid:1) ; indeed such a representation ρ is injective with discrete image.Thus a full invariant of conjugacy on Hom (cid:0) π ( S ) , Homeo + (cid:0) S (cid:1) (cid:1) would leadto a characterization of δ (cid:0) Hyp( S ) (cid:1) within Hom (cid:0) π ( S ) , G (cid:1) !3We will now develop this line of thought in the case in which S is of finitetopological type. We assume that S has a fixed orientation and let Σ denote acompact surface of genus g with n boundary components such that S = int(Σ).Then π ( S ) admits a presentation * a , b , . . . , a g , b g , c , . . . , c n : g Y i =1 [ a i , b i ] n Y j =1 c j = e + . (4.2)Here each c i is freely homotopic to the i -th component of ∂ Σ with orienta-tion compatible with the chosen orientation on Σ. Let now h be a completehyperbolic metric on S . We have then two possibilities for ρ h ( c i ):(1) ρ h ( c i ) is parabolic: it has a unique fixed point ξ i ∈ ∂ D and for the interior C i of an appropriate horocycle based at ξ i , the quotient h ρ h ( c i ) i\ C i is of finite area and embeds isometrically into ρ h (cid:0) π ( S ) (cid:1) \ D . It is aneighborhood of the i -th end of S .(2) ρ h ( c i ) is hyperbolic: it has an invariant axis a i ⊂ D which determinesa half plane H i ⊂ D such that ∂H i and a i have opposite orientation.The quotient h ρ h ( c i ) i\ H i embeds isometrically into ρ h (cid:0) π ( S ) (cid:1) \ D . It isof infinite area and a neighborhood of the i -th end.Let Λ h ⊂ ∂ D be the limit set of ρ h (cid:0) π ( S ) (cid:1) . Then either Λ h = ∂ D , equiva-lently ( S, h ) is of finite area, or Λ h = ∂ D , in which cases it is a Cantor set; theconnected components of ∂ D \ Λ h are then in bijective correspondence withthe set of elements in (cid:8) γ ∈ π ( S ) : γ is conjugate to a boundary loop c i s. t. ρ h ( c i ) is hyperbolic (cid:9) Thus, if h , h ∈ Hyp( S ) are such that h has finite area while h has infinitearea, then ρ h gives a minimal action of π ( S ) on ∂ D , while ρ h gives anaction on ∂ D which admits Λ h as minimal set. In particular ρ h , ρ h cannotbe conjugated in Homeo + (cid:0) S (cid:1) . Let us however consider the diffeomorphism F := f h ◦ f − h : D → D . Proposition 4.1.
Assume that h is of finite area. Then F extends to acontinuous map ϕ : ∂ D → ∂ D which is weakly monotone. This proposition is a slight generalization of a classical result stating thatif h , h have finite area, then since the isomorphism ρ h ◦ ρ − h is “type pre-serving”, f h ◦ f − h extends to a homeomorphism of ∂ D .To explain the statements of the proposition, recall that the circle S ∼ = ∂ D is equipped with its canonical positive orientation and this gives a naturalnotion for triples of points to be positively oriented. We have the following Definition 4.2.
An (arbitrary) map ϕ : S → S is weakly monotone ifwhenever x, y, z ∈ S are such that ϕ ( x ) , ϕ ( y ) , ϕ ( z ) are distinct, then ( x, y, z )and (cid:0) ϕ ( x ) , ϕ ( y ) , ϕ ( z ) (cid:1) have the same orientation.4 Typically the map in Proposition 4.1 is collapsing a connected componentin ∂ D \ Λ h corresponding to γ ∈ π ( S ) to the corresponding fixed point in ∂ D of the parabolic element ρ h ( γ ). We have for every x ∈ ∂ D ϕ ◦ ρ h ( γ )( x ) = ρ h ( γ ) ◦ ϕ ( x )and we say that ϕ semiconjugates ρ h to ρ h . It is in order to reverse thisprocess that in the definition of weakly monotone map one allows discontinuousmaps. For instance ϕ − ( x ) is always an interval and if we set ψ ( x ) equal tothe left endpoint of ϕ − ( x ), then ψ is weakly monotone and ρ h ( γ ) ◦ ψ ( x ) = ψ ◦ ρ h ( γ )( x ) . Thus given now arbitrary homomorphisms ρ , ρ : Γ → Homeo + (cid:0) S (cid:1) definedon a group Γ, we say that ρ and ρ are semiconjugate if there exists ϕ : S → S weakly monotone such that for every γ ∈ Γ ϕ ◦ ρ ( γ ) = ρ ( γ ) ◦ ϕ . It is now clear that semiconjugation is an equivalence relation. In our specificsituation we have then
Corollary 4.3. (1)
For any h , h ∈ Hyp( S ) , ρ h and ρ h are semiconju-gate. (2) If h ∈ Hyp( S ) and ρ ∈ Hom (cid:0) π ( S ) , G (cid:1) is semiconjugate to ρ h , then ρ ∈ δ (cid:0) Hyp( S ) (cid:1) .Proof. The first assertion follows from the discussion above. We will nowindicate the main points entering in the proof of the second one.Let ϕ : ∂ D → ∂ D be weakly monotone with ϕ ◦ ρ ( γ ) = ρ h ( γ ) ◦ ϕ (4.3)and assume, in virtue of the first part, that h has finite area. Since ρ h (cid:0) π ( S ) (cid:1) acts minimally on ∂ D , we must have that im ϕ = ∂ D . It is easy to see that fora weakly monotone map this implies that ϕ is continuous. Then we deducefrom (4.3) that ρ is injective and with discrete image. Thus Γ := ρ (cid:0) π ( S ) (cid:1) isa finitely generated discrete subgroup of PSU(1 ,
1) and hence Γ \ D is topolog-ically of finite type.One uses then ϕ to check that the isomorphism ρ : π ( S ) → Γ sends h c i i , foreach i , isomorphically into the fundamental group of a boundary componentof Γ \ D and that each boundary component is so obtained.Then an appropriate version of the Nielsen realization implies that ρ isimplemented by a diffeomorphism S → Γ \ D , by means of which we producethe hyperbolic structure h ′ for which ρ = ρ h ′ .5 The discussion of the preceding section shows that semiconjugation is a natu-ral notion of equivalence for group actions by homeomorphisms of the circle,at least in the framework of the questions regarding hyperbolic structures. Adifferent context is provided by a paraphrase of a famous theorem of Poincar´econcerning rotation numbers of homeomorphisms, namely two orientation pre-serving homeomorphisms of the circle are semiconjugate if and only if they havethe same rotation number.Remarkably, there is an invariant generalizing the rotation number of asingle homeomorphism to arbitrary group actions and which is a completeinvariant of semiconjugacy: it is the bounded Euler class, introduced by Ghysin [42] and whose main features we now describe briefly.For this let us recall that bounded cohomology can be defined by restrictingto bounded cochains in the (inhomogeneous) bar resolution. Let A = R or Z ,and G be any group. Denote by C n ( G, A ) the space of function from G n to A and by C nb ( G, A ) := (cid:8) f ∈ C n ( G, A ) : sup g =( g , ··· ,g n ) ∈ G n | f ( g ) | < ∞ (cid:9) thesubspace of bounded functions. Defining the boundary map d n : C n ( G, A ) → C n +1 ( G, A )by d n f ( g , · · · , g n +1 ) = f ( g , · · · , g n +1 )+ n X i =1 ( − i f ( g , · · · , g i − , g i g i +1 , g i +2 , · · · , g n +1 )+ ( − n +1 f ( g , · · · , g n ) , we obtain the complex (cid:0) C • ( G, A ) , d • (cid:1) , whose cohomology is the group coho-mology H • ( G, A ), and the sub-complex (cid:0) C • b ( G, A ) , d • (cid:1) , whose cohomology isthe bounded cohomology H • b ( G, A ) of G .Bounded cohomology behaves very differently from usual cohomology, forexample, the second bounded cohomology H ( F r , R ) of a nonabelian free groupis infinite dimensional. This different behavior will allow us to define boundedanalogues of the invariants introduced in §
3, which are meaningful when S isnoncompact, and give finer information even in the case when S is compact(see e.g. Corollary 4.5).Recall that, in the notation of § ǫ for the Eulerclass e ∈ H (cid:0) Homeo + (cid:0) S (cid:1) , Z (cid:1) was given by f ◦ g = f ◦ g ◦ T ǫ ( f,g ) , where f and g are the unique lifts to R of f, g ∈ Homeo + (cid:0) S (cid:1) such that0 ≤ f (0) , g (0) < , T : R → R is defined by T ( x ) := x + 1.Since f is increasing and commutes with T , we have f (cid:0) g (0) (cid:1) ∈ (cid:2) f (0) , f (0) + 1 (cid:1) and since f g (0) ∈ [0 , ǫ ( f, g ) ∈ { , } and hence in particular ǫ is a bounded cocycle. The class e b ∈ H (cid:0) Homeo + (cid:0) S (cid:1) , Z (cid:1) so obtained is called the bounded Euler class and given any homomorphism ρ : Γ → Homeo + (cid:0) S (cid:1) , ρ ∗ ( e b ) ∈ H (Γ , Z ) is called the bounded Euler class ofthe action given by ρ . We have then the following Theorem 4.4 ([42]).
The bounded Euler class of a homomorphism ρ : Γ → Homeo + (cid:0) S (cid:1) is a full invariant of semiconjugation. The relation with the classical rotation number is then the following. Recallthat the translation number τ ( ϕ ) ∈ R of a homeomorphism ϕ ∈ H + Z ( R ) is givenby τ ( ϕ ) := lim n →∞ ϕ n (0) n . Then τ has the following remarkable properties (compare to the properties of˜ r in § τ is continuous;(2) τ ( ϕ ◦ T m ) = τ ( ϕ ) + m , for m ∈ Z ;(3) τ ( ϕ k ) = kτ ( ϕ );(4) (cid:12)(cid:12) τ ( ϕψ ) − τ ( ϕ ) − τ ( ψ ) (cid:12)(cid:12) ≤
1, for all ϕ, ψ ∈ H + Z ( R ).In the language of bounded cohomology, this says that τ is a continuous ho-mogeneous quasimorphism. Then for f ∈ Homeo + (cid:0) S (cid:1) the rotation numberof f is rot ( f ) := τ ( f ) mod Z which is well defined in view of (2).Given now f ∈ Homeo + (cid:0) S (cid:1) , consider h f : Z → Homeo + (cid:0) S (cid:1) n f n to obtain an invariant h ∗ f ( e b ) ∈ H ( Z , Z ). Writing the long exact sequence inbounded cohomology [41, Proposition 1.1] associated to0 / / Z / / R / / R / Z / / / / Hom( Z , R / Z ) δ / / H ( Z , Z ) / / (cid:0) δ − h ∗ f ( e b ) (cid:1) (1) = rot ( f ) . (4.4)It should be noticed in passing that the definition of rot ( f ) involves taking alimit, while the left hand side of (4.4) only involves purely algebraic construc-tions. The proof of the not straightforward implication of Ghys’ theorem goes asfollows. Let ρ , ρ : Γ → Homeo + ( S ) be homomorphisms, with ρ ∗ ( e b ) = ρ ∗ ( e b ). Hence ρ ∗ ( e ) = ρ ∗ ( e ) and, by replacing Γ with a suitable centralextension by Z we may assume that ρ and ρ lift to homomorphisms e ρ , e ρ : Γ → H + Z ( R ). But then e ρ i ( γ ) = ρ i ( γ ) T c i ( γ ) and the hypothesis that ρ , ρ have the same bounded Euler class is equivalentto saying that we may choose the lifts e ρ and e ρ such that c − c : Γ → Z is bounded. Thus: e ϕ ( x ) := sup γ ∈ Γ (cid:8) e ρ ( γ ) − e ρ ( γ )( x ) : γ ∈ Γ (cid:9) < + ∞ is well defined for every x and gives a monotone map R → R commuting with T and satisfying e ϕ e ρ ( η ) = e ρ ( η ) e ϕ , for all η ∈ Γ. This shows that ρ and ρ are semiconjugate.In order to complete one of the descriptions of the image of Hyp( S ) underthe map δ in Hom (cid:0) π ( S ) , G (cid:1) let now S be again an oriented surface of finitetopological type, where we do not exclude the case in which S is compact. Wehave seen that for any two h , h ∈ Hyp( S ), the homomorphisms ρ h and ρ h are semiconjugate in Homeo + (cid:0) S (cid:1) and hence, by the easy direction of Ghys’theorem, we have that ρ ∗ h ( e b ) = ρ ∗ h ( e b ) . Let κ b S ∈ H (cid:0) π ( S ) , Z (cid:1) denote the class so obtained. Then In fact we have used H ( Z , Z ) = 0 and H ( Z , R ) = 0. The first equality follows from thefact that there are no (nonzero) bounded homomorphisms into R while the second followsfrom the elementary fact that if ψ : Z → R is a quasimorphism and α := lim n →∞ ψ ( n ) n ,then ψ is at bounded distance from the homomorphism n nα . Corollary 4.5. δ (cid:0) Hyp( S ) (cid:1) = (cid:8) ρ ∈ Hom (cid:0) π ( S ) , G (cid:1) : ρ ∗ ( e b ) = κ b S (cid:9) . Proof.
The inclusion ⊂ has already been discussed.If now ρ ∗ ( e b ) = κ b S , then Ghys’ theorem implies that ρ is semiconjugate toan element in δ (cid:0) Hyp( S ) (cid:1) and the assertion follows from Corollary 4.3. In this section we describe two ways in which one can associate a (real) numberto the bounded Euler class; this will give the two invariants mentioned in thetitle. The fact that they coincide is then an essential result containing a lot ofinformation.Recall that S is a surface of finite topological type and hence we mayconsider it as the interior of a compact surface Σ with boundary ∂ Σ. Let now ρ : π (Σ) → Homeo + (cid:0) S (cid:1) be a homomorphism and ρ ∗ ( e b ) ∈ H (cid:0) π (Σ) , Z (cid:1) its bounded Euler class. We proceed now to define the bounded Euler number of ρ . First we use that the classifying map Σ → Bπ (Σ) is a homotopy equiv-alence in order to obtain a natural isomorphism H (cid:0) π (Σ) , Z (cid:1) → H (cid:0) Σ , Z (cid:1) bymeans of which we consider, keeping the same notation, the class ρ ∗ ( e b ) as abounded singular class on Σ. (See [51] for the definition of singular boundedcohomology.) The inclusion ∂ Σ ֒ → Σ gives in a straightforward way a longexact sequence in bounded cohomology with coefficients in A = Z , R , whoserelevant part for us readsH ( ∂ Σ , A ) / / H (Σ , ∂ Σ , A ) f A / / H (Σ , A ) / / H ( ∂ Σ , A )which gives for A = Z / / H (Σ , ∂ Σ , Z ) f Z / / H (Σ , Z ) / / H ( ∂ Σ , Z )and for A = R / / H (Σ , ∂ Σ , R ) f R / / H (Σ , R ) / / ( ∂ Σ , A ) = 0 for A = R , Z ;(2) H ( ∂ Σ , R ) = 0.9As a result we have that if we consider ρ ∗ ( e b ) as a real bounded class on Σ, itcorresponds to a unique relative class f − R (cid:0) ρ ∗ ( e b ) (cid:1) ∈ H (Σ , ∂ Σ , R ) . The latter can then be seen as an ordinary singular relative class and hencecan be evaluated on the relative fundamental class, thus leading to the boundedEuler number : e b ( ρ ) := (cid:10) f − R (cid:0) ρ ∗ ( e b ) (cid:1) , [Σ , ∂ Σ] (cid:11) . (4.5)Two important remarks are in order here. First, the definition of thisinvariant not only involves π ( S ) but also the surface S itself; this is essentialif this invariant is to detect hyperbolic structures on S (see Remark 2.7).Second, let us denote by ρ ∗ ( e b R ) ∈ H (cid:0) Homeo + (cid:0) S (cid:1) , R (cid:1) the real boundedclass obtained by considering the cocycle ǫ as taking values in R . Then e b ( ρ )depends in fact only on the real class ρ ∗ ( e b R ); the extent to which this (real)class determines ρ (up to semiconjugation), is completely understood (see [12]).The bounded Euler number e b ( ρ ) is in general not an integer. Remarkably,one can give an explicit formula for the ”fractional part” of e b ( ρ ); indeed,combining the long exact sequence associated to ∂ Σ → Σ together with theone associated to the short exact sequence0 / / Z / / R / / R / Z / / e b ( ρ ) = − n X i =1 rot ρ ( c i ) mod Z . In fact, using this, one can establish a general formula for e b ( ρ ): Theorem 4.6 ([18]).
Let S be an oriented surface of finite topological typewith presentation of its fundamental group π ( S ) = * a , b , . . . , a g , b g , c , . . . , c n : g Y i =1 [ a i , b i ] n Y j =1 c j = e + as defined in (4.2) . Let ρ : π ( S ) → Homeo + (cid:0) S (cid:1) be a homomorphism and let τ denote the translation quasimorphism. Then (1) If S is compact (that is n = 0 ), then e ( ρ ) = e b ( ρ ) = τ g Y i =1 (cid:2) ρ ( a i ) , ρ ( b i ) (cid:3) e ! . if S is noncompact (that is n ≥ ), then e b ( ρ ) = − n X i =1 τ (cid:0) ˜ ρ ( c i ) (cid:1) , where ˜ ρ : π ( S ) → H + Z ( R ) denotes a homomorphism lifting ρ . Now we will turn to the description of the bounded Toledo number. Itsdefinition is based on the use of a very general operation in bounded coho-mology called ”transfer”, together with a description of the second boundedcohomology of G = PSU(1 , < G be a lattice in G . One has the isomorphismH • b (Γ , R ) ∼ = H • cb (cid:0) G, L ∞ (Γ \ G ) (cid:1) (4.6)analogous to the Eckmann–Shapiro isomorphism in ordinary cohomology. HereH • cb denotes the bounded continuous cohomology for whose definition thereader is referred to [83] or also [18, § G , but at the expense of replacing the trivialΓ-module R by the quite intractable G -module L ∞ (Γ \ G ). This principle isvery general and does not require Γ to be a lattice, but this hypothesis willnow allow us to ”simplify” the coefficients: indeed, let µ be the G -invariantprobability measure on Γ \ G . Then L ∞ (Γ \ G ) −→ R f Z Γ \ G f ( x )d µ ( x ) (4.7)is a morphism of G -modules, where R is then the trivial G -module. Composingthe induction isomorphism (4.6) with the morphism in cohomology induced bythe morphism of coefficients (4.7) and specializing to degree 2 leads to a map,called the transfer map T b : H (Γ , R ) → H ( G, R )which is linear and norm decreasing. The interest of this construction lies inthe fact that, while H (Γ , R ) is infinite dimensional, say when G is a real rankone group, the space H ( G, R ) is finite dimensional if G is a connected Liegroup and in fact one dimensional for G = PSU(1 , § c is bounded by , as the area ofgeodesic triangles in D is bounded by π , and therefore we can use c to definea bounded continuous class κ b G ∈ H ( G, R ) called the bounded K¨ahler class.We have then: Proposition 4.7.
