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Proof: Let (z, x, y) be an element of T 3+ = ∂ ∞ π 1 (Σ) 3+ . The two transverse flags ξ(x) and ξ(y) define a decomposition R n = L 1 (x, y) ⊕ L 2 (x, y)
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- Mathematics; Number Theory; Geometry ; Analysis; Algebra; Mathematics, general
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- 0073-8301
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- 1618-1913
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- 10.1007/s10240-007-0009-5
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Abstract
CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE by FRANÇOIS LABOURIE ABSTRACT This article relates representations of surface groups to cross ratios. We first identify a connected component of the space of representations into PSL(n, R) –known as the n-Hitchin component – to a subset of the set of cross ratios 1,h h on the boundary at infinity of the group. Similarly, we study some representations into C (T) Diff (T) associated to cross ratios and exhibit a “character variety” of these representations. We show that this character variety contains all n-Hitchin components as well as the set of negatively curved metrics on the surface. 1Introduction Let Σ be a closed surface of genus at least 2. The boundary at infinity ∂ π (Σ) of the fundamental group π (Σ) is a one dimensional compact connected ∞ 1 1 Hölder manifold – hence Hölder homeomorphic to the circle T – equipped with an action of π (Σ) by Hölder homeomorphisms. A cross ratio on ∂ π (Σ) is a Hölder function B defined on ∞ 1 4∗ 4 ∂ π (Σ) = (x, y, z, t) ∈ ∂ π (Σ) x = t and y = z , ∞ 1 ∞ 1 invariant under the diagonal action of π (Σ) and which satisfies some algebraic rules. Roughly speaking, these rules encode some symmetry and normalisation properties as well as multiplicative cocycle identities in some of the variables: SYMMETRY: B(x, y, z, t) = B(z, t, x, y), NORMALISATION: B(x, y, z, t) = 0 ⇔ x = y or z = t, NORMALISATION: B(x, y, z, t) = 1 ⇔ x = z or y = t, COCYCLE IDENTITY: B(x, y, z, t) = B(x, y, z, w)B(x, w, z, t), COCYCLE IDENTITY: B(x, y, z, t) = B(x, y, w, t)B(w, y, z, t). The period of a non trivial element γ of π (Σ) with respect to B is the following real number − + log|B(γ ,γ y,γ , y)|:= (γ), The author thanks the Tata Institute for fundamental research for hospitality and the Institut Universitaire de France for support. DOI 10.1007/s10240-007-0009-5 140 FRANÇOIS LABOURIE + − where γ (respectively γ ) is the attracting (respectively repelling) fixed point of γ on ∂ π (Σ) and y is any element of ∂ π (Σ); as the notation suggests, the period ∞ 1 ∞ 1 is independent of y –see Paragraph 3.3. These definitions are closely related to those given by Otal in [27, 28] and discussed by Ledrappier in [25] from various viewpoints and by Bourdon in [4] in the context of CAT(−1)-spaces. Whenever Σ is equipped with a hyperbolic metric, ∂ π (Σ) is identified ∞ 1 with RP and inherits a cross ratio from this identification. Thus, every discrete faithful homomorphism from π (Σ) to PSL(2, R) gives rise to a cross ratio – called hyperbolic –on ∂ π (Σ). The period of an element for a hyperbolic cross ∞ 1 ratio is the length of the associated closed geodesic. These hyperbolic cross ratios satisfy the following further relation (1) 1 − B(x, y, z, t) = B(t, y, z, x). Conversely, every cross ratio satisfying Relation (1) is hyperbolic. The purpose of this paper is to • generalise the above construction to PSL(n, R) • give an n-asymptotic infinite dimensional version. A correspondence between cross ratios and Hitchin representations. — Throughout this paper, a representation from a group to another group is a class of homomorphisms up to conjugation .Wedenote by Rep(H, G) = Hom(H, G)/G, the space of representations from H to G, that is the space of homomorphisms from H to G identified up to conjugation by an element of G.When G is a Lie group, Rep(π (Σ), G) has been studied from many viewpoints; classical references are [2, 14, 15, 20, 29]. In [23], we define an n-Fuchsian homomorphism to be a homomorphism ρ from π (Σ) to PSL(n, R) which may be written as ρ = ι ◦ ρ ,where ρ is 1 0 0 a discrete faithful homomorphism from π (Σ) to PSL(2, R) and ι is the irreducible homomorphism from PSL(2, R) to PSL(n, R). In [21], Hitchin proves the following remarkable result: every connected component of the space of completely reducible representations from π (Σ) to PSL(n, R) which contains an n-Fuchsian representation is diffeomorphic to a ball. Such a connected component is called a Hitchin component and denoted by Rep (π (Σ), PSL(n, R)). We emphasise that this terminology is slightly non standard. CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 141 A representation which belongs to a Hitchin component is called an n-Hitchin rep- resentation.In other words, an n-Hitchin representation is a representation that may be deformed into an n-Fuchsian representation. In [23], we give a geometric description of Hitchin representations, which is completed by Guichard [18] (cf. Section 2). In particular, we show in [23] that if ρ is a Hitchin representation and γ a nontrivial element of π (Σ),then ρ(γ) is real split (Theorem 1.5 of [23]). This allows us to define the width of a non trivial element γ of π (Σ) with respect to a Hitchin representation ρ as λ (ρ(γ)) max w (γ) = log , λ (ρ(γ)) min where λ (ρ(γ)) and λ (ρ(γ)) are the eigenvalues of respectively maximum and max min minimum absolute values of the element ρ(γ). Generalising the situation for n = 2, briefly described in the previous para- graph, we identify n-Hitchin representations with certain types of cross ratios. More precisely, for every integer p,let ∂ π (Σ) be the set of pairs ∞ 1 ∗ (e, u) = ((e , e ,..., e ), (u , u , ..., u )), 0 1 p 0 1 p of ( p + 1)-tuples of points in ∂ π (Σ) such that e = e = u and u = u = e , ∞ 1 j i 0 j i 0 whenever j>i > 0. p p Let B be a cross ratio and let χ be the map from ∂ π (Σ) to R defined ∞ 1 ∗ by χ (e, u) = det ((B(e , u , e , u )). i j 0 0 i, j>0 A cross ratio B has rank n if n n • χ (e, u) = 0,for all (e, u) in ∂ π (Σ) , ∞ 1 B ∗ n+1 n+1 • χ (e, u) = 0,for all (e, u) in ∂ π (Σ) . ∞ 1 B ∗ Rank 2 cross ratios are precisely cross ratios satisfying Relation (1) (see Prop- osition 4.1). Our main result is the following. Theorem 1.1. — There exists a bijection from the set of n-Hitchin representations to the set of rank n cross ratios. This bijection φ is such that for any nontrivial element γ of π (Σ) (γ) = w (γ), B ρ where (γ) is the period of γ with respect to B = φ(ρ),and w (γ) is the width of γ B ρ with respect to ρ. 142 FRANÇOIS LABOURIE Cross ratios and representations are related through the limit curve in P(R ) – see Paragraph 2.3. In [22], we use the relation between