Let G = PSU(1 , ֒ → Homeo + ( ∂ D ) be the natural inclu-sion. Then ( G, R ) = R κ b G ; (2) The restriction e b R | G to G of the real bounded Euler class equals thebounded K¨ahler class κ b G in H ( G, R ) . The first assertion is in fact a very special case of a more general resultand we will treat this later in its proper context; suffices it to say here that wealready know that the comparison mapH ( G, R ) → H ( G, R ) = R κ G is surjective as κ b G is sent to κ G ; the kernel of this map is then described bythe space of continuous quasimorphisms on PSU(1 ,
1) and it is easy to see thatthey must be bounded. Hence the comparison map is injective.For the second statement one needs an explicit relation between the cocycle ǫ used to define the Euler class and the orientation cocycle on S . Recall thatthe orientation cocycle or : S × S × S → Z is defined byor( x, y, z ) := x, y, x are cyclically positively oriented0 if at least two coordinates coincide − x, y, x are cyclically negatively oriented.A formula relating the orientation cocycle directly to hyperbolic geometry isgiven by or( x, y, z ) = 1 π Z ∆( x,y,z ) ω D where now x, y, z ∈ ∂ D and ∆( x, y, z ) denotes the oriented geodesic idealtriangle with vertices x, y, z . Here we have taken ∂ D as a model of S withthe identification Z \ R → ∂ D t e πıt and we denote again by ǫ the corresponding cocycle on Homeo + ( ∂ D ). Lemma 4.8 ([61]).
For f, g ∈ Homeo + ( ∂ D ) ǫ ( f, g ) = −
12 or (cid:0) , f (1) , f g (1) (cid:1) + dβ ( f, g ) , where β ( f ) := ( if f (1) = 1 − if f (1) = 1 . g , g ) c ( g , g ) = 12 π Z ∆(0 ,g ,g g ω D and ( g , g ) π Z ∆(1 ,g ,g g ω D are cohomologous in the complex of bounded (Borel) cochains; since the secondcocycle is then essentially or (cid:0) , g (1) , g g (1) (cid:1) , Lemma 4.8 allows to conclude.Now we are in the position to define the bounded Toledo number . Define,using Proposition 4.7(1), the linear form t b : H (Γ , R ) → R ,T b ( α ) = t b ( α ) κ b G . Then given a surface S of finite topological type as before, fix a hyperbolization,i.e. a homomorphism corresponding to a complete hyperbolic structure on S , h : π ( S ) → G with image a lattice Γ = h (cid:0) π ( S ) (cid:1) in G . Given now anyhomomorphism ρ : π ( S ) → Homeo + (cid:0) S (cid:1) , we define the bounded Toledonumber of ρ as T b ( ρ, h ) := t b (cid:0) ( ρ ◦ h − ) ∗ ( e b R ) (cid:1) . Observe that the hyperbolization h is involved in the definition, but we willsee that T b ( ρ, h ) is independent of h as a consequence of the relation betweenthe bounded Toledo and the bounded Euler numbers.Concerning this relation consider the following diagramH (cid:0) π ( S ) , R (cid:1) ∼ = / / H (Σ , R ) H (Σ , ∂ Σ , R ) f R o o h · , [Σ ,∂ Σ] i (cid:15) (cid:15) H (Γ , R ) h ∗ O O t b (cid:15) (cid:15) RR where, as before, h : π ( S ) → Γ is a hyperbolization with finite area.
Theorem 4.9 ([18, Theorem 3.3]).
For every α ∈ H (cid:0) π ( S ) , R (cid:1) ∼ = H (Σ , R ) ,we have that t b (cid:0) ( h ∗ ) − ( α ) (cid:1)(cid:12)(cid:12) χ ( S ) (cid:12)(cid:12) = (cid:10) f − R ( α ) , [Σ , ∂ Σ] (cid:11) . α = ρ ∗ ( e b ) this provides the desired equality between thebounded Toledo number and the bounded Euler number. In computing bounded cohomology one faces a priori the same difficultiesas for the usual cohomology, namely that the bar resolution contains manycoboundaries; the ideal situation then would be if one had a complex givingbounded cohomology and where all differentials are zero. While this can beachieved for various ordinary cohomology theories, we do not know of an ana-logue of either Hodge theory or Van Est isomorphism for bounded cohomology.What can be achieved for the moment is a good model in degree two. Thisfollows from the theory developed in [19, 20, 83] and of which we recall a fewconsequences in our case at hand.
Proposition 4.10.
Let G = PSU(1 , and L < G a closed subgroup whoseaction on ∂ D × ∂ D is ergodic. Then there is a canonical isomorphism H ( L, R ) ∼ = Z L ∞ alt (cid:0) ( ∂ D ) (cid:1) L . Here Z L ∞ alt (cid:0) ( ∂ D ) (cid:1) L := (cid:8) f : ( ∂ D ) → R : f is measurable, essentially bounded,alternating, L -invariant and f ( x , x , x ) − f ( x , x , x ) + f ( x , x , x ) − f ( x , x , x ) = 0for a. e. ( x , x , x , x ) ∈ ( ∂ D ) (cid:9) . In particular for L = G = PSU(1 ,
1) it is plain that Z L ∞ alt (cid:0) ( ∂ D ) (cid:1) L isone-dimensional, generated by the orientation cocycle. In addition one canverify that under the isomorphism in Proposition 4.10, κ b G is sent to or. Thisimplies immediately the following Corollary 4.11. k κ b G k = . This, in turn, together with the fact that T b is norm decreasing, implies: Corollary 4.12.
For every α ∈ H (Γ , R ) , (cid:12)(cid:12) t b ( α ) (cid:12)(cid:12) ≤ k α k . Another feature of this model for bounded cohomology is that the transferT b takes a particularly simple and useful form:4 Proposition 4.13 ([83]).
Let Γ be a lattice and µ the G -invariant probabilitymeasure on Γ \ G . Then T b : H (Γ , R ) → H ( G, R ) is given by the map Z L ∞ alt (cid:0) ( ∂ D ) (cid:1) Γ → Z L ∞ alt (cid:0) ( ∂ D ) (cid:1) G α T b ( α ) , where T b ( α )( x, y, z ) = Z Γ \ G α ( gx, gy, gz ) dµ ( g ) . In particular Z Γ \ G α ( gx, gy, gz ) dµ ( g ) = t b ( α )2 or( x, y, z ) for almost every ( x, y, z ) ∈ ( ∂ D ) . With this at hand we can now deduce a characterization of the boundedK¨ahler class which lies at the heart of our approach:
Theorem 4.14.
Let Γ < G be a lattice. For every α ∈ H (Γ , R ) (cid:12)(cid:12) t b ( α ) (cid:12)(cid:12) ≤ k α k with equality if and only if α is proportional to the restriction κ b G | Γ to Γ of thebounded K¨ahler class.Proof. Taking up the formula in Proposition 4.13 in terms of measurable co-cycles, (cid:12)(cid:12) t b ( α ) (cid:12)(cid:12) = 2 k α k reads Z Γ \ G α ( gx, gy, gz ) dµ ( x ) = k α k ∞ or( x, y, z ) , or Z Γ \ G (cid:0) k α k ∞ or( gx, gy, gz ) − α ( gx, gy, gz ) (cid:1) dµ ( g ) = 0 . For positively oriented triples ( x, y, z ) this implies that for almost every g k α k ∞ or( gx, gy, gz ) = α ( gx, gy, gz )and hence α = k α k ∞ or in Z L ∞ alt (cid:0) ( ∂ D ) (cid:1) G .Let κ b S, R ∈ H (cid:0) π ( S ) , R (cid:1) denote the class obtained by considering κ b S ∈ H (cid:0) π ( S ) , Z (cid:1) (see the end of § κ b S, R .5 Corollary 4.15.
Let S be of finite topological type realized as the interior of Σ . Then for every α ∈ H (cid:0) π ( S ) , R (cid:1) = H (Σ , R ) (cid:12)(cid:12)(cid:10) f − R ( α ) , [Σ , ∂ Σ] (cid:11)(cid:12)(cid:12) ≤ k α k (cid:12)(cid:12) χ ( S ) (cid:12)(cid:12) , with equality if and only if α is a multiple of κ b S, R . In this section we fulfill the promise to give explicit equations for the image inHom (cid:0) π ( S ) , G (cid:1) of Hyp( S ) under the map δ , in the case where S is a surfaceof finite topological type.Let thus ρ : π ( S ) → Homeo + (cid:0) S (cid:1) be a homomorphism; then we havethe following result which is a first characterization of the maximality of thebounded Euler number of ρ . Corollary 4.16.
We have (cid:12)(cid:12) e b ( ρ ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) χ ( S ) (cid:12)(cid:12) , where equality holds if and only if ρ ∗ ( e b R ) = ± κ b S, R .Proof. Combine Theorem 4.9 with Corollary 4.15.In fact, if S has n punctures and is of genus g , then we have that e b ( ρ ) =2 g − n if and only if ρ ∗ ( e b R ) = κ b S, R ; observe that for every h ∈ δ (cid:0) Hyp( S ) (cid:1) ,we have κ b S, R = h ∗ ( e b R ), so that ρ and h have the same real bounded Euler class.Keeping in mind that ρ ∗ ( e b ) ∈ H (cid:0) π ( S ) , Z (cid:1) determines ρ up to semiconjugacy,this is a rather strong conclusion and in fact we have the following Theorem 4.17.
Let ρ : π ( S ) → Homeo + (cid:0) S (cid:1) be a homomorphism with ρ ∗ ( e b R ) = κ b S, R . Then ρ is semiconjugate to an element in δ (cid:0) Hyp( S ) (cid:1) . In particular δ (cid:0) Hyp( S ) (cid:1) = (cid:8) ρ ∈ Hom (cid:0) π ( S ) , Homeo + ( S ) (cid:1) : ρ ∗ ( e b R ) = κ b S, R (cid:9) . Note that Theorem 4.17 combined with Corollary 4.16 gives a generalizationof Matsumoto’s theorem proved in [81] for compact surfaces to surfaces offinite topological type (see also [61] for a different proof in the case when S isa compact surface).6 Theorem 4.18.
Let S be a surface of finite type and let ρ i : π ( S ) → Homeo + (cid:0) S (cid:1) , i = 1 , , be homomorphisms with (cid:12)(cid:12) e b ( ρ i ) (cid:12)(cid:12) = (cid:12)(cid:12) χ ( S ) (cid:12)(cid:12) . Then ρ and ρ are semiconjugate.In particular, every ρ : π ( S ) → Homeo + (cid:0) S (cid:1) with (cid:12)(cid:12) e b ( ρ ) (cid:12)(cid:12) = (cid:12)(cid:12) χ ( S ) (cid:12)(cid:12) isinjective with discrete image. Below we will give the proof of this theorem in the case in which ρ takesvalues in G = PSU(1 , . Proof.
Let h, ρ : π ( S ) → G be homomorphisms, and suppose that h is ahyperbolization of finite area and ρ satisfies the hypotheses of the theorem.Then ρ ∗ ( e b R ) = h ∗ ( e b R ) . Consider now the exact sequenceHom (cid:0) π ( S ) , R / Z (cid:1) b / / H (cid:0) π ( S ) , Z (cid:1) / / H (cid:0) π ( S ) , R (cid:1) in order to conclude that there is a homomorphism χ : π ( S ) → R / Z with ρ ( e b ) − h ∗ ( e b ) = b ( χ ) . (4.8)We now proceed to show that χ is trivial. Since χ | [Γ , Γ] = 0, we deduce that (cid:0) ρ | [Γ , Γ] (cid:1) ∗ ( e b ) = (cid:0) h | [Γ , Γ] (cid:1) ∗ ( e b )and hence, by Ghys’ theorem, there exists a weakly monotone map ϕ : ∂ D → ∂ D with ϕ (cid:0) ρ ( η ) x (cid:1) = h ( η ) (cid:0) ϕ ( x ) (cid:1) , for all η ∈ (cid:2) π ( S ) , π ( S ) (cid:3) and all x ∈ ∂ D .If now ϕ had a point of discontinuity, there would be a nonempty openinterval in the complement of im ϕ and in particular im ϕ = ∂ D ; but h (cid:0) π ( S ) (cid:1) and hence h (cid:0) [ π ( S ) , π ( S )] (cid:1) act minimally on ∂ D and im ϕ is invariant underthe latter subgroup, which is a contradiction. Therefore ϕ is continuous andsurjective. This implies in a straightforward way that ρ (cid:0) [ π ( S ) , π ( S )] (cid:1) is adiscrete subgroup of G ; since the limit set of ρ (cid:0) [ π ( S ) , π ( S )] (cid:1) is either a Cantorset or ∂ D , we deduce that ρ (cid:0) [ π ( S ) , π ( S )] (cid:1) , and hence ρ (cid:0) π ( S ) (cid:1) , is Zariskidense. Thus ρ (cid:0) π ( S ) (cid:1) is either dense or discrete in G but since it normalizes anontrivial discrete subgroup and since G is simple, ρ (cid:0) π ( S ) (cid:1) must be discrete.7Restricting the equality ρ ( e b ) − h ∗ ( e b ) = b ( χ ) to a cyclic subgroup wededuce that rot ρ ( γ ) − rot h ( γ ) = χ ( γ )for all γ ∈ π ( S ). But since h ( γ ) has at least one fixed point in ∂ D , we havethat rot h ( γ ) = 0 for all γ ∈ π ( S ), and hence rot ρ ( γ ) = χ ( γ )for all γ ∈ π ( S ). In particular, ker ρ ⊂ ker χ and hence ρ | ker ρ is semiconjugateto h | ker ρ , which implies that h (ker ρ ) has a fixed point in ∂ D . But since h is ahyperbolization and ker ρ is normal in π ( S ), we deduce that h (ker ρ ) is trivialand hence ker ρ is trivial, thus showing that ρ is injective.Now we show that χ is trivial. Let γ ∈ π ( S ). We distinguish then threecases:(1) ρ ( γ ) is hyperbolic or parabolic . Hence ρ ( γ ) has a fixed point in ∂ D andhence χ ( γ ) = rot ρ ( γ ) = 0;(2) ρ ( γ ) is elliptic and rot ρ ( γ ) = χ ( γ ) / ∈ Q / Z . Then ρ ( γ ) is conjugate in G to an irrational rotation contradicting the fact that ρ (cid:0) π ( S ) (cid:1) is discrete;(3) ρ ( γ ) is elliptic and rot ρ ( γ ) = χ ( γ ) ∈ Q / Z . Let n ∈ N be such that nχ ( γ ) = 0. Hence rot ρ ( γ n ) = χ ( γ n ) = 0 and, since ρ ( γ n ) is elliptic, it ishence the identity. Thus γ n ∈ ker ρ = e and since π ( S ) has no torsion, γ = e .Thus we conclude that χ is trivial, ρ ∗ ( e b ) = h ∗ ( e b ) and hence ρ is semiconju-gate to h .Now we will put to use the explicit formula for e b ( ρ ) together with theabove result in order to restore, in a sense, the setting of the case of surfaceswithout boundary and interpret δ (cid:0) Hyp( S ) (cid:1) as a union of connected compo-nents. Namely let us introduceHom C (cid:0) π ( S ) , G (cid:1) = (cid:8) ρ ∈ Hom (cid:0) π ( S ) , G (cid:1) : ρ ( c i ) has at least one fixed point in ∂ D (cid:9) = (cid:8) ρ : π ( S ) → G : (cid:0) tr ρ ( c i ) (cid:1) ≥ , for 1 ≤ i ≤ n (cid:9) which is a real semialgebraic subset of Hom (cid:0) π ( S ) , G (cid:1) . Clearly for ρ ∈ Hom C (cid:0) π ( S ) , G (cid:1) we have that rot ρ ( c i ) = 0, and hence taking into account that ρ e b ( ρ ) iscontinuous, we have the following Corollary 4.19. If ρ ∈ Hom C (cid:0) π ( S ) , G (cid:1) , then e b ( ρ ) = − n X i =1 τ (cid:0) ˜ ρ ( c i ) (cid:1) takes integer values and is constant on connected components. (cid:0) π ( S ) , G (cid:1) and when S is notcompact, the image of ρ e b ( ρ ) is the whole interval (cid:2) − | χ ( S ) | , | χ ( S ) | (cid:3) . Inany case we obtain finally: Theorem 4.20.
In the notation of Theorem 4.6, we have δ (cid:0) Hyp( S ) (cid:1) = ( ρ : π ( S ) → G : n X i =1 τ (cid:0)e ρ ( c i ) (cid:1) = 2 g − n ) . Thus δ (cid:0) Hyp( S ) (cid:1) is a union of connected components of Hom C (cid:0) π ( S ) , G (cid:1) and,in particular, a semialgebraic set. In connection with Corollary 4.15, we would like to present a different view-point, coming from Ch. Bavard [2] and developed by D. Calegari [22], whichrelates the second bounded cohomology of π ( S ) to the stable commutatorlength (scl) via quasimorphisms. In order to simplify the discussion, we as-sume in this section that Γ is a group withH (Γ , R ) = 0 . (4.9)This applies in particular to Γ = π ( S ), where S is a non-compact surface. Inthe notation of § (Γ , R ) admits then a representative ofthe form d f , with f ∈ C (Γ , R ). This leads us to make the following definition Definition 4.21.
A quasimorphism on Γ is a function f : Γ → R such that D ( f ) := sup a,b ∈ Γ (cid:12)(cid:12) f ( ab ) − f ( a ) − f ( b ) (cid:12)(cid:12) < + ∞ , and D ( f ) is called the defect of f .The vector space Q (Γ , R ) of all quasimorphisms on Γ contains always thesubspace ℓ ∞ (Γ , R ) of bounded functions, as well as the subspace of all homo-morphisms Hom(Γ , R ). It is clear that d induces an isomorphism of vectorspaces Q (Γ , R ) /ℓ ∞ (Γ , R ) ⊕ Hom(Γ , R ) ∼ = / / H (Γ , R )and the Gromov norm k α k of a class α ∈ H (Γ , R ) is given by k α k = inf (cid:8) D ( f ) : [ d f ] = α, f ∈ Q (Γ , R ) (cid:9) . There are various ways of choosing a ”special” quasimorphism representing agiven class. One is to fix a finite symmetric generating set of Γ and establish9the existence of a harmonic representative F for each class α ∈ H (Γ , R ): thisquasimorphism then minimizes the defect, that is k α k = D ( f ) , ([19]; see also [6] for a recent application).Another way to get a representative, this time canonical, is to considerhomogeneous quasimorphisms, that is quasimorphisms satisfying the condition f ( x n ) = nf ( x ) for all n ∈ Z , , x ∈ Γ . A simple argument (see [88]), shows that for f ∈ Q (Γ , R ), the limit F ( x ) := lim n →∞ f ( x n ) n exists for all x and gives a homogeneous quasimorphism F with the propertythat f − F ∈ ℓ ∞ (Γ , R ). Denoting by Q h (Γ , R ) the subspace of homogeneousquasimorphisms, it is thus clear that d induces an isomorphism of vectorspaces Q h (Γ , R ) / Hom(Γ , R ) ∼ = / / H (Γ , R ) . Obviously the defect D gives a norm on the left hand side, while the righthand side is endowed with the (canonical) Gromov norm. There is then thefollowing non-trivial relation between these two norms: Theorem 4.22 ([22]).
For every homogeneous quasimorphism f on Γ , D ( f ) ≤ (cid:13)(cid:13) d f (cid:13)(cid:13) ≤ D ( f ) . This inequality is based on the following relation between defect of a ho-mogeneous quasimorphism f and commutators D ( f ) = sup a,b ∈ Γ (cid:12)(cid:12) f (cid:0) [ a, b ] (cid:1)(cid:12)(cid:12) . Example 4.23. (1) The function ˜ r : ^ PU(1 , → R in § τ : H + Z ( R ) → R defined in § ^ PU(1 ,
1) with a subgroup of H + Z ( R ), we have D ( τ ) = D (cid:0) τ | ^ PU(1 , (cid:1) = 1 . We let as usual [Γ , Γ] denote the subgroup of Γ generated by the set (cid:8) [ x, y ] : x, y ∈ Γ (cid:9) of all commutators and let cl( γ ) denote the word length with respect0to this generating set. The stable commutator length is defined byscl( γ ) := lim n →∞ cl( γ n ) n , where the existence of the limit follows from the fact that the map n
7→ k γ n k is subadditive.The following result then puts the geometry of the commutator subgroup[Γ , Γ] in direct relation with quasimorphisms, in fact, homogeneous ones:
Theorem 4.24 ([2]).
For every γ ∈ [Γ , Γ] , we have scl( γ ) = sup (cid:26) | ϕ ( γ ) | D ( ϕ ) : ϕ ∈ Q h (Γ , R ) (cid:27) . Now every element γ ∈ [Γ , Γ], seen as a 1-chain, is an element in thevector space B (Γ , R ) of 1-boundaries in the bar resolution defining grouphomology. In [21] the author extends scl to a seminorm on the vector spaceB (Γ , R ) and obtains an extension of the Bavard duality theorem. We refer tothe monograph [22] for the details and interesting developments, and proceeddirectly to state a corollary of the main result in [21]. If now Γ = π ( S ), where S is a non-compact surface and π ( S ) = * a , b , . . . , a g , b g , c , . . . , c n : g Y i =1 [ a i , b i ] n Y j =1 c j = e + , then P ni =1 c i is clearly in B (Γ , R ) and Proposition 4.25 ([21]).
With the above assumptions and notation scl n X i =1 c i ! = − χ ( S )2 . If now ˜ ρ : Γ → ^ PU(1 ,
1) is the lift of a fixed hyperbolization ρ of S , we canpullback the translation quasimorphism and obtain rot ρ := τ ◦ ρ , which defineson Γ a homogeneous quasimorphism taking values in Z ; in fact, rot ρ changesby an element of Hom(Γ , Z ) if one takes a different lift of ρ . A corollary to themain result in [21] is then the following Theorem 4.26 ([21]).
For any homogeneous quasimorphism f on Γ , we havethe inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 f ( c i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ D ( f ) | χ ( S ) | , with equality if and only if f differs from rot ρ by an element of Hom(Γ , R ) . α ∈ H (cid:0) π ( S ) , R (cid:1) = H (Σ , R ), let d F be a representative of α , where F : Γ → R is a homogeneous quasimorphism. Then: Lemma 4.27. If f R : H (Σ , ∂ Σ , R ) → H (Σ , R ) is the isomorphism in § (cid:10) f − R ( α ) , [Σ , ∂ Σ] (cid:11) = − n X i =1 F ( c i ) . Taking into account the inequality D ( F ) ≤ k α k in Theorem 4.22, onesees that Theorem 4.26 implies Corollary 4.15. For our purposes however,both results contain the same information as far as the characterization ofequality is concerned. An intriguing question in this context is whether if Γis a finitely generated group, F : Γ → R a homogeneous quasimorphism and[ d F ] ∈ H (Γ , R ) the bounded class it defines, then the equality (cid:13)(cid:13) [ d F ] (cid:13)(cid:13) = 12 D ( F )holds.It would suffice to show this for nonabelian free groups; in this case allknown examples of quasimorphisms satisfy the above equality ([22, 90]). Part II
Higher Teichm¨uller Spaces
Our considerations started with the study of how the set of hyperbolic struc-tures on a surface S , of finite topological type, is related to the set of represen-tations of π ( S ) into G = PSU(1 , δ : Hyp( S ) → Hom (cid:0) π ( S ) , G (cid:1) constructed in § § (cid:0) π ( S ) , G (cid:1) → R with various incarnations and whose maximal fiber T − (cid:0) − χ ( S ) (cid:1) coincideswith the image of δ . As indicated in § π ( S ) with values in any Lie group G . This leads to thevague question of how much of the “PSU(1 ,
1) picture” generalizes to an arbi-trary Lie group G . Interestingly enough, there are two classes of (semi)simpleLie groups for which one can make this question precise in defining, in verydifferent ways, components (or specific subsets when S is not compact) of2Hom (cid:0) π ( S ) , G (cid:1) which should play the role of Teichm¨uller space: those twoclasses are on the one hand the split real groups (Definition 6.1) and, on theother hand, the Lie groups of Hermitian type (Definition 5.1).Because of the various properties these connected components share withTeichm¨uller space, we will call them higher Teichm¨uller spaces .We will now describe in some detail the class given by Lie groups of Her-mitian type, and the corresponding subset, namely the space of maximal rep-resentations
Hom max (cid:0) π ( S ) , G (cid:1) ⊂ Hom (cid:0) π ( S ) , G (cid:1) . We will discuss the other class in § Definition 5.1.
A Lie group G is of Hermitian type if it is connected semisim-ple with finite center without compact factors and the associated symmetricspace X has a G -invariant complex structure.In this setting we will be able to define a Toledo invariant (and a boundedToledo invariant when S is not compact) satisfying a Milnor–Wood type in-equality and this will lead us to consider the set Hom max (cid:0) π ( S ) , G (cid:1) of maximalrepresentations into G .In this section we will describe a certain number of fundamental geometricproperties of maximal representations and we will also have something to sayabout the structure of the set of such representations, all this in the contextwhere S is of finite topological type. Let G be of Hermitian type, X the associated symmetric space, h · , · i : X → Sym(
T X )the Riemannian metric and J : X → End(
T X )the complex structure. Then h J · , · i defines a G -invariant Hermitian metricwhose imaginary part ω X is a real G -invariant 2-form on X . By a generallemma of E. Cartan, the complex of G -invariant forms on any symmetric space3 X consists of closed forms. Thus ω X is closed and the above Hermitian metricis K¨ahler.Given S compact and ρ : π ( S ) → G a homomorphism, we can then proceedas in § f : e S = D → X define the Toledo invariant of ρ ,T( ρ ) = 12 π Z S f ∗ ( ω X ) . For the cohomological interpretation we can proceed as for PSU(1 , G acts properly on X , we have the Van Est isomorphismΩ ( X ) G ∼ = H ( G, R ) . To the K¨ahler form ω X we associate the continuous (inhomogeneous) 2-cocycle c ( g , g ) = 12 π Z ∆( x ,g x ,g g x ) ω X where x ∈ X is a fixed base point and ∆( x, y, z ) denotes a smooth simplexwith geodesic sides connecting the vertices x, y, z ; of course such a simplexis not unique, but any two such simplices with fixed vertices have the sameboundary and hence, since ω X is closed, by Stokes’ theorem the integral doesnot depend on it. We denote by κ G ∈ H ( G, R ) the class defined by c and callit the K¨ahler class. Then, given ρ : π ( S ) → G , we have the equalityT( ρ ) = (cid:10) ρ ∗ ( κ G ) , [ S ] (cid:11) . (5.1)In fact, the cocycle c defining κ G turns out to be bounded; this is a consequenceof a precise study of the K¨ahler area of triangles with geodesic sides, due toDomic and Toledo [31], and Clerc and Orsted [27], and which we will describelater in more details. Here we deduce that c defines a bounded class κ b G ∈ H ( G, R ), called the bounded K¨ahler class and we have the following Theorem 5.2 (Clerc–Orsted, [27]).
Assume that the metric on X is normal-ized so that its minimal holomorphic sectional curvature is − . Then the valueof the Gromov norm of the bounded K¨ahler classK¨ahler class!bounded is k κ b G k = 12 rank X , where rank X is the rank of the symmetric space X . When X is irreducible, Ω ( X ) G = R ω X and hence the comparison mapH ( G, R ) → H ( G, R ) (5.2)4is surjective; in general, Ω ( X ) G is spanned by the pullbacks under the projec-tions of the K¨ahler forms of the irreducible factors of X and thus the compari-son map (5.2) is surjective as well; since G has finite center (see Definition 5.1),it is injective in all cases.Let now S be of finite topological type realized as the interior of an orientedcompact surface Σ with boundary and ρ : π ( S ) → G a homomorphism. Thenwe can proceed as in the definition of the bounded Euler number and, in thenotation of § ρ ) := (cid:10) f − R (cid:0) ρ ∗ ( κ b G ) (cid:1) , [Σ , ∂ Σ] (cid:11) , where we recall that f R : H (Σ , ∂ Σ , R ) → H (Σ , R )is the isomorphism given by the natural inclusion.For G = PSU(1 ,
1) we saw in § S is noncompact. In the case ofPSU(1 ,
1) we have at our disposal the relation with rotation numbers and thetranslation quasimorphism; these structures were given to us for free from thefact that PSU(1 ,
1) acts by orientation preserving homeomorphisms of the cir-cle. For G of Hermitian type one can construct (more sophisticated) analoguesof each of these objects; in particular the integral structure on (bounded) coho-mology and the analogues of rotation number can be described quite explicitlyand this is what we turn to now.We denote by H ( G, Z ) and H ( G, Z ) the (bounded) Borel cohomology,which is defined by considering the complex of (bounded) Borel functions from G n to Z . We refer the reader to [18, §§ Lemma 5.3.
The comparison map H ( G, Z ) → H ( G, Z ) is an isomorphism.Proof. The long exact sequences in cohomology associated to0 / / Z / / R / / R / Z / / c ( G, R / Z ) / / = (cid:15) (cid:15) H ( G, Z ) / / (cid:15) (cid:15) H ( G, R ) / / (cid:15) (cid:15) H ( G, R / Z ) = (cid:15) (cid:15) c ( G, R / Z ) / / H ( G, Z ) / / H ( G, R ) / / H ( G, R / Z ) . The fact that the third vertical arrow is an isomorphism and the 5-term lemmaallow to conclude.Actually we will turn to an explicit implementation of the isomorphism inLemma 5.3. This will also give an alternative treatment of some material in[18, § ( G, Z ) ∼ = Hom (cid:0) π ( G ) , Z (cid:1) (5.3)is valid for any connected Lie group G ; this follows easily from the fact thatH ( G, Z ) classifies equivalence classes of topological central extensions of G by Z , together with some covering theory; moreover this isomorphism is natural.If now K < G is a maximal compact subgroup, then by the Iwasawa decom-position
K ֒ → G is a homotopy equivalence and hence π ( K ) = π ( G ); this,together with (5.3), implies that the restriction mapH ( G, Z ) → H ( K, Z )is an isomorphism. Taking into account that continuous cohomology of com-pact groups with real coefficients is trivial, we obtain, considering the longexact sequence associated to the coefficient sequence, thatHom c ( K, R / Z ) ǫ / / H ( K, Z )is an isomorphism. As a result we obtain an isomorphismHom c ( K, R / Z ) −→ H ( G, Z ) ∼ = H ( G, Z ) χ κ and we say that κ corresponds to χ and viceversa.Now we will assume that G is real algebraic and semisimple. In this casewe have at our disposal the refined Jordan decomposition namely every g ∈ G is a product g = g e g h g u of pairwise commuting elements, where g e is contained in a compact subgroup, g h is in the connected component of the identity of a maximal real split torusand g u is unipotent. Given then χ : K → R / Z
6a continuous homomorphism and denoting by C ( h ) the conjugacy class of anelement h ∈ G , define for g ∈ Gχ ext ( g ) := χ (cid:0) C ( g e ) ∩ K (cid:1) Then, according to [7], χ ext is indeed a well defined continuous class functionon G extending χ ; moreover it satisfies χ ext ( g n ) = nχ ext ( g )for all n ∈ Z and g ∈ G . Let g χ ext : e G → R denote the unique continuous liftto the universal covering e G of G , vanishing at e ; finally we denote by χ ∗ : π ( G ) = π ( K ) → Z the morphism on the level of fundamental groups induced by χ . The followingresult then gives a precise description of the isomorphism in Lemma 5.3. Theorem 5.4 ([18]). (1)
The function g χ ext : e G → R is a homogeneousquasimorphism and the map Hom c ( K, R / Z ) → QH hc ( e G, R ) Z χ g χ ext establishes an isomorphism with the space of continuous homogeneousquasimorphisms sending π ( G ) to Z ; in fact χ ∗ = g χ ext | π ( G ) . (2) If κ ∈ H ( G, Z ) is the class corresponding to χ , and denote by [ · ] theinteger part, then ( g, h ) → (cid:2) g χ ext ( gh ) (cid:3) − (cid:2) g χ ext ( g ) (cid:3) − (cid:2) g χ ext ( h ) (cid:3) descends to a well defined Z -valued bounded cocycle on G × G representingthe class κ . Remark 5.5.
The only not obvious statement in Theorem 5.4 is the assertionthat g χ ext is a quasimorphism; in fact, this follows easily from the fact (provedin [7]) that for every k ∈ N , g χ ext is bounded on the elements in e G which areproducts of k commutators. We mention in addition that Theorem 5.4 is alsoconsequence of a different and more general approach taken in [18].In our context, rotation numbers arise in the following way. Let κ ∈ H ( G, Z ) and, for every g ∈ G consider, as in §
4, the homomorphism h g : Z → Gn g n h ∗ g ( κ ) ∈ H ( Z , Z ) andfinally by means of the canonical isomorphism0 / / Hom( Z , R / Z ) δ / / H ( Z , Z ) / / R / Z , rot κ ( g ) := δ − (cid:0) h ∗ g ( κ ) (cid:1) (1) . From standard homological considerations using the naturality of all con-structions involved, one can deduce that if χ ∈ Hom c ( K, R / Z ) corresponds tothe class κ ∈ H ( G, Z ) then rot κ ( g ) = g χ ext (˜ g ) mod Z where ˜ g ∈ e G is any lift of g ∈ G . Now we fix κ ∈ H ( G, Z ) and let as before S be a surface of finite topological type realized as the interior of Σ. Given ahomomorphism ρ : π ( S ) → G , we define thenT κ (Σ , ρ ) = (cid:10) f − R (cid:0) ρ ∗ ( κ ) (cid:1) , [Σ , ∂ Σ] (cid:11) . Of course if ∂ Σ = ∅ , that is if S is compact, the definition takes the simplifiedform T κ ( S, ρ ) = (cid:10) ρ ∗ ( κ ) , [ S ] (cid:11) . Then we have the following:
Theorem 5.6 ([18]).
Let S be of finite topological type with presentation π ( S ) = * a , b , . . . , a g , b g , c , . . . , c n : g Y i =1 [ a i , b i ] n Y j =1 c j = e + and ρ : π ( S ) → G a homomorphism. Let κ ∈ H ( G, Z ) and χ ∈ Hom c ( K, R / Z ) the corresponding homomorphism. (1) Assume that S is compact. Then T κ ( S, ρ ) = − χ ∗ g Y i =1 [ ρ ( a i ) , ρ ( b i )] ˜ ! , where χ ∗ : π ( G ) = π ( K ) → Z is the morphism induced by χ and [ · , · ] ˜ is the commutator map introduced in § (2) Assume that S is not compact. If e ρ : π ( S ) → e G is a lift of ρ to e G , then T κ (Σ , ρ ) = − n X j =1 g χ ext (cid:0)e ρ ( c j ) (cid:1) . Let S be of finite topological type and ρ : π ( S ) → G a homomorphism. Basedon our considerations in the case of PSU(1 ,
1) in § κ b G ∈ H ( G, R ) by Clerc andOrsted, we can conclude Corollary 5.7.