cross ratios and Hitchin representations to study the energy functional. In collaboration with McShane in [24], we use these relations to generalise McShane’s identities [26]. In [22], cross ratios also appear in relation with maximal representations, as discussed in the work of Burger, Iozzi and Wienhard [8, 9] and Bradlow, García-Prada, Gothen, Mundet i Riera [6, 5, 7, 12, 17]. In [11], Fock and Goncharov give a combinatorial description of Hitchin representations. A “character variety” containing all Hitchin representations. —Since Hitchin repre- sentations are irreducible – cf. Lemma 10.1 in [23] – the natural embedding of PSL(n, R) in PSL(n + 1, R) does not give rise to an embedding of the corres- ponding Hitchin component. Therefore, there is no natural algebraic way – by an injective limit procedure say – to build a limit when n goes to infinity of Hitchin components. However, it follows from Theorem 1.1 that Hitchin components sit in the space of cross ratios. The second construction of this article refines this observation and shows that all Hitchin components lie in a “character variety” of π (Σ) into an infinite dimensional group H(T). 1,h The group H(T) is defined as follows. Let C (T) be the vector space of C -functions with Hölder derivatives on the circle T,and letDiff (T) be the group of C -diffeomorphisms with Hölder derivatives of T.Weobserve that Diff (T) acts 1,h naturally on C (T) and set 1,h h H(T) = C (T) Diff (T). In Paragraph 6.1.2, we give an interpretation of H(T) as a group of Hölder homeomorphisms of the 3-dimensional space J (T, R) of 1-jets of real valued func- tions on the circle. Our first result singles out a certain class of homomorphisms from π (Σ) to H(T) with interesting topological properties. Namely, we define in Paragraph 7.2, ∞-Hitchin homomorphisms from π (Σ) to H(T). As part of the definition, the quo- tient J (T)/ρ(π (Σ)) is compact for an ∞-Hitchin homomorphism ρ. In Para- graph 9.2.2, we associate a real number (γ) – called the ρ-length of γ –to every nontrivial element γ of π (Σ) and every ∞-Hitchin representation ρ.The ρ-length – considered as a map from π (Σ) \{id} to R –isthe spectrum of ρ. In Paragraph 9.2.1, we also associate to every ∞-Hitchin homomorphism a cross ratio and we relate the spectrum with the periods of this cross ratio. We denote by Hom the set of all ∞-Hitchin homomorphisms. Let Z(H(T)) be the centre of H(T) –isomorphic to R. Our first result describes the action on H(T) on Hom . H CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 143 Theorem 1.2. — The set Hom is open in Hom(π (Σ), H(T)).The group H 1 H(T)/Z(H(T)), acts properly on Hom and the quotient Rep = Hom(π (Σ), H(T))/H(T), is Hausdorff. Moreover, two ∞-Hitchin homomorphisms with the same spectrum and the same cross ratio are conjugated. Our next result exhibits an embedding of Hitchin components into this char- acter variety Rep . Theorem 1.3. — There exists a continuous injective map ψ from every Hitchin com- ponent into Rep such that if ρ is an n-Hitchin representation, then for any γ in π (Σ) (2) (γ) = w (γ), ψ(ρ) ρ where is the ψ(ρ)-length and w is the width with respect to ρ.Moreover, thecross ψ(ρ) ρ ratios associated to ρ and ψ(ρ) coincide. 1,h This suggests that the group C (T) Diff (T) contains an n-asymptotic version of PSL(n, R). This character variety contains yet another interesting space. Let M be the space of negatively curved metrics on the surface Σ, up to diffeomorphisms isotopic to the identity. For every negatively curved metric g and every non trivial element γ of π (Σ), wedenoteby (γ) the length of the closed geodesic freely homotopic 1 g to γ . Theorem 1.4. — There exists a continuous injective map ψ from M to Rep ,such that for any γ in π (Σ) (γ) = (γ), g ψ( g) where is the ψ( g)-length of γ . ψ( g) The injectivity in this theorem uses a result by Otal [27]. Theorems 1.3 and 1.4 are both consequences of Theorem 11.3, a general conjugation result. We finish this introduction with a question about our construction. Let H be the closure of the union of images of Hitchin components H = Rep (π (Σ), PSL(n, R)). ∞ 1 Does H contain the space of negatively curved metrics M ? How is it charac- terised? I thank the referee for a careful reading and very helpful comments con- cerning the structure of this article. 144 FRANÇOIS LABOURIE CONTENTS 1 Introduction .... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 139 2 Curves and hyperconvex representations . . ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 145 2.1 Hyperconvex and Hitchin representations . ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 145 2.2 Hyperconvex representations and Frenet curves . .. ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 146 2.2.1 A smooth map ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 147 3 Cross ratio, definitions and first properties ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 147 3.1 Cross ratios and weak cross ratios .. ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 147 4 Examples of cross ratio .... .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 149 4.1 Cross ratio on the projective line ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 149 4.2 Weak cross ratios and curves in projective spaces . . ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 151 4.3 Negatively curved metrics and comparisons ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 153 4.4 A symplectic construction ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 154 4.4.1 Polarised cross ratio . ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 155 4.4.2 Curves .. ... .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 155 4.4.3 Projective spaces . ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 156 4.4.4 Appendix: Periods, action difference and triple ratios .... ... ... ... ... ... ... ... ... ... ... .... 157 5 Hitchin representations and cross ratios .. ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 159 5.1 Rank-n weak cross ratios . ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 159 5.1.1 Nullity of χ . .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 160 5.2 Hyperconvex curves and rank n cross ratios ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 161 5.3 Proof of Theorem 5.3 ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 161 5.3.1 Cross ratio associated to curves .... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 161 5.3.2 Strict cross ratio and Frenet curves. ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 162 5.3.3 Curves associated to cross ratios . .. ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 163 5.4 Hyperconvex representations and cross ratios .. ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 165 5.5 Proof of the main Theorem 1.1 . ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 166 6Thejetspace J (T, R) . ... .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 166 6.1 Description of the jet space . . ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 166 6.1.1 Projections .. .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 167 1,h h 6.1.2 Action of C (T) Diff (T) .. ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 167 6.1.3 Foliation . ... .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 168 6.1.4 Canonical flow .. ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 168 6.1.5 Contact form .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 168 ∞ ∞ 6.2 A geometric characterisation of C (T) Diff (T) .. ... ... .... ... ... ... ... ... ... ... ... ... ... .... 169 6.2.1 A preliminary proposition .. ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 169 6.2.2 Proof of Proposition 6.1 .... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 170 ∞ ∞ 6.3 PSL(2, R) and C (T) Diff (T) . ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 171 6.3.1 Proof of Proposition 6.3 .... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 171 6.3.2 Alternate description of J .. ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 172 7 Anosov and ∞-Hitchin homomorphisms . ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 172 7.1 ∞-Fuchsian homomorphism and H-Fuchsian actions ... ... .... ... ... ... ... ... ... ... ... ... ... .... 172 7.2 Anosov homomorphisms . ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 173 8 Stability of ∞-Hitchin homomorphisms .. ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 174 8.1 Filtered spaces and holonomy . .. ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 175 8.1.1 Laminations and filtered space . ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 175 8.1.2 Holonomy theorem . ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 177 8.1.3 Completeness of affine structure along leaves . ... ... .... ... ... ... ... ... ... ... ... ... ... .... 179 8.2 A stability lemma for Anosov homomorphisms ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 180 8.2.1 Minimal action on the circle . .. ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 180 8.2.2 Proof of Lemma 8.10 . .. ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 181 9 Cross ratios and properness of the action . ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 182 9.1 Corollaries of the Stability lemma 8.10 . ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 182 9.2 Cross ratio, spectrum and Ghys deformations . ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 183 9.2.1 Cross ratio associated to ∞-Hitchin representations . .... ... ... ... ... ... ... ... ... ... ... .... 183 9.2.2 Spectrum ... .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 184 9.2.3 Coherent representations and Ghys deformations ... .... ... ... ... ... ... ... ... ... ... ... .... 185 9.3 Action of H(T) on Hom ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 186 h h 9.3.1 Characterisation of Ω (T) Diff (T) . ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 186 h h 9.3.2 First step: Conjugation in Ω (T) Diff (T) . ... ... .... ... ... ... ... ... ... ... ... ... ... .... 187 h h 9.3.3 Second step: From Ω (T) Diff (T) to H(T) ... ... .... ... ... ... ... ... ... ... ... ... ... .... 187 9.3.4 Proof of Theorem 9.10 . ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 189 CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 145 1,h h 10 A characterisation of C (T) Diff (T) .. ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 190 10.1 C -exact symplectomorphisms .. ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 190 10.2 π-symplectic homeomorphisms and π-curves .. ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 192 10.3 Width . ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 194 11 A conjugation theorem . ... .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 196 11.1 Contracting the leaves (bis) .. ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 198 11.2 A conjugation lemma .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 199 11.3 Proof of Theorem 11.3 .. ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 201 11.3.1 A preliminary lemma ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 201 11.3.2 Proof of Theorem 11.3 . ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 202 12 Negatively curved metrics . .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 204 12.1 The boundary at infinity and the geodesic flow ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 204 12.2 Proof of Theorem 12.1 .. ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 205 13 Hitchin component . ... ... .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 205 13.1 Hyperconvex curves . .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 206 13.2 Proof of Theorem 13.1 .. ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 206 13.2.1 Proof . ... ... .... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 207 13.2.2 Compact quotient . .. ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 208 13.2.3 Contracting the leaves .. ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... 211 2 Curves and hyperconvex representations We recall results and definitions from [23]. 2.1 Hyperconvex and Hitchin representations Definition 2.1 (Fuchsian and Hitchin homomorphisms). — An n-Fuchsian homomorphism from π (Σ) to PSL(n, R) is a homomorphism ρ which may be written as ρ = ι ◦ ρ ,where ρ 1 0 0 is a discrete faithful homomorphism with values in PSL(2, R) and ι is the irreducible homomorph- ism from PSL(2, R) to PSL(n, R). A homomorphism is Hitchin if it may be deformed into an n-Fuchsian homomorphism. Definition 2.2 (Hyperconvex map). — A continuous map ξ from a set S to P(R ) is hyper- convex if, for any pairwise distinct points (x ,..., x ) with p ≤ n, the following sum is direct 1 p ξ(x ) +··· + ξ(x ). 1 p In the applications, S will be a subset of T. Definition 2.3 (Hyperconvex representation and limit curve). — A homomorphism ρ from π (Σ) to PSL(n, R) is n-hyperconvex, if there exists a ρ-equivariant hyperconvex map from ∂ π (Σ) in P(R ). Such a map is actually unique and is called the limit curve of the homo- ∞ 1 morphism. Arepresentation is n-Fuchsian (respectively Hitchin, hyperconvex)if, as a class,it con- tains an n-Fuchsian (respectively Hitchin, hyperconvex) homomorphism. As an example, we observe that the Veronese embedding is a hyperconvex curve equivariant under all Fuchsian homomorphisms. Therefore, a Fuchsian representation is hyperconvex. More generally, we prove in [23] 146 FRANÇOIS LABOURIE Theorem. — Every Hitchin homomorphism is discrete, faithful and hyperconvex. If ρ is a Hitchin homomorphism and if γ in π (Σ) different from the identity, then ρ(γ) is real split with distinct eigenvalues. We explain later a refinement of this result (Theorem 2.6). Conversely, complet- ing our work, O. Guichard [18] has shown the following result Theorem 2.4 (Guichard). — Every hyperconvex representation is Hitchin. 