The Toledo invariant T( ρ ) defined in (5.1) satisfies the in-equality (cid:12)(cid:12) T( ρ ) (cid:12)(cid:12) ≤ rank X (cid:12)(cid:12) χ ( S ) (cid:12)(cid:12) with equality if and only if ρ ∗ ( κ b G ) = ± rank X κ b S, R , where κ b S, R is the bounded real class defined in the context of Corollary 4.15.Proof. Corollary 4.15 with α = ρ ∗ ( κ b G ) implies that (cid:12)(cid:12) T( ρ ) (cid:12)(cid:12) ≤ (cid:13)(cid:13) ρ ∗ ( κ b G ) (cid:13)(cid:13) (cid:12)(cid:12) χ ( S ) (cid:12)(cid:12) . Using the fact that ρ ∗ is norm decreasing and the value of k κ b G k (see Theo-rem 5.2) we obtain that (cid:12)(cid:12) T( ρ ) (cid:12)(cid:12) ≤ rank X (cid:12)(cid:12) χ ( S ) (cid:12)(cid:12) . Equality implies that (cid:12)(cid:12) T( ρ ) (cid:12)(cid:12) = 2 (cid:13)(cid:13) ρ ∗ ( κ b G ) (cid:13)(cid:13) (cid:12)(cid:12) χ ( S ) (cid:12)(cid:12) = rank X (cid:12)(cid:12) χ ( S ) (cid:12)(cid:12) . It follows from the first equality that ρ ∗ ( κ b G ) = λ κ b S, R for some λ ∈ R (Corollary 4.15), and from the second equality that | λ | =rank X .Thus we now introduce the following Definition 5.8.
A representation ρ : π ( S ) → G is called maximal ifT( ρ ) = rank X (cid:12)(cid:12) χ ( S ) (cid:12)(cid:12) . The basic example of a family of maximal representations is obtained viaa geometric fact of fundamental importance called the polydisk theorem. Re-call that in a symmetric space of noncompact type there are maximal flatsubspaces, they are all G -conjugate and their common dimension is the rank r = rank X of X . When X is Hermitian symmetric, a geometric version of9a fundamental result of Harish-Chandra says that the complexification of amaximal flat is a maximal polydisk, or, in other words, that the image un-der the exponential map of the complexified tangent space of a maximal flatis a totally geodesic holomorphic copy of D r . The fact that the normalizedmetric on X is taken to be of minimal holomorphic sectional curvature − ϕ : D r → X is isometric. To such a map corresponds a homomorphismΦ : SU(1 , r → G with respect to which ϕ is equivariant. Given then r hyperbolizations ρ , . . . , ρ r : π ( S ) → SU(1 , ρ ( γ ) = Φ (cid:0) ρ ( γ ) , . . . , ρ r ( γ ) (cid:1) is then maximal. The main point is the fact thatΦ ∗ ( κ b G ) = κ bSU(1 , r . This follows from a Lie algebra computation which givesΦ ∗ ( ω X ) = ω D r and the naturality of the isomorphismsΩ ( X ) G ∼ = H ( G, R ) ∼ = H ( G, R ) . Observe that we could have taken an antiholomorphic embedding ϕ : D r → X ,in which case ρ ( γ ) = Φ (cid:0) ρ ( γ ) , . . . , ρ r ( γ ) (cid:1) has then T( ρ ) = − rank X (cid:12)(cid:12) χ ( S ) (cid:12)(cid:12) as Toledo invariant.Now, the class κ b G is not always integral, but there is a specific naturalnumber n X depending on the root system of G such that κ = n X κ b G is anintegral class. From this and Theorem 5.6 we deduce: Corollary 5.9 ([18]).
The map ρ T( ρ ) on Hom (cid:0) π ( S ) , G (cid:1) is continuous. (1) If S is compact, it takes values in n X Z and is constant on connectedcomponents. (2) If S is not compact, its range is (cid:2) − rank X (cid:12)(cid:12) χ ( S ) (cid:12)(cid:12) , rank X (cid:12)(cid:12) χ ( S ) (cid:12)(cid:12)(cid:3) . Proof.
The continuity follows from the formulas in Theorem 5.6. Then (1) isclear and (2) follows from the fact that, since π ( S ) is free, then Hom (cid:0) π ( S ) , G (cid:1) ∼ = G g + n − is connected and therefore the intermediate value theorem implies thestatement.In the study of representations of π ( S ), where S is of finite topologicaltype, we were several times led to consider the corresponding “completed”compact surface Σ with boundary, for which of course we have π ( S ) = π (Σ).In the light of the Fenchel–Nielsen approach to Teichm¨uller theory, it is naturalto ask what happens to Toledo invariants when one glues together two surfacesalong a component of their boundary. The answer is given by the following Proposition 5.10 ([18]).
Let Σ be a compact oriented surface with boundaryand ρ : π (Σ) → G a homomorphism. (1) If Σ = Σ ∪ C Σ is the connected sum of two subsurfaces Σ and Σ along a separating loop C , then T(Σ , ρ ) = T(Σ , ρ ) + T(Σ , ρ ) , where ρ i is the restriction of ρ to π (Σ i ) . (2) If Σ ′ is the surface obtained by cutting Σ along a non-separating loop C and i : Σ ′ → Σ is the canonical map, then T(Σ ′ , ρ i ∗ ) = T(Σ , ρ ) . Remark 5.11.
This result holds also for the invariants T κ (Σ , ρ ) for κ ∈ H ( G, Z ) introduced in § ρ ∈ Hom (cid:0) π (Σ) , G (cid:1) is maximal if and only if ρ i ∈ Hom (cid:0) π (Σ i ) , G (cid:1) are maximalfor i = 1 , Theorem 5.12 ([18]).
Maximal representations are injective and with discreteimage.
Remark 5.13.
As soon as the Lie group G is not locally isomorphic toPSU(1 , ρ : π ( S ) → G is maximal and rank X is the real rank of G , then ρ ∗ ( κ b G ) = rank X κ b S, R . causal representations which satisfy that ρ ∗ ( κ b G ) = λ κ b S, R for some λ = 0; details will appear in [91].While Theorem 5.12 holds for all surfaces of finite type, one has a substan-tially stronger result when S is compact, namely: Theorem 5.14 ([18]).
Let S be a compact surface, ρ : π ( S ) → G a maximalrepresentation and X the symmetric space associated to G . Then there areconstants A > and B ≥ such that A − k γ k − B ≤ d X (cid:0) ρ ( γ ) x , x (cid:1) ≤ A k γ k + B , where x ∈ X is a basepoint and k · k is a word metric on π ( S ) . This result is a consequence of the fact that maximal representations are
Anosov (see § Almost from its beginning in the 80’s, research on maximal representationswas driven by “irreducibility” questions. For instance D. Toledo, using toolsfrom the Gromov–Thurston proof of Mostow rigidity for real hyperbolic man-ifolds, showed in [95] that a maximal representation from a compact surfacegroup into SU( n,
1) leaves invariant a complex geodesic, or equivalently its im-age is contained in a conjugate of S (cid:0) U( n − × U(1 , (cid:1) . Then L. Hern´andezshowed in [58] that if SU( n,
2) (for n ≥
2) is the target group, the image mustbe contained in a conjugate of S (cid:0) U( n − × U(2 , (cid:1) . In [10] S. Bradlow,O. Garc´ıa-Prada and P. Gothen then showed that a reductive maximal rep-resentation with target SU( p, q ), with p ≤ q , is contained in a conjugate ofS (cid:0) U( p, p ) × U( q − p ) (cid:1) using methods from the theory of Higgs bundles.In its most general form the problem presents itself naturally in the follow-ing way: given a maximal representations ρ : π ( S ) → G where G := G ( R ) ◦ consists of the real points of the connected component of a semisimple alge-braic group G defined over R , determine the Zariski closure L := ρ (cid:0) π ( S ) (cid:1) Z of the image of ρ . In [18] we gave a complete answer to this question and mostof this section is devoted to the description of the result and the ingredientsof the proof.Recall that every Hermitian symmetric space X (of noncompact type) isbiholomorphic to a bounded domain D ⊂ C n . While this is the natural gener-alization of the Poincar´e disk, the question of the generalization of the upperhalf plane leads to the notion of tube type domain. We say that X (or D )2is of tube type if it is biholomorphic to a domain of the form V + ı Ω, where V is a real vector space and Ω ⊂ V is an open convex proper cone in V .The groups corresponding to irreducible Hermitian symmetric spaces of tubetype are Sp(2 n, R ), SU( p, p ), SO ∗ (2 n ) (for n even), SO(2 , n ) and one of thetwo exceptional ones. There are many known characterizations of tube typedomains, mainly in terms of special geometric structures, or the topology oftheir Shilov boundary, and we will add a new one in Theorem 5.28.With the notion of tube type at hand, the structure of the Zariski closureof the image of a maximal representation is described by the following Theorem 5.15 ([17]).
Let G := G ( R ) ◦ be a Lie group of Hermitian type withassociate symmetric space X . Let ρ : π ( S ) → G be a maximal representationand L := ρ (cid:0) π ( S ) (cid:1) Z the Zariski closure of its image. Then: (1) the Lie group L := L ( R ) ◦ is reductive with compact centralizer in G ; (2) the semisimple part of L is of Hermitian type; (3) the Hermitian symmetric space Y associated to L is of tube type and thetotally geodesic embedding Y ֒ → X is tight. In statement (3) the embedding Y ֒ → X is not necessarily holomorphic butit is tight, a notion involving the area of geodesic triangles in Y and X withrespect to ω X . We will elaborate on this notion in § X , maximal tubetype subdomains exists, they are all conjugate and of rank equal to the rankof X . We have then: Corollary 5.16 ([18]).
Let ρ : π ( S ) → G be a maximal representation. Thenthere is a maximal tube type subdomain which is ρ (cid:0) π ( S ) (cid:1) -invariant. A special case of Theorem 5.15 is when ρ has Zariski dense image in G ,in which case Y = X and hence X is of tube type. This result is optimal, inthe sense that every tube type domain admits a maximal representation withZariski dense image. In order to be more specific, we recall that a diagonaldisk in X is a holomorphic totally geodesic embedding d : D → X obtained as the composition of a diagonal embedding D → D r (where r =rank X ) and a maximal polydisk embedding D r → X . Let∆ : SU(1 , → G denote the homomorphism corresponding to d . Let now ρ : π ( S ) → SU(1 , Theorem 5.17 ([18]).
Assume that X is of tube type. Then there exists apath of homomorphisms ρ t : π ( S ) → G , for t ≥ , such that (1) ρ t is maximal for all t ≥ and ρ = ∆ ◦ ρ ; (2) ρ t has Zariski dense image for t > . Remark 5.18.
Using the structure theory developed in [17] which is describedin the following section, Kim and Pansu [63] recently showed that for funda-mental groups of compact surfaces the global rigidity result for maximal rep-resentation into non-tube type Hermitian Lie groups given by Theorem 5.15arises only in this context. For a precise statement of their result see [63,Corollary 2].In the next section we describe the various ingredients entering the proofof the structure theorem (Theorem 5.15).
Maximal representations are a special case of more general type of homomor-phism, namely tight homomorphisms ; they are defined on any locally compactgroup L and take values in a Lie group of Hermitian type G . Definition 5.19.
A continuous homomorphism ρ : L → G is tight if (cid:13)(cid:13) ρ ∗ ( κ b G ) (cid:13)(cid:13) = k κ b G k . By inspecting the proof of the Milnor–Wood type inequality in Corol-lary 5.7, one verifies easily that maximal representations are tight.In the case in which also L is a Lie group of Hermitian type and Y is theassociated symmetric space, then a continuous homomorphism h : L → G gives rise to a totally geodesic map f : Y → X . The geometric condition on f for ρ to be tight is then sup ∆ ⊂Y Z ∆ f ∗ ω X = sup ∆ ⊂ X Z ∆ ω X (5.4)and we call f tight if it satisfies (5.4). A useful observation is that if ρ : π ( S ) → L is a homomorphism such that h ◦ ρ is maximal, then h is tight. Forthe converse, we need to introduce an additional notion. Namely, recall thatthe space H ( L, R ) ∼ = H ( L, R ) ∼ = Ω ( Y ) L Y of the K¨ahler form of theirreducible factors of Y . The open cone generated by the linear combinationwith strictly positive coefficients of these forms is called the cone of positiveK¨ahler classes and denoted by H ( L, R ) > . Definition 5.20.
A continuous homomorphism h : L → G is positive if h ∗ ( κ b G ) ∈ H ( L, R ) > . With these definitions we have then:
Proposition 5.21 ([17]). If ρ : π ( S ) → L is maximal and h : L → G is tightand positive, then h ◦ ρ : π ( S ) → G is maximal. This is particularly useful in combination with the following geometric ex-amples.
Proposition 5.22 ([17]).
Let Y and X be Hermitian symmetric spaces withnormalized metrics and let f : Y → X be a holomorphic and isometric map.Then f is tight if and only if rank X = rank Y , in which case it is also positive.In particular: (1) a maximal polydisk t : D r → X is tight and positive; (2) the inclusion T ֒ → X of a maximal tube type subdomain is tight andpositive; (3) a diagonal disk d : D → X is tight and positive. We stress the fact that all Hermitian spaces involved carry the normalizedmetric, that is the one with minimal holomorphic sectional curvature − Proposition 5.23.
The n -dimensional irreducible representation ρ n : SL(2 , R ) → Sp(2 n, R ) is tight and corresponds to a holomorphic map only when n = 1 . The main structure theorem concerning tight homomorphisms is then thefollowing:
Theorem 5.24 ([17]).
Let L be a locally compact second countable group, G = G ( R ) ◦ a Lie group of Hermitian type and let ρ : L → G be a continuoustight homomorphism. Then: (1) the Zariski closure H := ρ ( L ) Z is reductive; the centralizer of H := H ( R ) ◦ in G is compact; (3) the semisimple part of H is of Hermitian type and the associated sym-metric space Y admits a unique H -invariant complex structure such thatthe inclusion H ֒ → G is tight and positive. Setting L = π ( S ) and assuming ρ to be maximal, the above result accountsthen for most of the statements in the structure Theorem 5.15 except theone, essential, that Y is of tube type. This is specific to the hypothesis that L = π ( S ) is a surface group.An important ingredient of the structure theorem for tight homomorphismsis the work of Clerc and Orsted on the characterization of “ideal triangleswith maximal symplectic area” [27]. To describe some important features, wewill assume for simplicity that G is simple (of Hermitian type), that is theassociated symmetric space X is irreducible. Let D ⊂ C n be the boundeddomain realization of X ; there is an explicit realization of D , called Harish-Chandra realization , and in which the Bergmann kernel K D : D × D → C ∗ of D can be computed rather explicitly. We let G act on D via the isomorphism X → D ; the Hermitian metric defined by K D is of course G -invariant, and ithas the interesting feature that its K¨ahler form comes from an integral classin H ( G, Z ). However, this metric is not normalized. One introduces then thenormalized Bergmann kernel k D := K /n D D where n D = n X is a specific integer (see Corollary 5.9 and the discussionpreceding it). This leads to the normalized metric whose K¨ahler form is ω D = ı∂∂ log k D ( z, z ) . In fact, on the specific formulas for k D one sees that it is defined and notvanishing on D of course and even on a certain open dense subset D (2) ⊂ D of pairs of points satisfying a certain transversality condition. The domain D (2) being star shaped, we let arg k D denote the unique continuous determinationof the argument of k D on D (2) vanishing on the diagonal of D × D . Then wehave the following formula for the area of a triangle with geodesic sides:
Lemma 5.25 ([31, 27]).
For x, y, z ∈ D Z ∆( x,y,z ) ω D = − (cid:0) arg k D ( x, y ) + arg k D ( y, z ) + arg k D ( z, x ) (cid:1) . D (3) of triplesof pairwise transverse points, namely the Bergmann cocycle β D ( x, y, z ) = − π (cid:0) arg k D ( x, y ) + arg k D ( y, z ) + arg k D ( z, x ) (cid:1) . Its role is to extend in a meaningful way the notion of area to “ideal triangles”.This function is continuous on D (3) , G -invariant and satisfies an obvious co-cycle property. The following is then a summary of some work of Clerc andOrsted. Theorem 5.26 ([27]).
Let ˇ S and rank D be respectively the Shilov boundaryand the real rank of D . (1) Then −
12 rank D ≤ β D ( x, y, z ) ≤
12 rank D , with strict inequality if ( x, y, z ) ∈ D ; (2) we have that β D ( x, y, z ) = rank D if and only if x, y, z ∈ ˇ S and there existsa diagonal disk d : D → D with d (1) = x , d ( ı ) = y and d ( −
1) = z . In the sequel we call the restriction of the Bergmann cocycle to the Shilovboundary
Maslov cocycle and we call a triple of points ( x, y, z ) on the Shilovboundary maximal if β D ( x, y, z ) = rank D . Observe that a maximal triple is always contained in the boundary of a max-imal tube type subdomain of D .One of the corollaries of the above result is the computation of the Gromovnorm of κ b G (Theorem 5.2). This is based on the following Corollary 5.27 ([13, 18]).
Under the canonical map H • (cid:0) L ∞ ( ˇ S • ) G (cid:1) → H • cb ( G, R ) the class defined by β D goes to κ b G . Finally we turn to the ingredient which leads to the conclusion in Theo-rem 5.15 that Y is of tube type. For this we construct an invariant of triples ofpoints on ˇ S which we call the Hermitian triple product and whose definitiongoes as follows; recall that the Bergmann kernel satisfies the relation K D ( gz, gw ) = j ( g, z ) K D ( z, w ) j ( g, w ) , j ( g, z ) is the complex Jacobian of g at the point z ∈ D . Then we defineon the set ˇ S (3) of pairwise transverse points the Hermitian triple product hh x, y, z ii := K D ( x, y ) K D ( y, z ) K D ( z, x ) mod R × . Recall that ˇ S is of the form G/Q , where Q is a maximal parabolic subgroup of G , and is hence in a natural way the set of real points of a complex projectivevariety. Theorem 5.28 ([16]).