2.2 Hyperconvex representations and Frenet curves Definition 2.5 (Frenet curve and osculating flag). — Ahyperconvex curve ξ defined from T n−1 1 2 n−1 to RP is a Frenet curve if there exists a family of maps (ξ ,ξ , ..., ξ ) defined on T, called the osculating flag curve, such that p n • for each p,the curve ξ takes values in the Grassmannian of p-planes of R , p p+1 • for every x in T, ξ (x) ⊂ ξ (x), • for every x in T, ξ(x) = ξ (x), • for every pairwise distinct points (x , ..., x ) in T and positive integers (n , ..., n ) such 1 l 1 l i=l that n ≤ n, then the following sum is direct i=1 n n i l (3) ξ (x ) + ··· + ξ (x ), i l i=l • finally, for every x in T and positive integers (n , ..., n ) such that p = n ≤ n,then 1 l i i=1 i=l n p (4) lim ξ ( y ) = ξ (x). ( y ,...,y )→x,y all distinct 1 l i i=1 n−1 We call ξ the osculating hyperplane. Remarks 1. By Condition (4), the osculating flag of a Frenet hyperconvex curve is contin- 1 p uous and the curve ξ completely determines ξ . 1 ∞ p 2. Furthermore, if ξ is C ,then ξ (x) is completely generated by the deriva- tives of ξ at x up to order p − 1. 3. In general a Frenet hyperconvex curve is not C . 4. However by Condition (4), its image is a C -submanifold and the tangent line 1 2 to ξ (x) is ξ (x). We list several properties of hyperconvex representations proved in [23]. Theorem 2.6. — Let ρ be an hyperconvex homomorphism from π (Σ) to PSL(E) with limit curve ξ.Then: CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 147 1. The limit curve ξ is a hyperconvex Frenet curve. 2. The osculating hyperplane curve ξ is hyperconvex. 3. The osculating flag curve of ξ is Hölder. + + 4.Finally, if γ is the attracting fixed point of γ in ∂ π (Σ),then ξ(γ ), (respectively ∞ 1 ∗ + ∗ ξ (γ )) is the unique attracting fixed point of ρ(γ) in P(E) (respectively P(E )). 2.2.1 A smooth map As a consequence of Theorem 2.6, we obtain the following Proposition 2.7. — Let ρ be a hyperconvex representation. Let 1 n−1 ξ = (ξ , ..., ξ ), be the limit curve of ρ.Then, there exist 1 2 n ∗n • A C embedding with Hölder derivatives η = (η ,η ) from T to P(R ) × P(R ), 1 2 • two representations ρ and ρ from π (Σ) in Diff (T),the group of C -diffeomorphisms 1 2 1 of T with Hölder derivatives, • a Hölder homeomorphism κ of T, such that 1 n−1 1. η (T) = ξ (∂ π (Σ)) and η (T) = ξ (∂ π (Σ)), 1 ∞ 1 2 ∞ 1 2.each map η is ρ equivariant, i i 3.the sum η (s) + η (t) is direct when κ(s) = t, 1 2 4. κ intertwines ρ and ρ . 1 2 1 n−1 1 Proof. — By the Frenet property, ξ (∂ π (Σ)) and ξ (∂ π (Σ)) are C one- ∞ 1 ∞ 1 dimensional manifolds. Let η (respectively η ) be the arc-length parametrisation of 1 2 1 n−1 ξ (∂ π (Σ)) (respectively ξ (∂ π (Σ))). Let ∞ 1 ∞ 1 −1 1 n−1 −1 κ = (η ) ◦ ξ ◦ (ξ ) ◦ η . 2 1 The result follows. 3 Cross ratio, definitions and first properties 3.1 Cross ratios and weak cross ratios Let S be a metric space equipped with an action of a group Γ by Hölder homeo- morphisms and 4∗ 4 S ={(x, y, z, t) ∈ S | x = t and y = z}. 148 FRANÇOIS LABOURIE Definition 3.1 (Cross ratio). — A cross ratio on S is a Hölder R-valued function B 4∗ on S , invariant under the diagonal action of Γ and which satisfies the following rules (5) B(x, y, z, t) = B(z, t, x, y), (6) B(x, y, z, t) = 0 ⇔ x = y or z = t, (7) B(x, y, z, t) = 1 ⇔ x = z or y = t, (8) B(x, y, z, t) = B(x, y, z, w)B(x, w, z, t), (9) B(x, y, z, t) = B(x, y, w, t)B(w, y, z, t). Definition 3.2 (Weak cross ratio). — If B – non necessarily Hölder – satisfies all relations except (7), we say that B is a weak cross ratio. If a weak cross ratio satisfies Relation (7),we say that the weak cross ratio is strict. A weak cross ratio that is strict and Hölder is a genuine cross ratio. In this article, we shall only consider the case where S = T,or S = ∂ π (Σ) ∞ 1 equipped with the action of π (Σ). In [24], we extend and specialise this definition to subsets of T. We give examples and constructions in Section 4. Definition 3.3 (Period). — Let B be a weak cross ratio on ∂ π (Σ) and γ be a nontriv- ∞ 1 + − ial element in π (Σ).The period (γ) is defined as follows. Let γ (respectively γ )be the 1 B attracting (respectively repelling) fixed point of γ on ∂ π (Σ).Let y be an element of ∂ π (Σ). ∞ 1 ∞ 1 Let − + (10) (γ, y) = log|B(γ ,γ y,γ , y)|. Relation (8) and the invariance under the action of γ imply that (γ) = (γ, y) does not depend B B −1 on y. Moreover, by Equation (5), (γ) = (γ ). B B Remarks 1. The definition given above does not coincide with the definition given by Otal in [28] and studied by Ledrappier, Bourdon and Hamenstädt in [25, 4, 19]. We in particular warn the reader that our cross ratios are not de- termined by their periods contrarily to Otal’s [28]. Apart from the choice of multiplicative cocycle identities rather than additive, we more crucially do not require that B(x, y, z, t) = B( y, x, t, z). This allows more flexibility as we shall see in Definition 9.9 and breaks the period rigidity. 2. However, if B(x, y, z, t) is a cross ratio according to our definition, so is ∗ ∗ B (x, y, z, t) = B( y, x, t, z), and finally log|BB | is a cross ratio according to Otal’s definition. We explain the relation with negatively curved metrics discovered by Otal in Section 4.3. CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 149 3. Ledrappier’s article [25] contains many viewpoints on the subject, in particu- lar the link with Bonahon’s geodesic currents [3] as well as an accurate bibli- ography. 4. Triple ratios. The remark made in this paragraph is not used in the article. Let B be a cross ratio. For every quadruple of pairwise distinct points (x, y, z, t), the expression B(x, y, z, t)B(z, x, y, t)B( y, z, x, t), is independent of the choice of t. We call such a function of (x, y, z) a triple ratio. Indeed, in some cases, it is related to the triple ratios introduced by A. Goncharov in [16]. A triple ratio satisfies the (multiplicative) cocycle iden- tity and hence defines a bounded cohomology class in H (π (Σ)). 4 Examples of cross ratio In this section, we recall the construction of the classical cross ratio on the pro- jective line and explain that various structures such as curves in projective spaces and negatively curved metrics on surfaces give rise to cross ratios. In some cases, the periods are computed explicitly. We also explain a general symplectic construction. 