The function hh · , · ii : ˇ S (3) → R × \ C × is a G -invariant multiplicative cocycle and, for an appropriate real structure on R × \ C × , it is a real rational function. Moreover the following are equivalent: (1) D is not of tube type; (2) ˇ S (3) is connected; (3) the Hermitian triple product is not constant.Sketch of the proof that Y is of tube type. Let ρ : π ( S ) → G := G ( R ) ◦ bea maximal representation. Using the structure theorem for tight homomor-phisms we may assume that ρ (cid:0) π ( S ) (cid:1) is Zariski dense in G and hence Y = X .We have to show that X is of tube type. Realize π ( S ) as a lattice in PSU(1 , ρ has Zariski dense image, the ac-tion on the Shilov boundary ˇ S is strongly proximal. This together with theamenability of the action of π ( S ) on ∂ D via the chosen hyperbolization im-plies (according to [80]) the existence of an equivariant measurable map (seee.g. [13] for a description of the construction) ϕ : ∂ D → ˇ S into the Shilov boundary. From the maximality assumption we deduce that ρ ∗ ( κ b G ) = rank D κ b S, R and hence that β D (cid:0) ϕ ( x ) , ϕ ( y ) , ϕ ( z ) (cid:1) = rank D β D ( x, y, z )for almost every ( x, y, z ). Thus we obtained that for almost every x, y, z ∈ ∂ D K D (cid:0) ϕ ( x ) , ϕ ( y ) (cid:1) K D (cid:0) ϕ ( y ) , ϕ ( z ) (cid:1) K D (cid:0) ϕ ( z ) , ϕ ( x ) (cid:1) = e πın D β D (cid:0) ϕ ( x ) ,ϕ ( y ) ,ϕ ( z ) (cid:1) mod R × + = e ± πın D rank D mod R × + and as a result the square hh · , · ii of the Hermitian triple product is equalto 1 on (Ess Im ϕ ) (3) ⊂ ˇ S (3) , where Ess Im ϕ ⊂ ˇ S is the essential image of ϕ .8But, being invariant under ρ , this set is Zariski dense in ˇ S hence the rationalfunction hh · , · ii on ˇ S (3) is identically equal to 1. If now D were not of tubetype, ˇ S (3) would be connected and hence hh · , · ii would be identically equalto 1 on ˇ S (3) , which is a contradiction. We have seen that if ρ : π ( S ) → G is a maximal representation into a groupof Hermitian type and h : π ( S ) → PSU(1 ,
1) is a hyperbolization of S of finitearea, then there exists a measurable ρ ◦ h − -equivariant map ϕ : ∂ D → ˇ S andfurthermore β D (cid:0) ϕ ( x ) , ϕ ( y ) , ϕ ( z ) (cid:1) = r D β D ( x, y, z ) (5.5)for almost every ( x, y, z ) ∈ ( ∂ D ) . Here β D is the Maslov cocycle on the Shilovboundary of the bounded symmetric domain D ; observe that β D is just ofthe orientation cocycle.In the case where G = PSU(1 ,
1) we have see that h and ρ are semiconjugateby using Ghys’ theorem; an alternative approach would be to use the equality(5.5) to show that ϕ coincides almost everywhere with a weakly monotonemap; this has been carried out in [61]. In fact, this way of “improving” theregularity of ϕ works in general and the basic idea is presented in [14]. Oneconsiders the essential graph of ϕ Ess Gr( ϕ ) ⊂ ∂ D × ˇ S which is by definition the support of the direct image of the Lebesgue measureon ∂ D under the map ∂ D → ∂ D × ˇ Sx (cid:0) x, ϕ ( x ) (cid:1) . Then one shows that there are exactly two sections ϕ − and ϕ + of the projectionof Ess Gr( ϕ ) on ∂ D such that:(1) ϕ − and ϕ + are strictly equivariant;(2) ϕ − is right continuous while ϕ + is left continuous;(3) Ess Gr( ϕ ) = (cid:8)(cid:0) x, ϕ − ( x ) (cid:1) , (cid:0) x, ϕ + ( x ) (cid:1) : x ∈ ∂ D (cid:9) ;(4) for every positive triple x, y, z ∈ ∂ D , both triples ϕ + ( x ) , ϕ + ( y ) , ϕ + ( z )and ϕ − ( x ) , ϕ − ( y ) , ϕ − ( z ) are maximal.This generalizes exactly the PSU(1 ,
1) picture and, remarkably, the disconti-nuities of ϕ − and ϕ + are simple.One can summarize the situation as follows:9 Theorem 5.29 ([18]).
The representation ρ : π ( S ) → G is maximal if andonly if there exists a left continuous map ϕ : ∂ D → ˇ S such that (1) ϕ is strictly ρ ◦ h − -equivariant; (2) ϕ maps every positively oriented triple in ∂ D to a maximal triple on ˇ S . The first obvious consequence is the following result on the existence offixed points:
Corollary 5.30 ([18]).
Let ρ : π ( S ) → G be maximal. Then: (1) if γ is freely homotopic to a boundary component, ρ ( γ ) has a fixed pointin ˇ S ; (2) if γ is not conjugate to a boundary component, then ρ ( γ ) has (at least)two fixed points in ˇ S , which are transverse. We use again the standard presentation of π ( S ) (see (4.2)) and define thefollowing subset of Hom ˇ S (cid:0) π ( S ) , G (cid:1) :Hom ˇ S (cid:0) π ( S ) , G (cid:1) = (cid:8) ρ ∈ Hom (cid:0) π ( S ) , G (cid:1) ; for every 1 ≤ i ≤ n,ρ ( c i ) has at least one fixed point in ˇ S (cid:9) Then Hom ˇ S (cid:0) π ( S ) , G (cid:1) is a semialgebraic subset of Hom (cid:0) π ( S ) , G (cid:1) and wehave from Corollary 5.30 thatHom max (cid:0) π ( S ) , G (cid:1) ⊂ Hom ˇ S (cid:0) π ( S ) , G (cid:1) . Theorem 5.31 ([18]).
Assume that D is of tube type. Then the Toledo in-variant ρ T(Σ , ρ ) is locally constant on Hom ˇ S (cid:0) π ( S ) , G (cid:1) . In particular, thesubset of maximal representations Hom max (cid:0) π ( S ) , G (cid:1) is a union of connectedcomponents of Hom ˇ S (cid:0) π ( S ) , G (cid:1) and therefore semialgebraic. This result is essentially a consequence of the formulas in § κ (Σ , ρ ), κ ∈ H ( G, Z ) together with the lemma that if Q is thestabilizer in G of a point in ˇ S , and if D is of tube type, then the restrictionmap H ( G, R ) → H ( Q, R )is identically zero.We end by mentioning a result which gives additional invariants for maxi-mal representations. Recall that for χ ∈ Hom c ( K, R / Z ) we have introduced aclass function χ ext : G → R / Z extending χ . We have: Theorem 5.32 ([18, Theorem 13]).
Let χ ∈ Hom c ( K, R / Z ) and fix ρ : π ( S ) → G maximal. For every maximal ρ : π ( S ) → G , the map R χ ( S ) : π ( S ) −→ R / Z γ χ ext (cid:0) ρ ( γ ) (cid:1) − χ ext (cid:0) ρ ( γ ) (cid:1) is a homomorphism. (2) If D is of tube type, R χ ( S ) takes values in e − G Z / Z and Hom max (cid:0) π ( S ) , G (cid:1) → Hom (cid:0) π ( S ) , e − G Z / Z (cid:1) is constant on connected components. (Here e G is an explicit constantdepending on G , not just on the symmetric space associated to G , e. g e SL(2 , R ) = 2 .) Hitchin representations and positive representations are defined when G is asplit real Lie group. Definition 6.1.
A real simple Lie groups G is split if its real rank equals thecomplex rank of its complexification G C , i. e. the maximal torus is diagonaliz-able over R . Let S be a compact surface and G a split real simple adjoint group, e.g. G = PSL( n, R ), PSp(2 n, R ), PO( n, n + 1) or PO( n, n ), Hitchin [60] singled outa connected componentHom Hit (cid:0) π ( S ) , G (cid:1) ⊂ Hom (cid:0) π ( S ) , G (cid:1) which he called Teichm¨uller component , now it is usually called
Hitchin com-ponent .In order to define the Hitchin component we recall that the Lie algebra g ofa split real simple adjoint Lie group G contains a (up to conjugation) uniqueprincipal three dimensional simple Lie algebra. That is an embedded subalge-bra isomorphic to sl (2 , R ) which is the real form of a subalgebra sl (2 , C ) ⊂ g C given (via the theorem of Jacobson–Morozov) by a regular nilpotent elementin g C . Here g C denotes the complexification of g and a nilpotent element isregular if its centralizer is of dimension equal to the rank of g C . For moredetails on principal three dimensional subalgebra we refer the reader to [96] orKostant’s original papers [65, 66, 67].1The embedding sl (2 , R ) → g gives rise to an embedding π : SL(2 , R ) → G .Precomposition of π with a discrete (orientation preserving) embedding of π ( S ) into SL(2 , R ) defines a homomorphism ρ : π ( S ) → G , which we call aprincipal Fuchsian representation. The Hitchin component Hom Hit (cid:0) π ( S ) , G (cid:1) is defined as the connected component of Hom (cid:0) π ( S ) , G (cid:1) containing ρ . Byconstruction it contains a copy of Teichm¨uller space. Remark 6.2.
When G = PSL( n, R ), PSp(2 m, R ) or PO( m, m +1) the embed-ding π is given by the n -dimensional irreducible representation PSL(2 , R ) → PSL( n, R ), which is contained in PSp(2 m, R ) if n = 2 m and in PO( m, m + 1) if n = 2 m + 1. For G = PO( m, m ), the embedding π is given by the compositionof the 2 m − m, m −
1) withthe embedding PO( m, m −
1) into PO( m, m ). Remark 6.3.
When G is a finite cover of a split real simple adjoint Lie group G Ad , one can define the Hitchin components Hom Hit (cid:0) π ( S ) , G (cid:1) by taking liftsof the principal Fuchsian representations π ( S ) → G Ad and the correspondingconnected components. Equivalently one can define a Hitchin representation ρ : π ( S ) → G as a representation whose projection ρ : π ( S ) → G Ad is aHitchin representation.Hitchin studied the Hitchin component following an analytic approach,relying on the correspondence between irreducible representations of π ( S ) inPSL( n, C ) and stable Higgs bundles. One direction of this correspondence isdue to Corlette [29] and Donaldson [32], following ideas of Hitchin [59], theother direction is due to Simpson [92, 93].The correspondence requires to fix a complex structure j on S ; with thischoice, there is an isomorphism (see [60]) h j : Hom Hit (cid:0) π ( S ) , G (cid:1) /G → H (cid:0) S, ⊕ rk =1 Ω d k ( S,j ) (cid:1) , (6.1)where H (cid:0) S, Ω d ( S,j ) (cid:1) is the vector space of holomorphic differentials (with re-spect to the fixed complex structure j on S ) of degree d . The coefficients d k , k = 1 , · · · , r = rank( G ) are the degrees of a basis of the algebra of invariantpolynomials on g . In particular, this proves Theorem 6.4 ([60, Theorem A]).
Let S be a compact surface and G the ad-joint group of a split real Lie group. Then the Hitchin component Hom
Hit (cid:0) π ( S ) , G (cid:1) /G is homeomorphic to R | χ ( S ) | dim G . Hitchin pointed out that the analytic approach via Higgs bundles gives noindication about the geometric significance of the representations belonging tothis component. The only example supporting the idea that Hitchin compo-nents might parametrize geometric structures on S available at that time was2given in work of Goldman [47] and Choi and Goldman [26], who showed thatfor G = PSL(3 , R ) the Hitchin component parametrizes convex real projectivestructures on S . Now we have a better, but not yet satisfactory understand-ing of the geometric significance of the Hitchin components beyond PSL(3 , R ),which will be described in more detail in § ρ : π ( S ) → PSL(3 , R ) is a discrete embedding with the additionalproperty that for any γ ∈ π ( S ) − { } , the element ρ ( γ ) is diagonalizable withdistinct real eigenvalues. These properties have been generalized to all Hitchinrepresentations into PSL( n, R ) by Labourie [68]. Theorem 6.5 ([68]).
Let ρ : π ( S ) → PSL( n, R ) be a Hitchin representation.Then ρ is a discrete embedding, and for any γ ∈ π ( S ) − { } , the element ρ ( γ ) is diagonalizable with distinct real eigenvalues. Note that by Remark 6.2 similar results hold for Hitchin representationsinto Sp(2 m, R ) and SO( m, m + 1). In the case when S is a noncompact surface (of finite type), the generalizationof Hitchin’s work, which is based on methods from the theory of Higgs bundles,has only been partially carried out as it presents some additional analytic diffi-culties (see for example [5]). But when G is an adjoint split real Lie group and S is a noncompact surface, there is a completely different approach to define aspecial subset of Hom (cid:0) π ( S ) , G (cid:1) due to Fock and Goncharov [36] which leadsto the set of positive representations Hom pos (cid:0) π ( S ) , G (cid:1) ⊂ Hom (cid:0) π ( S ) , G (cid:1) .In order to describe the definition let us identify S with a punctured surfacesof the same topological type. As recalled in § ρ : π ( S ) → G corresponds to a flat principal G -bundle G ( ρ ) on S. The definition of thespace of positive representations relies on considering the space of framed G -bundles on S . Let B = G/B be the space of Borel subgroups of G .A framed G -bundle on S is a pair (cid:0) G ( ρ ) , β (cid:1) , where G ( ρ ) is a flat principal G -bundle on S and β is a flat section of the associated bundle G ( ρ ) × G B restrictedto the punctures. There is a natural forgetful map from the space of framed G -bundles to the space of flat principal G -bundles sending (cid:0) G ( ρ ) , β (cid:1) → G ( ρ ).Since there always exists a flat section of G ( ρ ) × G B over the punctures, thismap is surjective.Given an ideal triangulation of the surface S , i.e. a triangulation whosevertices lie at the punctures of S , one can use the information provided bythe section β to define a coordinate system on the space of framed G -bundles.Fock and Goncharov show that these coordinate systems form a positive atlas.3This means in particular that the coordinate transformations are given byrational functions, involving only positive coefficients. Hence the set of positiveframed G -bundles, i.e. the set where for a given triangulation all coordinatefunctions are positive real numbers, is well defined and independent of thechosen triangulation. The space of positive representations Hom pos (cid:0) π ( S ) , G (cid:1) is the image of the space of positive framed G -bundles under the forgetful map.The construction of the coordinates involves the notion of positivity inLie groups introduced by Lusztig [78, 79], to which we will come back in § G = PSL( n, R ) one can give an elementary description of thecoordinates in terms of projective invariants of triples and quadruples of flags(see [36, § G = PSL(2 , R ) the coordinates correspond toshearing coordinates constructed first by Thurston and Penner [87] and similarcoordinates constructed by Fock [35]. Theorem 6.6 ([36, Theorem 1.13, Theorem 1.9 and Theorem 1.10]).
Thespace
Hom pos (cid:0) π ( S ) , G (cid:1) /G of positive representations and R | χ ( S ) | dim G arehomeomorphic. Every positive representation is a discrete embedding, andevery nontrivial element γ ∈ π ( S ) which is not homotopic to a loop around aboundary component of S , is sent to a positive hyperbolic element. The notion of positive representations can be extended to the situationwhen S is compact, using the characterization of positive representations interms of equivariant positive boundary maps, which is described in the nextsection. Theorem 6.7 ([36, Theorem 1.15]).
When S is compact, Hom pos (cid:0) π ( S ) , G (cid:1) =Hom Hit (cid:0) π ( S ) , G (cid:1) . Remark 6.8.
In the case when G = PSL( n, R ) , PSp(2 n, R ) or PO( n, n + 1)this theorem also follows from Labourie’s work [68].When S is a noncompact surface, the set of positive framed G -bundlescarries many more interesting structures. It is a cluster variety, admits amapping class group invariant Poisson structure and natural quantizations.We will not discuss any of these interesting structures and refer the reader to[36] and [37] for further reading. In this section we discuss common structures as well as differences betweenthe higher Teichm¨uller spaces introduced above – comparing maximal repre-sentations on one hand and Hitchin representations on the other hand. We4explain that, when S is compact, representations in higher Teichm¨uller spacesfit into the context of Anosov structures, which is a more general concept (forthe definition see § The higher Teichm¨uller spaces we are discussing here were defined and stud-ied with very different methods and so far we see no unified approach to it.Nevertheless there is a common theme in all these works which highlights animportant underlying structure for all higher Teichm¨uller spaces: The exis-tence of very special boundary maps .Since π ( S ) is a word hyperbolic group, the boundary ∂π ( S ) of π ( S ) is awell defined compact metrizable space. When S is compact ∂π ( S ) identifiesnaturally with a topological circle S endowed with a canonical H¨older struc-ture. When S is not compact there is no natural identification of ∂π ( S ) witha subset of S (see the discussion in § G is a Lie group ofHermitian type, we saw in § S of the symmetric space associated to G gives rise to the notion of a maximaltriple of points in ˇ S ; and Theorem 5.29 characterizes maximal representationsas those which admit an equivariant boundary map ϕ : ∂π ( S ) → ˇ S , whichsends every positively oriented triple to a maximal triple in ˇ S .For Hitchin representations Labourie constructed special boundary mapsin [68]. In this case S is compact and ∂π ( S ) = S . A map ϕ : S → RP n − is said to be convex if for every n -tuple of distinct points x , · · · , x n ∈ S the images ϕ ( x ) , · · · , ϕ ( x n ) are in direct sum - or, equivalently, the map isinjective and any hyperplane in RP n − intersects ϕ ( S ) in at most ( n − n, R ) in terms ofconvex maps is due to a combination of the construction by Labourie and aresult by Guichard: Theorem 7.1 ([68, 52]).
A representation ρ : π ( S ) → PSL( n, R ) lies in theHitchin component if and only if there exists a ρ -equivariant continuous convexmap ϕ : ∂π ( S ) → RP n − . In the context of positive representations the notion of positivity for theboundary maps relies on Lusztig’s notion of positivity. Recall that a matrix5in GL( n, R ) is totally positive if all its minors are positive numbers. An uppertriangular matrix is positive if all not obviously zero minors are positive. Thenotion of positivity has been extended to all split real semisimple Lie groups G by Lusztig [78, 79]. This can be used to define a notion of positivity for k -tuplesin full flag varieties. Let B + be a Borel subgroup of G , B − an opposite Borelsubgroup and U the unipotent radical of B + . Then the set of Borel groups in B being opposite to B + can be identified with the orbit of B − under U . Thenotion of positivity gives us a well defined subset U ( R > ) ⊂ U . A k -tuple ofpoints ( B , · · · , B k ) in B is said to be positive if (up to the action of G ) it canbe written as (cid:0) B + , B − , u B − , · · · ( u · · · u k − ) B − (cid:1) , where u i ∈ U ( R > ) for all i = 1 , · · · , k − Definition 7.2.