4.1 Cross ratio on the projective line Let E be a vector space with dim(E) = 2.The classical cross ratio on P(E), identified with R∪{∞} using projective coordinates, is defined by (x − y)(z − t) b(x, y, z, t) = . (x − t)(z − y) The classical cross ratio is a cross ratio according to Definition (3.1). It also satisfies the following rule (11) 1 − b(x, y, z, t) = b(t, y, z, x). This extra relation completely characterises the classical cross ratio. Moreover, this (simple) relation is equivalent to a more sophisticated one which we may generalise to higher dimensions Proposition 4.1. — For a cross ratio B, Relation (11) is equivalent to (12) (B( f , v, e, u) − 1)(B( g, w, e, u) − 1) = (B( f , w, e, u) − 1)(B( g, v, e, u) − 1). 150 FRANÇOIS LABOURIE Furthermore, if a cross ratio B on S satisfies Relation (12), then there exists an injective Hölder map ϕ of S in RP , such that B(x, y, z, t) = b(ϕ(x), ϕ( y), ϕ(z), ϕ(t)). Proof. — Suppose first that B satisfies Relation (11). By Relation (8), it also sat- isfies (12): (B( f , v, e, u) − 1)(B( g, w, e, u) − 1) = B(u, v, e, f )B(u, w, e, g) by (11) = (B(u, v, e, w)B(u, w, e, f ))(B(u, v, e, g)B(u, w, e, v)) by (8) = (B(u, v, e, w)B(u, w, e, v))(B(u, v, e, g)B(u, w, e, f )) = B(u, w, e, f )B(u, v, e, g) by (8) = (B( f , w, e, u) − 1)(B( g, v, e, u) − 1) by (11). Conversely, suppose that a cross ratio B satisfies Relation (12). We first prove that (13) B( f , v, k, z) (B(v, w, e, u) − B( f , w, e, u))(B(z, w, e, u) − B(k, w, e, u)) = . (B(v, w, e, u) − B(k, w, e, u))(B(z, w, e, u) − B( f , w, e, u)) Indeed, we first have (B( f , w, e, u) − 1)(B( g, v, e, u) − 1) B( f , v, e, u) = + 1. B( g, w, e, u) − 1 Setting g = v,weobtain 1 − B( f , w, e, u) (14) B( f , v, e, u) = + 1 B(v, w, e, u) − 1 B(v, w, e, u) − B( f , w, e, u) = . B(v, w, e, u) − 1 We have B( f , v, e, z) (15) B( f , v, k, z) = by (9) B(k, v, e, z) B( f , v, e, u) B(k, z, e, u) = by (8) B( f , z, e, u) B(k, v, e, u) B( f , v, e, u)B(k, z, e, u) = . B(k, v, e, u)B( f , z, e, u) CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 151 Applying Relation (14) to the four right terms of Equation (15), we obtain Equa- tion (13). Let (w, e, u) be a triple of pairwise distinct points in S,and let ϕ be the map from S to RP –identified with R∪{∞} –defined by S → R∪{∞}, ϕ = ϕ (w,e,u) x → B(x, w, e, u), where by convention B(w, w, e, u) =∞. We now observe that ϕ has the same regularity as B. This is obvious in S\{w}, and for x in the neighbourhood of w we use that B(x, u, e, w) = . B(x, w, e, u) The normalisation Relation (7) implies that ϕ is injective. Indeed B(x, w, e, u) = B( y, w, e, u) ⇒ B(x, w, y, u) = 1 ⇒ x = y. By construction and Equation (13), we have B(x, y, z, t) = b(ϕ(x), ϕ( y), ϕ(z), ϕ(t)). This in turn implies that B satisfies Relation (11) and completes the proof of the proposition. 4.2 Weak cross ratios and curves in projective spaces We now extend the previous discussion to higher dimensions. Let E be an n- ∗ ∗ dimensional vector space. Let ξ and ξ be two curves from a set S to P(E) and P(E ) respectively, such that (16) ξ( y) ∈ ker ξ (z) ⇔ z = y. We observe that a hyperconvex Frenet curve ξ and its osculating hyperplane ξ satisfy Condition (16). In the applications, S will be a subset of T. Definition 4.2 (Weak cross ratio associated to curves). — For every x in S, we choose ∗ ∗ ˆ ˆ a nonzero vector ξ(x) (respectively ξ (x)) in the line ξ(x) (respectively ξ (x)). The weak cross ∗ 4∗ ratio associated to (ξ, ξ ) is the function B on S defined by ξ,ξ ∗ ∗ ˆ ˆ ˆ ˆ ξ(x), ξ ( y)ξ(z), ξ (t) B ∗(x, y, z, t) = . ξ,ξ ∗ ∗ ˆ ˆ ˆ ˆ ξ(z), ξ ( y)ξ(x), ξ (t) 152 FRANÇOIS LABOURIE Remarks ˆ ˆ • The definition of B does not depend on the choice of ξ and ξ . ξ,ξ • B ∗ is a weak cross ratio. ξ,ξ • Let x, z be distinct points in S,and V = ξ(x) ⊕ ξ(z).For any m in S,let ζ(m) = ker ξ (m) ∩ V.Let b be the classical cross ratio on P(V),then B ∗(x, y, z, t) = b (ξ(x), ζ( y), ξ(z), ζ(t)). ξ,ξ V • As a consequence, B ∗ is strict if for all quadruple of pairwise distinct points ξ,ξ (x, y, z, t), ∗ ∗ ker(ξ (z)) ∩ ξ(x) ⊕ ξ( y) = ker(ξ (t)) ∩ ξ(x) ⊕ ξ( y) . Finally, we have ∗ ∗ n ∗n Lemma 4.3. — Let (ξ, ξ ) and (η, η ) be two pairs of maps from S to P(R )×P(R ), ∗ ∗ satisfying Condition (16), such that ξ and η are hyperconvex. Assume that B ∗ = B ∗. η,η ξ,ξ Then there exists a linear map A such that ξ = A ◦ η. Proof. — We assume the hypotheses of the lemma. Let (x , x ,..., x ) be a tuple 0 1 n of n + 1 pairwise distinct points of S.Let u be a nonzero vector in ξ (x ) and let 0 0 ∗ ∗ U = (u , ..., u ) be the basis of E such that u ∈ ξ (x ) and u , u= 1.The hyper- ˆ ˆ ˆ ˆ 1 n i i 0 i convexity of ξ guaranty the existence of U . The projective coordinates of ξ( y) in the dual basis of U are ξ( y), uˆ (17) [... :ξ( y), uˆ: ...]= ... : : ... ξ( y), u ξ( y), uˆ u , uˆ i 0 1 = ... : : ... ξ( y), uˆ u , uˆ 1 0 i =[... : B ( y, x , x , x ) : ...]. ξ,ξ i 0 1 Symmetrically, let v be a non zero vector in η (x ) and let V = (vˆ , ..., vˆ ) be the 0 0 1 n ∗ ∗ basis of E such that vˆ ∈ η (x ) and v , vˆ= 1. i i 0 i Let A be the linear map that sends the dual basis of V to the dual basis of U . By Equation (17), for all y ∈ S, wehavethat ξ( y) = A(η( y)). The result now follows. CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 153 4.3 Negatively curved metrics and comparisons We now explain Otal’s construction of cross ratios associated to negatively curved metrics on surfaces [27, 28]. Let g be a negatively curved metric on Σ which we lift to the universal cover Σ of Σ.Let (a , a , a , a ) be a quadruple of pairwise distinct 1 2 3 4 points of ∂ π (Σ) = ∂ Σ.Let c be the unique geodesic from a to a . We choose ∞ 1 ∞ ij i j nonintersecting horoballs H centred at each point a and set i i O = Σ \ (H ∪ H ). ij i j Let be the length of the geodesic arc c ∩ O . Otal’s cross ratio of the quadruple ij ij ij (a , a , a , a ) is 1 2 3 4 O (a , a , a , a ) = − + − . g 1 2 3 4 12 23 34 41 This number is independent on the choice of the horoballs H and isacrossratio according to Otal’s definition. Definition 4.4 (Cross ratio associated to negatively curved metric). — The cross ratio associated to g is defined as follows. For a cyclically ordered 4-tuple of distinct, we take the expo- nential of Otal’s cross ratio O (a ,a ,a ,a ) g 1 2 3 4 B (a , a , a , a ) = e . g 1 2 3 4 For noncyclically ordered 4-tuple, we introduce a sign compatible with the sign of the classical cross ratio. Remarks 1. The function B satisfies the rules 3.1 and the period of an element γ is the length of the closed geodesic of Σ in the free homotopy class of γ . 2. The cross ratio of a negatively curved metric satisfies the extra symmetry B(x, y, z, t) = B( y, x, t, z). 3. If the metric g is hyperbolic, ∂ π (Σ) is identified with RP and the cross ∞ 1 ratio B coincides with the classical cross ratio. 