A map ∂π ( S ) → G/B is said to be positive if it sends everypositively oriented k -tuple in ∂π ( S ) to a positive k -tuple of flags in G/B . Remark 7.3.
A map ∂π ( S ) → G/B is positive if and only if it sends everypositively oriented triple in ∂π ( S ) to a positive tripe of flags in G/B . Theorem 7.4 ([36, Theorem 1.6]).
Let G be a split real simple Lie group and B < G a Borel subgroup. A representation ρ : π ( S ) → G is positive if andonly if there exists a ρ -equivariant positive map ϕ : ∂π ( S ) → G/B . Remark 7.5.
Note that Fock and Goncharov choose a different identificationof the boundary of π ( S ) with a subset of S . For their identification theboundary map is indeed continuous.Let us emphasize that Fock and Goncharov prove Theorem 7.4 when S is a noncompact surface. In the case when S is a compact surface, they usethe characterization of Theorem 7.4 in order to define positive representations ρ : π ( S ) → G by requiring the existence of a ρ -equivariant positive map ϕ : ∂π ( S ) = S → G/B . In order to prove the equality Hom
Hit (cid:0) π ( S ) , G (cid:1) =Hom pos (cid:0) π ( S ) , G (cid:1) for compact surfaces S , they observe first that the set ofpositive representation is an open subset of the Hitchin component. Then theystudy limits of positive representations in order to prove that it is also closed.Hence as a nonempty open and closed subset it is a connected component andthus coincides with the Hitchin component.For G = PSL( n, R ) there is an intimate relation between positive maps of ∂π ( S ) into the full flag variety G/B and convex maps into the the partial flagvariety RP n − . In particular, the projection of a positive map into the flagvariety to RP n − is a convex map, and convex maps (with some regularity)naturally lift to positive maps, see [36, Theorem 1.3] and [68, Chapter 5], [72,Appendix B] for details.6 The only simple groups which are both of Hermitian type as well as split realare the real symplectic groups, G = PSp(2 n, R ). When n ≥
2, Hitchin rep-resentations or positive representation, and maximal representation providedifferent generalizations of Teichm¨uller space in this situation. It is indeed notdifficult to see that the Hitchin component and the space of positive represen-tations are properly contained in the space of maximal representations.Moreover, for the symplectic group the properties of the boundary mapsrequired in Theorem 7.4 and in Theorem 5.29 are related in the following way.In this situation F = G/B is the flag variety consisting of full isotropic flagsand ˇ S = G/Q is the partial flag variety consisting only of Lagrangian (i.e.maximal isotropic) subspaces. Positive triples of flags in F in the sense ofDefinition 7.2 are mapped to maximal triples in the Shilov boundary in thesense of Theorem 5.29 under the natural projection F → ˇ S ( F , · · · , F n ) F n . The notion of Anosov structure is a dynamical analogue of the concept oflocally homogeneous (
G, X )-structures in the sense of Ehresmann, introducedby Labourie in [68] to study Hitchin representations into PSL( n, R ). Holonomyrepresentations of Anosov structure are called Anosov representations .The class of Anosov representations is much bigger than the higher Te-ichm¨uller spaces discussed above. Anosov representations of fundamentalgroups of surfaces exist into any semisimple Lie group, and they can be de-fined more generally also for fundamental groups of arbitrary closed negativelycurved manifolds. Nevertheless, when S is a compact surface, representationsin higher Teichm¨uller spaces are examples of Anosov representations and recentresults about Anosov representations provide important geometric informationabout higher Teichm¨uller spaces. From now on let S be a compact connected oriented surface with a fixedhyperbolic metric. Denote by T S its unit tangent bundle and by ϕ t thegeodesic flow on T S . The group π ( S ) acts as group of deck transformationson T e S , commuting with ϕ t .7Let G be a semisimple Lie group, given a representation ρ : π ( S ) → G weobtain a proper action of π ( S ) on T e S × G by γ ∗ ( x, g ) = (cid:0) T γ x, ρ ( γ ) g (cid:1) whose quotient π ( S ) \ ( T e S × G ) is the total space G ( ρ ) of a (flat) principal G -bundle over T S . (Note that the bundle G ( ρ ) defined here is the pullbackof the bundle G ( ρ ) over S , defined in § T S → S .) The geodesic flow lifts to a flow on G ( ρ ) defined (with a slightabuse of notation) by ϕ t ( x, g ) = ( ϕ t ( x ) , g ) on T e S × G. Let P + , P − < G be a pair of opposite parabolic subgroup of G . The uniqueopen G -orbit O ⊂
G/P + × G/P − inherits two foliations, whose correspondingdistributions we denote by E ± , i.e. ( E ± ) ( z + ,z − ) ∼ = T z ± G/P ± . Definition 8.1. [68] Let O ( ρ ) be the associated O -bundle of G ( ρ ). An Anosovstructure on O ( ρ ) is a continuous section σ such that(1) σ commutes with the flow, and(2) the action of the flow ϕ t on σ ∗ E + (resp. σ ∗ E − ) is contracting (resp.dilating), i.e. there exist constants A, a > • for any e in σ ∗ ( E + ) m and for any t > k ϕ t e k ϕ t m ≤ A exp( − at ) k e k m , • for any e in σ ∗ ( E − ) m and for any t > k ϕ − t e k ϕ − t m ≤ A exp( − at ) k e k m , where k · k is any continuous norm on O ( ρ ). Remark 8.2.
The definition of Anosov structure does not depend on thechoice of the hyperbolic metric on S . Definition 8.3.
A representation ρ : π ( S ) → G is said to be a ( P + , P − )-Anosov representation if O ( ρ ) carries an Anosov structure.The conditions on σ are equivalent to requiring that σ ( T S ) is a hyberbolicset for the flow ϕ t . Stability of hyperbolic sets implies stability for ( P + , P − )-Anosov representations: Proposition 8.4 ([68]).
The set of ( P + , P − ) -Anosov representations is openin Hom (cid:0) π ( S ) , G (cid:1) . Since we fixed a hyperbolic structure on S we can equivariantly identifythe boundary ∂π ( S ) with the boundary S = ∂ D of the Poincar´e disk as wedid in § Proposition 8.5 ([68]).
To every Anosov representation ρ : π ( S ) → G thereare associated continuous ρ -equivariant boundary maps ξ ± : S → G/P ± , with the property that for all t, t ′ ∈ S distinct, we have (cid:0) ξ + ( t ) , ξ − ( t ′ ) (cid:1) ∈ O .Moreover, for every element γ ∈ π ( S ) − { } with fixed points γ ± ∈ S thepoint ξ ± ( γ + ) is the unique attracting fixed point of ρ ( γ ) in G/P ± and ξ ± ( γ − ) is the unique repelling fixed point of ρ ( γ ) in G/P ± .Sketch of proof. Since the existence of these boundary maps play an importantrole in some results discussed above, let us sketch how these maps are obtained.Recall that T e S is naturally identified with the space of positively orientedtriples in S , via the map T e S → ( S ) (3 + ) v ( v + , v , v − ) , where v ± are the endpoints at ±∞ of the unique geodesic g v determined by v and v is the unique point which is mapped to the basepoint of v under theorthogonal projection to the geodesic g v and such that ( v + , v , v − ) is positivelyoriented.The existence of a continuous section σ of O ( ρ ) is equivalent to the existenceof a ρ -equivariant continuous map F : T e S → O . The Anosov condition (1) on σ is equivalent to F being ϕ t -invariant. In particular, the map F only dependsof ( v + , v − ) ∈ ( S × S ) − diag =: ( S ) (2) . Thus we have a map F = ( ξ + , ξ − ) : ( S ) (2) → G/P + × G/P − . It is not difficult to see that due to the contraction properties of the geodesicflow (see Anosov condition (2)) the map ξ + ( v + , v − ) only depends on v + and ξ − ( v + , v − ) only depends on v − , and that ξ ± satisfy the above properties.The property of a representation ρ : π ( S ) → G being ( P + , P − )-Anosov isindeed (almost) equivalent to the existence of such continuous boundary maps. Proposition 8.6 ([55]).
Let ρ : π ( S ) → G be a Zariski dense representationand assume that there exists ρ -equivariant continuous boundary maps ξ ± : S → G/P ± such that (1) for all t, t ′ ∈ S distinct, we have (cid:0) ξ + ( t ) , ξ − ( t ′ ) (cid:1) ∈ O , and (2) for all t ∈ S , the two parabolic subgroups stabilizing ξ + ( t ) and ξ − ( t ) contain a common Borel subgroup.Then ρ is a ( P + , P − ) -Anosov representation. The Anosov section σ can be easily reconstructed from the boundary mapusing the identification T e S ∼ = ( S ) (3 + ) .9Let us list several consequences of the existence of such boundary maps,for proofs see [68] and [55].(1) The representation ρ is faithful with discrete image.(2) For every γ ∈ π ( S ) − { } the holonomy ρ ( γ ) is conjugate to a (contract-ing) element in H = P + ∩ P − .(3) The orbit map π ( S ) → X , γ ρ ( γ ) x for some x ∈ X is a quasi-isometric embedding with respect to the word metric on π ( S ) and any G -invariant metric on the symmetric space X = G/K .(4) The representation ρ is well-displacing.The concepts of welldisplacing representations and quasiisometric embeddingsare discussed in more detail in § § n, R ), PSp(2 n, R ), PO( n, n + 1) areAnosov representations with P ± being minimal parabolic subgroups [68].Using the characterization of Hitchin representations via the existenceof convex curves, we are given ϕ : S → RP n − and we can take ϕ ∗ : S → ( RP n − ) ∗ to be the dual curve, i.e. ϕ ∗ ( t ) is the uniqueosculating hyperplane of the curve ϕ containing ϕ ( t ). We set ξ + = ϕ , ξ − = ϕ ∗ , then ( ξ + , ξ − ) : S → RP n − × ( RP n − ) ∗ satisfies the hypoth-esis of Proposition 8.6, thus ρ is Anosov with respect to the parabolicsubgroup stabilizing a line in R n . In order to see that Hitchin represen-tations are actually Anosov with respect to the minimal parabolic sub-group, note that for any point on the convex curve we can consider theosculating flag and obtain maps ξ ± = ξ : S → G/P min . The convexityof ϕ implies the transversality condition on ξ ± (see [68, Chapter 5])(2) Maximal representations are Anosov representation with P ± being sta-bilizers of points in the Shilov boundary ˇ S of the Hermitian symmetricspace [14, 15]. This means in particular that in this case the boundarymaps ξ ± = ϕ : S → ˇ S (Theorem 5.29) sending positively oriented triples0 to maximal triples are continuous. Since ( ξ + , ξ − ) satisfies the transver-sality conditions required in Proposition 8.6, maximal representationsare ( P + , P − )-Anosov. The automorphism groups of π ( S ) and of G act naturally on Hom (cid:0) π ( S ) , G (cid:1) Aut (cid:0) π ( S ) (cid:1) × Aut( G ) × Hom (cid:0) π ( S ) , G (cid:1) → Hom (cid:0) π ( S ) , G (cid:1) ( ψ, α, ρ ) α ◦ ρ ◦ ψ − . When we consider the quotient of the representation variety Hom (cid:0) π ( S ) , G (cid:1) /G this action descends to an action of the outer automorphism group Out (cid:0) π ( S ) (cid:1) =Aut (cid:0) π ( S ) (cid:1) / Inn (cid:0) π ( S ) (cid:1) on Hom (cid:0) π ( S ) , G (cid:1) /G . As we discussed in § S is closed, the group of orientation preserving outer automorphisms of π ( S )is isomorphic to the mapping class group Map( S ), and we will refer to thisaction as the action of the mapping class group.The components of Hom (cid:0) π ( S ) , G (cid:1) /G which form higher Teichm¨uller spaces,are preserved by this action. In the case when G = PSU(1 ,
1) the action ofthe mapping class group on Teichm¨uller space is properly discontinuous andthe quotient M ( S ) is the moduli space of Riemann surfaces.Given a higher Teichm¨uller space it is natural to consider its quotient bythe action of the mapping class group, to study how it relates to the modulispace of Riemann surface, and to investigate its possible compactifications.The first question here is whether the action of the mapping class group isproperly discontinuous on higher Teichm¨uller spaces.In order to answer this question the essential notion is that of a represen-tation being well-displacing. For this let us introduce the translation lengths τ and τ ρ of an element γ ∈ π ( S ) − { } : τ ( γ ) = inf p ∈ e S d ( p, γp ) τ ρ ( γ ) = inf z ∈ X d G ( z, ρ ( γ ) z ) , where d is the lift of a hyperbolic metric on S and d G is a G -invariant Rie-mannian metric on the symmetric space X . Definition 8.7.
A representation ρ : π ( S ) → G is well-displacing if thereexist constants A, B > γ ∈ π ( S ) A − τ ρ ( γ ) − B ≤ τ ( γ ) ≤ Aτ ρ ( γ ) + B. τ depends on the choice of hyperbolic metric on S .For any two choices the translations lengths are comparable, thus the definitionof well-displacing is independent of the chosen hyperbolic metric on S .It is shown in [71, 99, 57] that representations in higher Teichm¨uller spacesare well-displacing. Then a simple argument (see e.g. [71, 99]) shows thatthis implies that the action of the mapping class group on higher Teichm¨ullerspaces is properly discontinuous. Remark 8.8.
The notion of well-displacing is related to the notion of quasi-isometric embeddings. A representation ρ : π ( S ) → G is a quasiisometricembedding if there exist constants A, B > γ ∈ π ( S ) A − d G (cid:0) ρ ( γ ) z, z (cid:1) − B ≤ d ( γp, p ) ≤ A d G (cid:0) ρ ( γ ) z, z (cid:1) + B, for some p ∈ e S and some z ∈ X .Both notions can be defined more generally for representations of arbitraryfinitely generated groups. The relation between the two notions is studiedin [30]. Representations in higher Teichm¨uller spaces are also quasiisometricembeddings [71, 99, 57, 55]. The notion of well-displacement also playsan important role when trying to obtain a mapping class group invariant pro-jection from higher Teichm¨uller spaces to classical Teichm¨uller space. To de-scribe this approach recall that given a representation ρ : π ( S ) → G andhyperbolic metric h ∈ Hyp( S ), one can define the energy of a ρ -equivariantsmooth map f : e S → X into the symmetric space X = G/K as E ρ ( f, h ) := Z S || df || dvol , where || df || ( p ), p ∈ S is the norm of the linear map df p with respect to thehyperbolic metric on S and the G -invariant Riemannian metric on X .The map f is said to be harmonic if and only if it minimizes the energyin its ρ -equivariant homotopy class. Setting E ρ ( h ) := inf f E ρ ( f, h ), where f ranges over all ρ -equivariant smooth maps e S → X , we obtain a function E ρ : F ( S ) = Diff +0 ( S ) \ Hyp( S ) → R , called the energy functional associated to the representation ρ : π ( S ) → G .The energy functional is a smooth function on the Fricke space F ( S ). In thecase when G = PSU(1 ,
1) and ρ is a discrete embedding, it is known that E ρ has a unique minimum [34, 100], namely the hyperbolic structure determinedby ρ .In the general case, one would like to construct a mapping class groupinvariant projection from higher Teichm¨uller spaces to classical Teichm¨uller2space by showing that the energy functional has a unique minimum. As a firststep we have Theorem 8.9 ([71, Theorem 6.2.1]).
Let ρ : π ( S ) → G be a well-displacingrepresentation, then the energy functional E ρ is a proper function on F ( S ) . In [71] Labourie describes an approach to realize the Hitchin componentfor PSL( n, R ) as a vector bundle over Teichm¨uller space in a equivariant waywith respect to the mapping class group. Recall for this that the isomorphism(see (6.1)) h j : Hom Hit (cid:0) π ( S ) , PSL( n, R ) (cid:1) / PSL( n, R ) → H (cid:0) S, ⊕ nk =2 Ω k ( S,j ) (cid:1) , where H ( S, Ω k ( S,j ) ) is the vector space of holomorphic differentials (with re-spect to the fixed complex structure j on S ) of degree k , is not mapping classgroup invariant. Consider the vector bundle E n over Teichm¨uller space T ( S )realized as space of complex structures on S , where the fiber over the complexstructure j equals H (cid:0) S, ⊕ nk =3 Ω k ( S,j ) (cid:1) . Then E n has the same dimension asHom Hit (cid:0) π ( S ) , PSL( n, R ) (cid:1) / PSL( n, R ). Labourie defines the Hitchin map( j, ω ) h j (0 , ω ) , where j ∈ T ( S ) is a complex structure and ω ∈ H (cid:0) S, ⊕ nk =3 Ω k ( S,j ) (cid:1) . Thismap is equivariant with respect to the mapping class group action. Labourieproves that it is surjective [71, Theorem 2.2.1] and conjectures that H is ahomeomorphism, which would imply Conjecture ([71, Conjecture 2.2.3]).
The quotient of the Hitchin component
Hom
Hit (cid:0) π ( S ) , PSL( n, R ) (cid:1) / PSL( n, R ) by the mapping class group is a vectorbundle over the moduli space of Riemann surfaces with fiber being the space ofholomorphic k -differentials H (cid:0) S, ⊕ nk =3 Ω kS (cid:1) . In order to prove this conjecture it would be sufficient to show that for aHitchin representation ρ ∈ Hom
Hit (cid:0) π ( S ) , PSL( n, R ) (cid:1) / PSL( n, R ) the energyfunctional E ρ has a nondegenerate minimum.Conjecture 8.2.2 has been proved for n = 2 and n = 3. The proof forPSL(3 , R ) is independently due to Labourie [70] and Loftin [74]. They rely onthe description of Hom Hit (cid:0) π ( S ) , PSL(3 , R ) (cid:1) as deformation space of convexreal projective structures due to Choi and Goldman, and use the theory ofaffine spheres developed in [24, 25] in order to prove Theorem 8.10 ([70, 74]).