4. This construction is yet another instance of the “symplectic construction” ex- plained in Section 4.4. Indeed, the metric identifies the tangent space of the universal cover of a negatively curved manifold with the cotangent space. Therefore, this tangent space inherits a symplectic structure and its symplectic reduction with respect to the multiplicative R-action is the space of geodesics which thus admits a symplectic structure. Identified with the space of pairs of distinct points of the boundary at infinity, the space of geodesics also admits a product structure. The cross ratio defined according to Section 4.4 coin- cides with the cross ratio described above. 154 FRANÇOIS LABOURIE 5. More generally, Otal’s construction can be extended to Anosov flows on unit tangent bundle of surfaces of which geodesic flows of negatively curved met- rics are special cases [1]. For a cross associated to an Anosov flow, the period of an element γ is the length of the closed orbit in the free homotopy class of γ . 6. Conversely a cross ratio gives rise to a flow as it is explained in Section 3.4.1 of [22]. The flow identity is equivalent to the Cocycle Identity (8) in the 4∗ definition of cross ratios. More precisely, given a cross ratio B on ∂ π (Σ) ∞ 1 3+ and any real number t, we define in [22] a map ϕ from ∂ π (Σ) –see t ∞ 1 Definition (12.3) – to itself by ϕ (x , x , x ) = (x , x , x ), t − 0 + − t + where B(x , x , x , x ) = e . + 0 − t Equation (8) implies that t → ϕ is a one parameter group, that is ϕ = ϕ ◦ ϕ . t+s t s In [25], François Ledrappier gives an excellent overview of the various aspects of Otal’s cross ratios and in particular their enlightening interpretation as Bonahon’s geodesic currents [3]. 4.4 A symplectic construction All the examples of cross ratio that we have defined may be interpreted from the following “symplectic” construction. Let V and W be two manifolds of the same dimension and O be an open set of V×W equipped with an exact symplectic structure, or more generally an exact two-form ω. We assume furthermore that the two foliations coming from the product structure F = O ∩ (V×{w}), F = O ∩ ({v}× W), satisfy the following properties: 1. Leaves are connected. 2. The first cohomology groups of the leaves are reduced to zero. 3. ω restricted to the leaves is zero. + − 4. Squares are homotopic to zero, where squares are closed curves c = c ∪ c ∪ 1 1 + − ± c ∪ c with c along F . 2 2 i CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 155 Remark. —When ω is symplectic, every leaf F is Lagrangian by definition, and a standard observation shows that they carry a flat affine structure. If every leaf is simply connected and complete from the affine viewpoint, condition (4) above is satisfied: one may “straighten” the edges of the square, that is deforming them into geodesics of the affine structure, then use this straightening to define a homotopy. 4.4.1 Polarised cross ratio Let U ={(e, u, f , v) ∈ V × W × V × W | (e, u), ( f , u), (e, v), ( f , v) ∈ O}. Definition 4.5 (Polarised cross ratio). — The polarised cross ratio is the function defined B defined on U by 1 ∗ G ω B(e, u, f , v) = e . where G is a map from the square [0, 1] to O such that • the image of the vertexes (0, 0), (0, 1), (1, 1), (1, 0) are respectively (e, u), ( f , u), (e, v), ( f , v), + − • the image of every edge on the boundary of the square lies in a leaf of F or F . The definition of B does not depend on the choice of the specific map G.Let ψ be an diffeomorphism of O preserving ω and isotopic to the identity. Let α and ζ be two fixed points of ψ and c a curve joining α and ζ.Since ψ is isotopic to the identity, it follows that c ∪ ψ(c) bounds a disc D. 4.4.2 Curves Let φ = (ρ, ρ) ¯ be a representation from π (Σ) in the group of diffeomorphisms of O which preserve ω and are restrictions of elements of Diff(V)×Diff(W).Let (ξ, ξ ) be a φ-equivariant map of ∂ π (Σ) to O. ∞ 1 Then, we have the following immediate Proposition 4.6. — The function B ∗ defined by ξ,ξ ∗ ∗ B (x, y, z, t) = B(ξ(x)), ξ ( y), ξ(z), ξ (t)), ξ,ξ satisfies B ∗(x, y, z, t) = B(z, t, x, y), ξ,ξ B ∗(x, y, z, t) = B(x, y, z, w)B(x, w, z, t), ξ,ξ B ∗(x, y, z, t) = B(x, y, w, t)B(w, y, z, t). ξ,ξ 156 FRANÇOIS LABOURIE This function B may be undefined for x = y and z = t. By the above proposition, ξ,ξ it extends to a weak cross ratio provided that lim B ∗(x, y, z, t) = 0. ξ,ξ y→x 4.4.3 Projective spaces We concentrate on the case of projective spaces, although the construction can be extended to flag manifolds to produce a whole family of cross ratios associated to a hyperconvex curve. However in this case, we do not know how to characterise these cross ratios using functional relations, as we did in the case of curves in the projective space. As a specific example of the previous situation, we relate the polarised cross ratio to the weak cross ratio associated to curves. Let E be a vector space. Let 2∗ ∗ ⊥ P = P(E) × P(E )\{(D, P) | D ⊂ P }. 2∗ ⊥ ⊥ Using the identification of T P with Hom(D, P ) ⊕ Hom(P , D),let (D,P) Ω (( f , g), (h, j)) = tr( f ◦ j) − tr(h ◦ g). 2∗ Let L be bundle over P(n) ,whose fibre at (D, P) is L ={u ∈ D, f ∈ P | f , u= 1}/{+1,−1}. (D,P) −λ λ Note that L is a principal R-bundle whose R-action is given by λ(u, f ) = (e u, e f ), equipped with an action of PSL(n, R). The following proposition summarise properties of this construction Proposition 4.7. — The form Ω is symplectic and the polarised cross ratio associated to 2Ω is f , vˆ gˆ, uˆ (18) B(u, f , v, g) = , f , u g, u ˆ ˆ ˆ where, in general, h is a nonzero vector in h. Moreover, there exists a connection form β on L whose curvature is symplectic and equal to Ω , and such that, for any u non zero vector in a line D, the section ξ : P → (u, f ) such that u, f= 1, and f ∈ P, ∗ ⊥ is parallel for β above {D}× (P(E )\{P | D ⊂ P }). Finally, let A be an element of PSL(n, R).Let D (resp. D)bean A-eigenspace of dimension ⊥ 2∗ one for the eigenvalue λ (resp. µ). Then (D, D ) is a fixed point of A in P and he action of A on L ⊥ is the translation by (D,D ) log|λ/µ|. CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 157 0 ∗ Proof. — We consider the standard symplectic form Ω on E× E /{+1,−1}.We 0 0 observe that Ω = dβ where β (v, g) =u, g. A symplectic action of R is given by (u, f ) −λ λ λ. (u, f ) = (e u, e f ), with moment map µ((u, f )) = f , u. −1 Then L = µ (1). Therefore, we obtain that β = β | is a connection form for the 0 L R-action, whose curvature Ω is the symplectic form obtained by reduction of the Hamiltonian action of R. 2∗ We now compute Ω explicitly. Let (D, P) be an element of P .Let π be 2∗ ⊥ the projection onto P parallel to D.Weidentify T P with Hom(D, P ) ⊕ (D,P) ⊥ 2∗ Hom(P , D).Let ( f , gˆ) be an element of T P ,Let (u,α) ∈ D× P be an element (D,P) of L.Now, ( f (u), gˆ(α)) is an element of T L which projects to ( f , gˆ).Bydefinition (u,α) 2∗ of the symplectic reduction, if ( f , gˆ) and (h, l ) are elements of T P ,then (D,P) ˆ ˆ Ω (( f , gˆ), (h, l )) =l(α), f (u)− gˆ(α), h(u). Finally, let π be the projection onto P in the D direction. We consider the map ⊥ ⊥ Hom(P , D) → Hom(P , D) f → f = ( f ◦ π) In particular l(α), f (u)= tr(l ◦ f )α, u. The description of the parallel sections follows from the explicit formula for β .This description allows to compute the holonomy of this connection along squares. This proves Formula (18). The last point is obvious. 