The quotient
Map( S ) \ Hom
Hit (cid:0) π ( S ) , PSL(3 , R ) (cid:1) / PSL(3 , R ) is a vector bundle over the moduli space of Riemann surface with fiber beingthe space of cubic holomorphic differentials on the surface. § Mapping class group equivariant compactificationsof higher Teichm¨uller spaces are partially understood.A general construction to compactify the space of discrete, injective non-parabolic representations of a finitely generated group into G using the gen-eralized marked length spectrum is given in [85]. This construction applies togive compactifications of higher Teichm¨uller spaces. Boundary points in thiscompactification can be interpreted as actions on R -buildings [86].For the Hitchin component Hom Hit (cid:0) π ( S ) , PSL(3 , R ) (cid:1) / PSL(3 , R ) the iden-tification with the deformation space of convex real projective structures al-lows to obtain a better understanding of this compactification, see e.g. [62,75, 76, 28]. Through the study of degenerations of convex projective struc-tures, Cooper et al. [28] obtain a description of boundary points as mixturesof measured laminations and special Finsler metrics (Hex metrics) on S .Fock and Goncharov construct tropicalizations of the spaces of positiverepresentations which they expect to provide (partial) completions [36] when S is noncompact. But except for the case when G = PSL(2 , R ) (treated in [37]),they do not define a topology of the union of space of positive representationsand its tropicalized counterpart. Realizing ∂ D ⊂ CP , the restriction of the classical cross-ratio function on CP c ( x, y, t, z ) = x − yx − t z − tz − y gives a continuous real valued PSU(1 , ∂ D ) ∗ := (cid:8) ( x, y, z, t ) ∈ ( ∂ D ) : x = t y = z (cid:9) . This crossratio and several generalizations (see e.g. [84, 73]) play an importantrole in the study of negatively curved manifolds and hyperbolic groups.Given a hyperbolic element γ ∈ PSU(1 ,
1) its period is defined as l c ( γ ) = log c ( γ − , z, γ + , γz ) , where γ + is the unique attracting fixed point and γ − the unique repelling fixedpoint of γ in ∂ D and z ∈ ∂ D − { γ ± } is arbitrary. The period of γ equals thetranslation length τ ( γ ) = inf p ∈ D d D ( p, γp ).4 Given a discrete embedding ρ : π ( S ) → PSU(1 , c ρ := ϕ ∗ c : ( S ) ∗ → R be the pullback of c by some ρ -equivariant boundary map, be the associatedcrossratio function. Then c ρ contains all information about the marked lengthspectrum of S with respect to the hyperbolic metric defined by ρ . In particular,two discrete embeddings ρ , ρ are conjugate if and only if c ρ = c ρ .A generalized crossratio function is a π ( S )-invariant continuous functions( S ) ∗ = (cid:8) ( x, y, z, t ) ∈ ( S ) : x = t y = z (cid:9) → R satisfying the following relations [69, Introduction]:(1) (Symmetry) c ( x, y, z, t ) = c ( z, t, x, y )(2) (Normalization) c ( x, y, z, t ) = 0 if and only if x = y or z = tc ( x, y, z, t ) = 1 if and only if x = z or y = t (3) (Cocycle identity) c ( x, y, z, t ) = c ( x, y, z, w ) c ( x, w, z, t ) c ( x, y, z, t ) = c ( x, y, w, t ) c ( w, y, z, t ).Among such functions crossratios arising from a discrete embedding π ( S ) → PSU(1 ,
1) are uniquely characterized by the functional equation 1 − c ( x, y, z, t ) = c ( t, y, z, x ).The study of generalized crossratio functions associated to higher Teichm¨ullerspaces has been pioneered by Labourie. In particular, he associates a general-ized crossratio function to any Hitchin representation into PSL( n, R ) and showsthat crossratio functions arising from a Hitchin representation into PSL( n, R )are characterized by explicit functional equations [69].In [72] Labourie and McShane establish generalized McShane identities forthe crossratios associated to Hitchin representations into PSL( n, R ). Remark 8.11.
Related crossratio functions of four partial flags consisting ofa line and a hyperplane are used in the work of Fock and Goncharov [36] inorder to construct explicit coordinates for the space of positive representationsinto PSL( n, R ).In the context of maximal representations crossratio functions have beendefined and studied by Hartnick and Strubel [57]. They construct crossra-tio functions defined on a suitable subset ˇ S ∗ of the fourfold product of theShilov boundary of any Hermitian symmetric space of tube type. They showthat there is a unique such crossratio function which satisties some natural5functorial properties. Given a maximal representation ρ : π ( S ) → G a con-crete implementation of the continuous boundary map ϕ : S → ˇ S (see Theo-rem 5.29) allows to pullback this crossratio function to a generalized crossratiofunction on ( S ) ∗ . The well-displacing property of representations in higherTeichm¨uller spaces can be easily deduced from the existence of generalizedcrossratio functions. In all works investigating crossratio functions, the ex-istence of boundary maps with special positivity properties (as discussed in § We already mentioned that Hitchin had asked in [60] about the geometricsignificance of Hitchin components, and one might raise the same questionfor maximal representation, even though the picture there seems to be morecomplicated due to the fact that the space of maximal representations hassingularities and multiple components.Interpreting higher Teichm¨uller spaces as deformation spaces of geometricstructures is not just of interest in itself. Any such interpretation gives animportant tool to study these spaces, their quotients by the mapping classgroup, their relations to the moduli space of Riemann surfaces as well as theircompactifications. This is illustrated by the fact that the deeper understand-ing of these questions for the Hitchin component of PSL(3 , R ) relies on theTheorem by Choi and Goldman, which we already mentioned above: Theorem 8.12 ([47, 26]).
The Hitchin component
Hom
Hit (cid:0) π ( S ) , PSL(3 , R ) (cid:1) / PSL(3 , R ) parametrizes convex real projective structures on S . The original proof of this theorem relied on Goldman’s work on convexprojective structures on surfaces [47], which implied that the deformation spaceof these structures is an open domain in Hom
Hit (cid:0) π ( S ) , PSL(3 , R ) (cid:1) / PSL(3 , R ).Goldman and Choi [26] then proved that this subset is furthermore closed,establishing the above theorem.In terms of the properties of Hitchin representations we have discussed sofar, Theorem 8.12 is basically equivalent to the characterization of Hitchinrepresentations into PSL(3 , R ) by the existence of a convex map from S into RP (Theorem 7.1). We give a sketch of how Theorem 8.12 follows fromTheorem 7.1 when n = 3. Sketch of a proof of Theorem 8.12 assuming Theorem 7.1.
A convex real pro-jective structure on S is a pair ( N, f ), where N is the quotient Ω / Γ of a strictly6convex domain Ω in RP by a discrete subgroup Γ of PSL(3 , R ), and f : S → N is a diffeomorphism.Starting from a representation ρ : π ( S ) → PSL(3 , R ) in the Hitchin com-ponent, let Ω ξ ⊂ RP be the strictly convex domain bounded by the convexcurve ξ ( S ) ⊂ RP . Then ρ (cid:0) π ( S ) (cid:1) acts freely and properly discontinuouslyon Ω ξ . The quotient Ω ξ /ρ (cid:0) π ( S ) (cid:1) is a real projective convex manifold, dif-feomorphic to S . Conversely given a real projective structure on S , we can ρ -equivariantly identify S (identified with the boundary of π ( S )) with theboundary of Ω and get a convex curve ξ : S → ∂ Ω ⊂ RP .Inspired by this proof and with Theorem 7.1 at hand for arbitrary n , onemight try to follow a similar strategy on order to find geometric structuresparametrized by the Hitchin component for PSL( n, R ). This works for n = 4,where we obtain the following Theorem 8.13 ([53]).
The Hitchin component for
PSL(4 , R ) is naturallyhomeomorphic to the moduli space of properly convex foliated projective struc-tures on T S . Properly convex foliated projective structures are locally homogeneous (cid:0)
PSL(4 , R ) , RP (cid:1) -structures on T S satisfying the following additional conditions: • every orbit of the geodesic flow is locally a projective line, • every (weakly) stable leaf of the geodesic flow is locally a projective planeand the projective structure on the leaf obtained by restriction is convex.Using the convex curve provided by Theorem 7.1 one can consider the cor-responding discriminant surface ∆ ⊂ RP , i.e. the union of all its tangentlines. The complement RP − ∆ consists of two connected components, onboth of which ρ (cid:0) π ( S ) (cid:1) acts properly discontinuous. The quotient of one ofthe connected components by π ( S ) is homeomorphic to T S , equipped with aproperly convex foliated projective structure. The main work goes into estab-lishing the converse direction, i.e. showing that the holonomy representationof a properly convex foliated projective structure on T S lies in the Hitchincomponent – this is rather tedious. Remark 8.14.
The above theorem implies that the Hitchin component forPSp(4 , R ) is naturally homeomorphic to the moduli space of properly convexfoliated projective contact structures on the unit tangent bundle of S .For n ≥ ρ : π ( S ) → G in higher Teichm¨ullerspaces is to find domains of discontinuity for such representations in homoge-neous spaces, more precisely in generalized flag varieties associated to G , on7which π ( S ) is supposed to act with compact quotient. This problem becomesmore difficult the bigger G gets, since π ( S ) is a group of cohomological di-mension 2, whereas the dimension of the generalized flag varieties grows as G get bigger. So it comes a bit as a surprise that finding domains of discontinu-ity with compact quotient can be accomplished in the very general setting ofAnosov representations. Theorem 8.15 ([54, 55]).
Let G be a semisimple Lie group and assume thatno simple factor of G is locally isomorphic to PSL(2 , R ) . Let ρ : π ( S ) → G bea ( P + , P − ) -Anosov representation. Let P = M AN be the minimal parabolicsubgroup of G . Then there exists an open non-empty set Ω ρ ⊂ G/AN , onwhich π ( S ) acts freely, properly discontinuous and with compact quotient. Remark 8.16.
The homogeneous space
G/AN is the maximal compact quo-tient of G . In many cases the domain Ω ρ descends to a domain of discontinuityin G/P . Remark 8.17.
Theorem 8.15 holds more general for Anosov representationsof convex cocompact subgroups of Hadamard manifolds of strictly negativecurvature or even of hyperbolic groups. The reader interested in the moregeneral statement is referred to [55].The main tool in order to define the domain of discontinuity Ω ρ are the ρ -equivariant continuous boundary maps ξ ± : S → G/P ± associated to the( P + , P − )-Anosov representation (see Proposition 8.5).There is some evidence that – at least in the case of higher Teichm¨ullerspaces – the quotients Ω ρ /ρ (cid:0) π ( S ) (cid:1) are homeomorphic to the total spacesof bundles over S with compact fibers. This has been established for maxi-mal representation into Sp(2 n, R ) as well as for Hitchin representations intoSL(2 n, R ). Theorem 8.18 ([55]). (1)
The Hitchin component for
SL(2 n, R ) parametrizesreal projective structures on a compact manifold M , which is topologicallya O( n ) / O( n − -bundle over the surface S . (2) Maximal representations into
Sp(2 n, R ) parametrize real projective struc-tures on a compact manifold M homeomorphic to an O( n ) / O( n − -bundle over the surface S . Its isomorphism type depends on the con-nected component containing the representation. The Hitchin component is by definition a single connected component, butthe space of maximal representations is a priori only a union of connected8components, and their might be more than one. In many cases the exactnumber of connected components of the space of maximal representations hasbeen computed using methods from the theory of Higgs bundles [50, 40, 39,10, 11]. And the most interesting family in terms of the number of connectedcomponents are maximal representations into symplectic groups Sp(2 n, R ):there are 3 × g connected components when n ≥ × g + 2 g − n = 2 [50].Invariants to distinguish these connected components can be derived fromthe associated Higgs bundles, but topological invariants to distinguish thedifferent connected components also arise from considering maximal represen-tations as Anosov representations.Recall that in the definition of Anosov structures one considers the flat G -bundle G ( ρ ) over T S and the associated bundle O ( ρ ). The first part ofthe data of an Anosov structure is a section σ of O ( ρ ). Since O ( ρ ) is the G/H -bundle associated to G ( ρ ) its sections are in one-to-one correspondencewith reductions of the structure group of G ( ρ ) from G to H . In general thereis no canonical section, but in the case of Anosov structures we have Proposition 8.19 ([56]).
If a section σ of O ( ρ ) with the properties requiredin Definition 8.1 exists, then it is unique. As a consequence an Anosov representation ρ : π ( S ) → G gives a canonicalreduction of the G -principal bundle G ( ρ ) to an H -principal bundle. This H -bundle is in general not flat; its characteristic classes give topological invariantsof the Anosov representation ρ which live in H ∗ ( T S ).In the situation of maximal representations ρ : π ( S ) → Sp(2 n, R ), wehave that H = GL( n, R ), embedded into Sp(2 n, R ) as the stabilizer of twotransverse Lagrangian subspaces. The topological invariants of significanceare first and second Stiefel–Whitney classes, as well as an Euler class if n = 2. Theorem 8.20 ([56]).
The topological invariants distinguish the connectedcomponents of
Hom max (cid:0) π ( S ) , Sp(2 n, R ) (cid:1) \ Hom
Hit (cid:0) π ( S ) , Sp(2 n, R ) (cid:1) .Considering Hitchin representations as ( P min , P oppmin ) -Anosov representa-tions there is an additional first Stiefel–Whitney class, which distinguishes theconnected components of Hom
Hit (cid:0) π ( S ) , Sp(2 n, R (cid:1) . The invariants constructed using the Anosov property of maximal repre-sentations are in principle computable for a given representation ρ : π ( S ) → Sp(2 n, R ). Explicit computations for various representations allows us to de-scribe model representations in any connected component. This is of particularinterest for Sp(4 , R ) as there are 2 g − irreducible Fuchsian representation which were introduced todefine the Hitchin component, there are two other kinds of model representa-9tions: A twisted diagonal representation is a maximal representation ρ θ = ( ι ⊗ θ ) : π ( S ) → SL(2 , R ) × O( n ) ⊂ Sp(2 n, R ) , where ι : π ( S ) → SL(2 , R ) is a discrete embedding and θ : π ( S ) → O( n ) is anorthogonal representation; SL(2 , R ) × O( n ) sits in Sp(2 n, R ) as the normalizerof the diagonal embeddingSL(2 , R ) → SL(2 , R ) n ⊂ Sp(2 n, R ) . A hybrid representation is a maximal representation ρ k = ρ ∗ ρ : S = S ∪ γ S → Sp(2 n, R ) ,k = 3 − g, · · · , −
1, which is obtained by amalgamation of an irreducible Fuch-sian representation on π ( S ) and a suitable deformation of an (untwisted)diagonal representation on π ( S ). The subscript k indicates the Euler char-acteristic of S . The construction of hybrid representations relies on the ad-ditivity of the Toledo number and the Euler characteristic under gluing (seeProposition 5.10). Theorem 8.21 ([56]).
When n ≥ any maximal representation ρ : π ( S ) → Sp(2 n, R ) can be deformed either to an irreducible Fuchsian representation orto a twisted diagonal representation.When n = 2 there are g − connected components H k , k = 1 , · · · , g − of Hom max (cid:0) π ( S ) , Sp(4 , R ) (cid:1) in which every representation has Zariski denseimage. Representations in H k can be deformed to k -hybrid representations. The information about model representations in each connected componentcan be used to obtain further information about the holonomies of maximalrepresentations.For representations in the Hitchin components, Theorem 6.5 implies thatfor every γ ∈ π ( S ) − { } the image ρ ( γ ) is diagonalizable over R with dis-tinct eigenvalues. This does not hold for the other components of maximalrepresentations.The Anosov property implies that for every γ ∈ π ( S ) \ { e } the image ρ ( γ )is conjugate to an element in GL( n, R ) < Sp(2 n, R ). More precisely, we have: Theorem 8.22 ([56]).
Let H be a connected component of Hom max (cid:0) π ( S ) , Sp(2 n, R ) (cid:1) \ Hom
Hit (cid:0) π ( S ) , Sp(2 n, R ) (cid:1) , and let γ ∈ π ( S ) − { } be an element corresponding to a simple curve. Thenthere exist (1) a representation ρ ∈ H such that the Jordan decomposition of ρ ( γ ) in GL( n, R ) has a nontrivial parabolic component. a representation ρ ′ ∈ H such that the Jordan decomposition of ρ ( γ ) in GL( n, R ) has a nontrivial elliptic component. This results indicates that understanding the structure of the space of max-imal representations is much more complicated than understanding the struc-ture of Hitchin components, since already the conjugacy classes in which theholonomy of one element can lie in might differ from connected component toconnected component.
In the previous sections we already mentioned some open questions regardingthe quotients of higher Teichm¨uller spaces, their compactifications as well astheir geometric significance. In this section we want to conclude our surveywith mentioning some further directions in the study of higher Teichm¨ullerspaces which to our knowledge have not yet been explored.
As we pointed out in § § k -tuples in (partial) flag varieties which then extends to other groups whichare neither split real forms nor of Hermitian type. Such notions of positivitymight lead to discovering higher Teichm¨uller spaces for other Lie groups G ,which are again characterized by the existence of special boundary maps.A first family of groups to look at could be G = PO( p, q ), which is ofHermitian type if ( p, q ) = (2 , q ) and a split real form if ( p, q ) = ( n, n + 1) or( p, q ) = ( n, n ).Every time there is a notion of positivity or cyclic ordering, the images ofboundary maps tend to be more regular, namely rectifiable circles. This con-trasts with the case of quasifuchsian deformations into PSL(2 , C ) of compactsurface groups in PSL(2 , R ), where in fact the limit set, or – what is the same– the image of the boundary map, is a topological circle with Hausdorff di-mension larger than 1, unless the deformed group is Fuchsian. This suggests This is due to [8]: see also the footnote on p. 76 of Fricke’s address in Chicago in 1893,[38]. i ◦ ρ : π ( S ) → G ( C ) , where i : G = G ( R ) ◦ → G ( C ) is the natural inclusion and ρ : π ( S ) → G iseither a maximal representation into a group of Hermitian type or a Hitchinrepresentation into a real split Lie group. Observe that i ◦ ρ is Anosov for asuitable pair of parabolic subgroups and, as a result, small deformations of i ◦ ρ are as well. Fock and Goncharov describe explicit coordinate for the space of positive repre-sentations. For PSL( n, R ) these coordinates have a particular nice form. Basedon the explicit coordinate system they describe the cluster variety structureand quantizations of the space of positive representations.It would be interesting to construct similar explicit coordinate systems forthe space of maximal representations, in particular when G = Sp(2 n, R ). The-orem 8.22 gives a hint that constructing coordinates for the space of maximalrepresentations is more involved. The structure of the coordinates also needsto be more complicated as they have to model the singularities of the space ofmaximal representations.The additivity of the Toledo number on the other hand implies that thespace of maximal representations of a compact surface S can be built out ofthe space of maximal representations of a pair of pants.Having coordinates at hand, one might also ask for quantizations of thespace of maximal representations or try to express the symplectic form on thespace of maximal representations explicitly in coordinates. References [1] R. Baer. Erweiterung von Gruppen und ihren Isomorphismen.