4.4.4 Appendix: Periods, action difference and triple ratios The results of this paragraph are not used in the sequel. We come back to the general setting of the beginning of the section and make further definitions and re- marks. Let ψ be a diffeomorphism preserving ω and homotopic to the identity. Let ζ and α be two fixed points of ψ, c a curve joining α and ζ and D adiskbounded by c and ψ(c). Definition 4.8 (Action difference). — The action difference is ∆ (α, ζ, c) = exp ω . D 158 FRANÇOIS LABOURIE We have Proposition 4.9. — The quantity ∆ := ∆ (α, ζ, c) just depends on the homotopy class of c. Proof. — This follows from the invariance of ω under ψ.The reader should observe the analogy with the action difference for Hamiltonian diffeomorphisms. In our context, we have a preferred class of curves joining two points. Let α = (a, b) and ζ = (a, b) be two points of O. Since squares are homotopic to zero, we define the homotopy class c of curves from (a, b) to (a¯, b) to be curves homotopic a,b,a¯,b + − + + + − to c ∪ c ∪ ¯ c ,where c is a curve along F going from (a, b) to ( y, b), c acurve − − + ¯ ¯ along F going from ( y, b) to ( y, b),and c acurve along F going from ( y, b) to (a¯, b).Byconvention weset ∆ (α, ζ) = ∆ (α, ζ, c ). γ γ ¯ a,b,a¯,b Periods and action difference. — Using the notations of the previous paragraph, we have Proposition 4.10. — Let γ be an element of π (Σ) then, + − 2 + ∗ − − ∗ + B (γ , y,γ ,γ y) = ∆ (ξ(γ ), ξ (γ )), (ξ(γ ), ξ (γ )) . ξ,ξ φ(γ) In particular, + ∗ − − ∗ + (γ) = log ∆ (ξ(γ ), ξ (γ )), (ξ(γ ), ξ (γ )) . B ∗ ρ(γ) ξ,ξ Proof. —Let f = ( g, g) be a diffeomorphism of O preserving ω, restriction of an element of Diff(V)× Diff(W).Let (a, b) and (a, b) be two fixed points of f .Let as + − + before c = c ∪ c ∪ c composition of + + • c acurve along F from (a, b) to ( y, b), − − • c acurve along F from ( y, b) to ( y, b), + + ¯ ¯ • and c acurve along F from ( y, b) to (a, b). Assume (a, b) and (a, b) be fixed points of f .Then c∪γ(c) is a square. Let D be a disk whose boundary is this square. By definition ∆ ((a, b), (a, b)) = ω. The proposition follows from the definition of the weak cross ratio associated to (ξ, ξ ), when we take f = (ρ(γ), ρ (γ)) and + ∗ − + ∗ − (a, b, a¯, b) = (ξ(γ ), ξ (γ ), ξ(γ ), ξ (γ )). CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 159 Symplectic interpretation of triple ratio. — We explain quickly a similar construction for triple ratio. We consider a sextuplet (e, u, f , v, g, w) in V × W × V × W × V × W. Let now φ be a map from the interior of the regular hexagon H in V × W such that + − the image of the edges lies in F or F , and the (ordered) image of the vertexes are (e, u), ( f , u), ( f , v), ( g, v), ( g, w), (e, w) (Figure 1). Then, the following quantity does FIG. 1. — Triple ratio not depend on the choice of φ: 1 ∗ φ ω T(e, u, f , v, g, w) = e . Finally using the same notations as above, we verify that ∗ ∗ ∗ t(x, y, z) = T(ξ(x), ξ (z), ξ( y), ξ (x), ξ(z), ξ ( y)), is the triple ratio as defined in Paragraph 3.1. 5 Hitchin representations and cross ratios We prove in this section Theorem 1.1. This result is as a consequence of The- orem 5.3 that generalises Proposition 4.1 in higher dimensions. 5.1 Rank-n weak cross ratios We extend a definition given in the introduction. Let S be the set of pairs (e, u) = ((e , e ,..., e ), (u , u , ..., u )), 0 1 p 0 1 p of p + 1-tuples of points of a set S such that e = e = u , u = u = e , j i 0 j i 0 160 FRANÇOIS LABOURIE p p whenever j>i > 0.Let B be a weak cross ratio on S and let χ be the map from S to R defined by χ (e, u) = det ((B(e , u , e , u )). i j 0 0 i, j>0 Definition 5.1 (Rank-n weak cross ratio). — A weak cross ratio B has rank n if n n •∀(e, u) ∈ S , χ (e, u) = 0, ∗ B n+1 n+1 •∀(e, u) ∈ S , χ (e, u) = 0. ∗ B When the context makes it obvious, we omit the subscript B. Remarks • We prove in Paragraph 5.1.1 that for all weak cross ratios the nullity of χ (e, u) for a given e and u does not depend on e and u . 0 0 • The function χ never vanishes for a strict weak cross ratio B. Indeed, we have χ ((e, e, f ), (u, u, v)) = 1 B( f , v, e, u) = B( f , v, e, u) − 1. The remark now follows from the previous one. • A cross ratio has rank 2 if if and only if it satisfies Equation (12), or the equivalent Equation (11). Indeed 11 1 χ ((e, e, f , g), (u, u, v, w)) = 1 B( f , v, e, u) B( g, v, e, u) 1 B( f , w, e, u) B( g, w, e, u) 10 0 = 1 B( f , v, e, u) − 1 B( g, v, e, u) − 1 1 B( f , w, e, u) − 1 B( g, w, e, u) − 1 = (B( f , v, e, u) − 1)(B( g, w, e, u) − 1) − (B( f , w, e, u) − 1)(B( g, v, e, u) − 1). Therefore by the previous remarks, a cross ratio has rank 2 if and only if it satisfies Equation (12). 5.1.1 Nullity of χ We first prove the following result of independent interest. Proposition 5.2. — For a weak cross ratio, the nullity of χ ((e , e , ..., e ), (u , u ,..., u )), 0 1 n 0 1 n is independent of the choice of e and u . 0 0 CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 161 Proof. —Let f and v be arbitrary points of T such that 0 0 f = v = e , u = e = f . 0 0 i j 0 0 By the Cocycle Identities (8) and (9), we have B(e , u , e , u )B(e , u , e , v ) = B(e , u , e , v ) i j 0 0 i 0 0 0 i j 0 0 = B(e , u , f , v )B( f , u , e , v ). i j 0 0 0 j 0 0 Therefore χ ((e , e , ..., e ), (u , u ,..., u )) 0 1 n 0 1 n B( f , u , e , v ) 0 j 0 0 = χ (( f , e , ..., e ), (v , u , ..., u )). 0 1 n 0 1 n B(e , v , e , u ) i 0 0 0 i, j The proposition immediately follows. 5.2 Hyperconvex curves and rank n cross ratios The mainresultinthissection is ∗ ∗ Theorem 5.3. — Let ξ and ξ be two hyperconvex curves from T to P(E) and P(E ). Assume that ξ( y) ∈ ker ξ (x) if and only if x = y. Then the associated weak cross ratio B ∗ ξ,ξ has rank n. ∗ ∗ Moreover, if ξ and ξ are Frenet and if ξ is the osculating hyperplane of ξ,then B ∗ strict. ξ,ξ Conversely, let B be a rank n cross ratio on T. Then, there exist two hyperconvex curves ξ ∗ ∗ and ξ with values in P(E) and P(E ) respectively, unique up to projective transformations, such that B = B ∗.Moreover ξ( y) ∈ ker ξ (x) if and only if x = y. ξ,ξ We prove this theorem in Paragraph 5.3. 5.3 Proof of Theorem 5.3 5.3.1 Cross ratio associated to curves ∗ ∗ Let ξ and ξ be two hyperconvex curves with values in P(E) and P(E ) such that x = y ⇔ ξ( y) ∈ ker ξ (x). Let e and u be two distinct points of T.Let E be a nonzero vector in ξ(e ).Let U 0 0 0 0 0 ∗ ∗ be a covector in ξ (u ) such that U , E = 1.Wenow lift thecurves ξ and ξ with 0 0 0 ∗ ∗ ∗ ˆ ˆ values in P(E) and P(E ) to continuous curves ξ and ξ from T\{e , u } to E and E , 0 0 such that ˆ ˆ ξ (v), E = 1 =U , ξ(e). 