Math. Z. ,38(1):375–416, 1934.[2] C. Bavard. Longueur stable des commutateurs.
Enseign. Math. (2) , 37(1-2):109–150, 1991.[3] R. Benedetti and J.-J. Risler.
Real algebraic and semi-algebraic sets . Actualit´esMath´ematiques. [Current Mathematical Topics]. Hermann, Paris, 1990.[4] N. Bergeron and T. Gelander. A note on local rigidity.
Geom. Dedicata ,107:111–131, 2004. [5] I. Biswas, P. Ar´es-Gastesi, and S. Govindarajan. Parabolic Higgs bundlesand Teichm¨uller spaces for punctured surfaces. Trans. Amer. Math. Soc. ,349(4):1551–1560, 1997.[6] M. Bjorklund and T. Hartnick. Biharmonic functions on groups and limittheorems for quasimorphisms along random walks. to appear.[7] A. Borel. Class functions, conjugacy classes and commutators in semisim-ple Lie groups. In
Algebraic groups and Lie groups , volume 9 of
Aus-tral. Math. Soc. Lect. Ser. , pages 1–19. Cambridge Univ. Press, Cambridge,1997.[8] R. Bowen. Hausdorff dimension of quasicircles.
Publ. Math. Inst. Hautes´Etudes Sci. , 50(1):11–25, 1979.[9] S. B. Bradlow, O. Garc´ıa-Prada, and P. B. Gothen. Deformations of maximalrepresentations in Sp(4 , R ). Preprint, arXiv:0903.5496.[10] S. B. Bradlow, O. Garc´ıa-Prada, and P. B. Gothen. Surface group representa-tions and U( p, q )-Higgs bundles. J. Differential Geom. , 64(1):111–170, 2003.[11] S. B. Bradlow, O. Garc´ıa-Prada, and P. B. Gothen. Maximal surface grouprepresentations in isometry groups of classical Hermitian symmetric spaces.
Geom. Dedicata , 122:185–213, 2006.[12] M. Burger. An extension criterion for lattice actions on the circle. to appear in”Geometry, rigidity, and groups actions”, a Festschrift for Robert J. Zimmer,University of Chicago Press, to appear.[13] M. Burger and A. Iozzi. Bounded K¨ahler class rigidity of actions on Hermitiansymmetric spaces.
Ann. Sci. ´Ecole Norm. Sup. (4) , 37(1):77–103, 2004.[14] M. Burger, A. Iozzi, F. Labourie, and A. Wienhard. Maximal representationsof surface groups: Symplectic Anosov structures.
Quaterly Journal of Pureand Applied Mathematics , 1(3):555–601, 2005. Special Issue: In Memory ofArmand Borel, Part 2 of 3.[15] M. Burger, A. Iozzi, and A. Wienhard. Maximal representations and Anosovstructures. in preparation.[16] M. Burger, A. Iozzi, and A. Wienhard. Hermitian symmetric spaces and K¨ahlerrigidity.
Transform. Groups , 12(1):5–32, 2007.[17] M. Burger, A. Iozzi, and A. Wienhard. Tight embeddings of Hermitian sym-metric spaces.
Geom. Funct. Anal. , 19(3):678–721, 2009.[18] M. Burger, A. Iozzi, and A. Wienhard. Surface group representations withmaximal Toledo invariant.
Annals of Math. , 172:517–566, 2010.[19] M. Burger and N. Monod. Bounded cohomology of lattices in higher rank Liegroups.
J. Eur. Math. Soc. , 1(2):199–235, 1999.[20] M. Burger and N. Monod. Continuous bounded cohomology and applicationsto rigidity theory.
Geom. Funct. Anal. , 12:219–280, 2002.[21] D. Calegari. Faces of the scl norm ball.
Geom. Topol. , 13(3):1313–1336, 2009. [22] D. Calegari. scl , volume 20 of MSJ Memoirs . Mathematical Society of Japan,Tokyo, 2009.[23] R. D. Canary, D. B. A. Epstein, and P. Green. Notes on notes of Thurston.In
Analytical and geometric aspects of hyperbolic space (Coventry/Durham,1984) , volume 111 of
London Math. Soc. Lecture Note Ser. , pages 3–92. Cam-bridge Univ. Press, Cambridge, 1987.[24] S. Y. Cheng and S. T. Yau. On the regularity of the Monge-Amp`ere equationdet( ∂ u/∂x i ∂sx j ) = F ( x, u ). Comm. Pure Appl. Math. , 30(1):41–68, 1977.[25] S. Y. Cheng and S. T. Yau. Complete affine hypersurfaces. I. The completenessof affine metrics.
Comm. Pure Appl. Math. , 39(6):839–866, 1986.[26] S. Choi and W. M. Goldman. Convex real projective structures on closedsurfaces are closed.
Proc. Amer. Math. Soc. , 118(2):657–661, 1993.[27] J. L. Clerc and B. Orsted. The Gromov norm of the Kaehler class and theMaslov index.
Asian J. Math. , 7(2):269–295, 2003.[28] D. Cooper, K. Delp, D. Long, and M. Thistlethwaite. Convex projective struc-tures I : Small 2-orbifolds. in preparation.[29] K. Corlette. Flat G -bundles with canonical metrics. J. Differential Geom. ,28(3):361–382, 1988.[30] T. Delzant, O. Guichard, F. Labourie, and S. Mozes. Well displacing rep-resentations and orbit maps. Preprint, to appear in Proceedings of RobertZimmer’s birthday conference, http://xxx.lanl.gov/abs/0704.3499 , 2008.[31] A. Domic and D. Toledo. The Gromov norm of the K¨ahler class of symmetricdomains.
Math. Ann. , 276(3):425–432, 1987.[32] S. K. Donaldson. Anti self-dual Yang-Mills connections over complex algebraicsurfaces and stable vector bundles.
Proc. London Math. Soc. (3) , 50(1):1–26,1985.[33] B. Farb and D. Margalit. A primer on mapping class groups. Book in prepa-ration, ∼ margalit/primer/ , 2009.[34] A. E. Fischer and A. J. Tromba. A new proof that Teichm¨uller space is a cell. Trans. Amer. Math. Soc. , 303(1):257–262, 1987.[35] V. V. Fock. Dual Teichm¨uller spaces. http://lanl.arxiv.org/abs/dg-ga/9702018 .[36] V. V. Fock and A. B. Goncharov. Moduli spaces of local systems and higherTeichm¨uller theory.
Publ. Math. Inst. Hautes ´Etudes Sci. , 103:1–211, 2006.[37] V. V. Fock and A. B. Goncharov. Dual Teichm¨uller and lamination spaces.In
Handbook of Teichm¨uller theory. Vol. I , volume 11 of
IRMA Lect. Math.Theor. Phys. , pages 647–684. Eur. Math. Soc., Z¨urich, 2007.[38] R. Fricke. Automorphe Functionen und Zahlentheorie. In E. H. Moore,O. Bolza, H. Maschke, and H. S. White, editors,
Mathematical Papers read atthe International Mathematical Congress held in connection with the world’scolumbian exposition Chicago 1893 , pages 72–91. Macmillan and co., 1896. [39] O. Garc´ıa-Prada, P. B. Gothen, and I. M. i Riera. Higgs bundlesand surface group representations in real symplectic groups. Preprint http://lanl.arxiv.org/abs/0809.0576 .[40] O. Garc´ıa-Prada and I. M. i Riera. Representations of the fundamental groupof a closed oriented surface in Sp(4 , R ). Topology , 43(4):831–855, 2004.[41] S. M. Gersten. Bounded cocycles and combings of groups.
Internat. J. AlgebraComput. , 2(3):307–326, 1992.[42] ´E. Ghys. Groupes d’hom´eomorphismes du cercle et cohomologie born´ee. In
The Lefschetz centennial conference, Part III, (Mexico City 1984), Contemp.Math. , volume 58, pages 81–106. American Mathematical Society, RI, 1987.[43] W. M. Goldman. Discontinuous groups and the Euler class. Thesis, Universityof California at Berkeley, 1980.[44] W. M. Goldman. Characteristic classes and representations of discrete groupsof Lie groups.
Bull. Amer. Math. Soc. , 6(1):91–94, 1982.[45] W. M. Goldman. Geometric structures on manifolds and varieties of represen-tations. In
Geometry of group representations (Boulder, CO, 1987) , volume 74of
Contemp. Math. , pages 169–198. Amer. Math. Soc., Providence, RI, 1988.[46] W. M. Goldman. Topological components of spaces of representations.
Invent.Math. , 93(3):557–607, 1988.[47] W. M. Goldman. Convex real projective structures on compact surfaces.
J.Differential Geom. , 31(3):791–845, 1990.[48] W. M. Goldman and M. W. Hirsch. Flat bundles with solvable holonomy.
Proc. Amer. Math. Soc. , 82(3):491–494, 1981.[49] W. M. Goldman and J. Millson. Local rigidity of discrete groups acting oncomplex hyperbolic space.
Invent. Math. , 88:495–520, 1987.[50] P. B. Gothen. Components of spaces of representations and stable triples.
Topology , 40(4):823–850, 2001.[51] M. Gromov. Volume and bounded cohomology.
Publ. Math. Inst. Hautes´Etudes Sci. , 56:5–99 (1983), 1982.[52] O. Guichard. Composantes de Hitchin et repr´esentations hyperconvexes degroupes de surface.
J. Differential Geom. , 80(3):391–431, 2008.[53] O. Guichard and A. Wienhard. Convex foliated projective structures and theHitchin component for PSL ( R ). Duke Math. J. , 144(3):381–445, 2008.[54] O. Guichard and A. Wienhard. Domains of discontinuity for surface groups.
C. R. Acad. Sci. Paris, S´er. I , 40(347):1057–1060, 2009.[55] O. Guichard and A. Wienhard. Domains of discontinuity with compact quo-tients. in preparation, 2009.[56] O. Guichard and A. Wienhard. Topological invariants of anosov representa-tions. Preprint, http://lanl.arxiv.org/abs/0907.0273 , 2009. [57] T. Hartnick and T. Strubel. Cross ratios associated with maximal represen-tations. Preprint, http://lanl.arxiv.org/abs/0908.4101 , 2009.[58] L. Hern´andez Lamoneda. Maximal representations of surface groups inbounded symmetric domains. Trans. Amer. Math. Soc. , 324:405–420, 1991.[59] N. J. Hitchin. The self-duality equations on a Riemann surface.
Proc. LondonMath. Soc. (3) , 55(1):59–126, 1987.[60] N. J. Hitchin. Lie groups and Teichm¨uller space.
Topology , 31(3):449–473,1992.[61] A. Iozzi. Bounded cohomology, boundary maps, and representations intoHomeo + ( S ) and SU (1 , n ). In Rigidity in Dynamics and Geometry, Cam-bridge, UK, 2000 , pages 237–260. Springer Verlag, 2002.[62] I. Kim. Compactification of strictly convex real projective structures.
Geom.Dedicata , 113:185–195, 2005.[63] I. Kim and P. Pansu. Density of Zariski density for surface groups. Preprint2009.[64] H. Kneser. Die kleinste Bedeckungszahl innerhalb einer Klasse von Fl¨achen-abbildungen.
Math. Ann. , 103(1):347–358, 1930.[65] B. Kostant. The principal three-dimensional subgroup and the Betti numbersof a complex simple Lie group.
Amer. J. Math. , 81:973–1032, 1959.[66] B. Kostant. Lie group representations on polynomial rings.
Amer. J. Math. ,85:327–404, 1963.[67] B. Kostant. On Whittaker vectors and representation theory.
Invent. Math. ,48(2):101–184, 1978.[68] F. Labourie. Anosov flows, surface groups and curves in projective space.
Invent. Math. , 165(1):51–114, 2006.[69] F. Labourie. Cross ratios, surface groups, PSL( n, R ) and diffeomorphisms ofthe circle. Publ. Math. Inst. Hautes ´Etudes Sci. , 106:139–213, 2007.[70] F. Labourie. Flat projective structures on surfaces and cubic holomorphicdifferentials.
Pure Appl. Math. Q. , 3(4, part 1):1057–1099, 2007.[71] F. Labourie. Cross ratios, Anosov representations and the energy functionalon Teichm¨uller space.
Ann. Sci. ´Ec. Norm. Sup´er. (4) , 41(3):437–469, 2008.[72] F. Labourie and G. McShane. Cross ratios and identities for higher Te-ichm¨uller-Thurston theory.
Duke Math. J. , 149(2):279–345, 2009.[73] F. Ledrappier. Structure au bord des vari´et´es `a courbure n´egative. In
S´eminaire de Th´eorie Spectrale et G´eom´etrie, No. 13, Ann´ee 1994–1995 , vol-ume 13 of
S´emin. Th´eor. Spectr. G´eom. , pages 97–122. Univ. Grenoble I, Saint,1995.[74] J. C. Loftin. Affine spheres and convex RP n -manifolds. Amer. J. Math. ,123(2):255–274, 2001. [75] J. C. Loftin. The compactification of the moduli space of convex RP surfaces.I. J. Differential Geom. , 68(2):223–276, 2004.[76] J. C. Loftin. Flat metrics, cubic differentials and limits of projectiveholonomies.
Geom. Dedicata , 128:97–106, 2007.[77] W. L. Lok. Deformations of locally homogeneous spaces and Kleinian groups.Doctoral Thesis, Columbia University, 1984.[78] G. Lusztig. Total positivity in reductive groups. In
Lie theory and geometry ,volume 123 of
Progr. Math. , pages 531–568. Birkh¨auser Boston, Boston, MA,1994.[79] G. Lusztig. Total positivity and canonical bases. In
Algebraic groups and Liegroups , volume 9 of
Austral. Math. Soc. Lect. Ser. , pages 281–295. CambridgeUniv. Press, Cambridge, 1997.[80] G. A. Margulis.
Discrete subgroups of semisimple Lie groups , volume 17 of
Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematicsand Related Areas (3)] . Springer-Verlag, Berlin, 1991.[81] S. Matsumoto. Some remarks on foliated S bundles. Invent. Math. , 90:343–358, 1987.[82] J. Milnor. On the existence of a connection with curvature zero.
Com-ment. Math. Helv. , 32:215–223, 1958.[83] N. Monod.
Continuous bounded cohomology of locally compact groups . Number1758 in Lecture Notes in Math. Springer-Verlag, 2001.[84] J.-P. Otal. Le spectre marqu´e des longueurs des surfaces `a courbure n´egative.
Ann. of Math. (2) , 131(1):151–162, 1990.[85] A. Parreau. D´eg´en´erescences de sous-groupes discrets de groupes de liesemisimples et actions de groupes sur des immeubles affines. Ph.D. Thesis,Universit´e Paris Sud, 2000.[86] F. Paulin. D´eg´en´erescence de sous-groupes discrets des groupes de Lie semi-simples.
C. R. Acad. Sci. Paris S´er. I Math. , 324(11):1217–1220, 1997.[87] R. C. Penner. The decorated Teichm¨uller space of punctured surfaces.
Comm.Math. Phys. , 113(2):299–339, 1987.[88] G. P´olya and G. Szeg˝o.
Problems and theorems in analysis. I . Classics inMathematics. Springer-Verlag, Berlin, 1998. Series, integral calculus, theoryof functions, Translated from the German by Dorothee Aeppli, Reprint of the1978 English translation.[89] M. S. Raghunathan.
Discrete subgroups of Lie groups . Springer-Verlag, NewYork, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68.[90] P. Rolli. Quasi-morphisms on free groups. Preprint http://arxiv.org/abs/0911.4234 .[91] G. B. Simon, M. Burger, T. Hartnick, A. Iozzi, and A. Wienhard. Causalrepresentations. in preparation, 2009. [92] C. T. Simpson. Constructing variations of Hodge structure using Yang-Millstheory and applications to uniformization. J. Amer. Math. Soc. , 1(4):867–918,1988.[93] C. T. Simpson. Higgs bundles and local systems.
Publ. Math. Inst. Hautes´Etudes Sci. , 75:5–95, 1992.[94] W. Thurston.
Three-dimensional geometry and topology. Vol. 1 , volume 35of
Princeton Mathematical Series . Princeton University Press, Princeton, NJ,1997. Edited by Silvio Levy.[95] D. Toledo. Representations of surface groups in complex hyperbolic space.
J.Diff. Geom. , 29(1):125–133, 1989.[96] `E. Vinberg, editor.
Lie groups and Lie algebras, III , volume 41 of
Encyclopae-dia of Mathematical Sciences . Springer-Verlag, Berlin, 1994. Structure ofLie groups and Lie algebras, A translation of
Current problems in mathemat-ics. Fundamental directions. Vol. 41 (Russian), Akad. Nauk SSSR, Vsesoyuz.Inst. Nauchn. i Tekhn. Inform., Moscow, 1990 [MR 91b:22001], Translationby V. Minachin [V. V. Minakhin], Translation edited by A. L. Onishchik and`E. B. Vinberg.[97] A. Weil.
Introduction `a l’´etude des vari´et´es k¨ahl´eriennes . Publications del’Institut de Math´ematique de l’Universit´e de Nancago, VI. Actualit´es Sci.Ind. no. 1267. Hermann, Paris, 1958.[98] H. Whitney. Elementary structure of real algebraic varieties.
Ann. of Math.(2) , 66:545–556, 1957.[99] A. Wienhard. The action of the mapping class group on maximal representa-tions.
Geom. Dedicata , 120:179–191, 2006.[100] M. Wolf. The Teichm¨uller theory of harmonic maps.
J. Differential Geom. ,29(2):449–479, 1989.[101] J. W. Wood. Bundles with totally disconnected structure group.