0 0 162 FRANÇOIS LABOURIE Then the associated cross ratio is ˆ ˆ B( f , v, e , u ) =ξ (v), ξ( f ). 0 0 n+1 n From this expression it follows that χ = 0 and that by hyperconvexity χ = 0. 5.3.2 Strict cross ratio and Frenet curves ∗ ∗ We now prove that B is strict if ξ and ξ are Frenet and if ξ is the osculating ξ,ξ hyperplane of ξ . ∗ 1 Since ξ(T) and ξ (T) are both C submanifolds, there exist homeomorphisms σ and σ of T so that η = (ξ ◦ σ, ξ ◦ σ ), is a C map. The following preliminary result is of independent interest Proposition 5.4. — The two-form ω = η Ω is symplectic. Proof. —The regularity of η implies that η Ω is continuous. We begin by an n n∗ ⊥ n observation. Let D be a line in R , P alinein R ,such that D ⊕ P = R .Let W be a two-plane containing D.Let W = T P(W) ⊂ T P(R ). D D Let V an n − 2-plane contained in P .Let ⊥ ∗n V = T P(W ) ⊂ T P(R ). P P By the definition of Ω in Section 4.4.3, if V ⊕ W = R , then Ω | = 0. ˆ ˆ V⊕W In thecaseof hyperconvexcurves, 1 n−1 2 n−2 1 n−1 T ξ (∂ π (Σ)) = ξ (x), T ξ (∂ π (Σ)) = ξ (x). ξ (x) ∞ 1 ξ (x) ∞ 1 2 n−2 n ∗ Since by hyperconvexity ξ (x) ⊕ ξ ( y) = R for x = y, we conclude that η Ω is symplectic. The following regularity result will be used in the sequel Proposition 5.5. — If moreover the osculating flags of ξ and ξ are Hölder, so is ω. CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 163 Proof. — If we furthermore assume that the osculating flags of ξ and ξ are Hölder, then we can choose the homeomorphisms σ to be Hölder. Indeed, σ and ∗ ∗ σ are the inverse of the arc parametrisation of the submanifolds ξ(T) and ξ (T). Therefore, σ and σ are Hölder. Since the tangent spaces to these submanifolds are Hölder, η is C with Hölder derivatives. It follows that ω is Hölder. As a corollary, we obtain ∗ ∗ Proposition 5.6. — Let ξ and ξ be Frenet curves such that ξ is the osculating hyperplane of ξ.Then B is strict. Proof. —Let (x, y, z, t) be a quadruple of pairwise distinct points. Let η˙ = η ◦ ∗ −1 2∗ (σ, σ ) .Let Q be the square in ∂ π (Σ) whose vertexes are ∞ 1 (x, y), (z, y), (x, t), (z, t). By Formula (18), we know that 2 η(Q ) |B (x, y, z, t)|= e . Since ω is symplectic, and Q has a nonempty interior, we have that ω = 0. η( ˙ Q ) Hence B (x, y, z, t) = 1. 5.3.3 Curves associated to cross ratios We prove in this paragraph Proposition 5.7. — Let B be a rank n cross ratio on T. Then there exists hyperconvex ∗ n ∗n curves (ξ, ξ ) with values in P(R ) × P(R ), unique up to projective equivalence, so that B = B ∗. ξ,ξ Proof. —Let B be a rank n cross ratio. Let us fix (e, u) = ((e , e ,..., e ), 0 1 n n∗ (u , u , ..., u )) in T .Weconsider 0 1 n T\{u }→ R , ξ : f → (B( f , u , e , u ), ..., B( f , u , e , u )). 1 0 0 n 0 0 Since (19) χ (e, u) = 0, n ∗ ∗ ˆ ˆ the set (ξ(e ), ..., ξ(e )) is a basis of R .Let (E ,..., E ) be its dual basis. We now 1 n 1 n consider the map ∗n T\{e}→ R , ξ : i=n v → B(v, e , e , u )E . i 0 0 i=1 i 164 FRANÇOIS LABOURIE We now prove that ˆ ˆ (20) B( f , v, e , u ) =ξ( f ), ξ (v). 0 0 We first observe that ˆ ˆ (21) ξ(e ), ξ (v)= B(e , v, e , u ). i i 0 0 In particular ˆ ˆ (22) ξ(e ), ξ (u )= B(e , u , e , u ), i j i j 0 0 and thus ∗ ∗ ˆ ˆ (ξ (u ), ..., ξ (u )), 1 n ∗n is the canonical basis of R . As a consequence ˆ ˆ (23) ξ( f ), ξ (u )= B( f , u , e , u ). j j 0 0 n ∗ ˆ ˆ Let M be the n × n-matrix whose coefficients are ξ(e ), ξ (u ).For anyany f anf v, i j n+1 let eˆ = (e ,..., e , f ) and uˆ = (u , ..., u , v).Let also M be the degenerate (n + 1) × 0 n 0 n ˆ ˆ (n + 1)-matrix whose coefficients are ξ(e ), ξ (u ) with the convention that e = f i j n+1 and u = v. By Equation (22), n+1 n n (24) det(M ) = χ (e, u) = 0. The equation n+1 χ (e, u) = 0, ˆ ˆ yields, after developing the determinant along the last line, B( f , v, e , v )χ (e, u) = F(..., B( f , u , e , u ), ..., B(e , v, e , u ), ...), 0 0 i 0 0 i 0 0 where the right hand term is polynomial in B( f , u , e , u ) and B(e , v , e , u ).The i 0 0 i i 0 0 n+1 same argument applied to the determinant of M yields ∗ n ∗ ∗ ˆ ˆ ˆ ˆ ˆ ˆ ξ( f ), ξ (v) det(M ) = F(...,ξ( f ), ξ (u ),...,ξ(e ), ξ (v),...). j i Therefore, we complete the proof of Equation (20) using Equations (21), (23) and (24). ∗ n ∗n ˆ ˆ Let now ξ and ξ be defined from S = T\{e , u } to P(R ) and P(R ) as the 0 0 projections of the map ξ and ξ . By the Normalisation Relation (6) and Equation (20) x = y ⇔ ξ( y) ∈ ker ξ (x). Equation (20) applied four times yields that B = B ∗ . Finally since χ never vanishes, ξ,ξ ξ is hyperconvex as well as ξ . By Lemma 4.3, this shows that the curves ξ and ξ are unique up to projec- tive transformations, and in particular do not depend on the choice of e and u.We CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 165 can therefore extend ξ and ξ to T by a simple gluing argument and the proposition follows. 5.4 Hyperconvex representations and cross ratios Using Theorem 2.6, the following proposition relates hyperconvex representa- tions and cross ratios. Proposition 5.8. — Let ρ be a a n-hyperconvex representation with limit curve ξ and os- culating hyperplane ξ .Then B = B is a cross ratio defined on ∂ π (Σ) and its periods ρ ξ,ξ ∞ 1 are λ (ρ(γ)) max (γ) = w (γ) := log , B ρ λ (ρ(γ)) min where λ (ρ(γ)) and λ (ρ(γ)) are the eigenvalues of the real split element ρ(γ) having maximal max min and minimal absolute values respectively. Proof. — By Theorem 5.3, B is rank n weak crossratio.Furthermore,since ξ and ξ are Hölder, B is Hölder. By Proposition 5.6, B is strict. It remains to compute ρ ρ the periods. By Theorem 2.6, if γ is the attracting fixed point of γ in ∂ π (Σ),then ∞ 1 + ∗ − ξ(γ ),(resp. ξ (γ )) is the unique attracting (respectively repelling) fixed point of ρ(γ) in P(E).In particular + + ˆ ˆ ρ(γ)ξ(γ ) = λ ξ(γ ), max − − ˆ ˆ ρ(γ)ξ(γ ) = λ ξ(γ ). min Therefore, − + −1 (γ) = log|B(γ , y,γ ,γ y)| − ∗ + ∗ −1 ˆ ˆ ˆ ˆ ξ(γ ), ξ ( y)ξ(γ ), ξ (γ y) = log − ∗ −1 + ∗ ˆ ˆ ˆ ˆ ξ(γ ), ξ (γ y)ξ(γ ), ξ ( y) − ∗ + ∗ ∗ ˆ ˆ ˆ ˆ ξ(γ ), ξ ( y)ξ(γ ), ρ(γ) ξ ( y) = log − ∗ ∗ + ∗ ˆ ˆ ˆ ˆ ξ(γ ), ρ(γ) ξ ( y)ξ(γ ), ξ ( y) − ∗ + ∗ ˆ ˆ ˆ ˆ ξ(γ ), ξ ( y)ρ(γ)ξ(γ ), ξ ( y) = log − ∗ + ∗ ˆ ˆ ˆ ˆ ρ(γ)ξ(γ ), ξ ( y)ξ(γ ), ξ ( y) max = log . min Remark. —Let ρ be the contragredient representation of ρ defined by ∗ −1 ∗ ρ (γ) = (ρ(γ )) . 166 FRANÇOIS LABOURIE ∗ ∗ ∗ Let B be a cross ratio. We define B by B (x, y, z, t) = B( y, x, t, z).Then (B ) = B . ρ ρ By the above proposition, the cross ratios B and B have the same periods, however ρ ρ ρ and ρ are not necessarily conjugated. 5.5 Proof of the main Theorem 1.1 Theorem 1.1 stated in the introduction is a consequence of Proposition 5.8 for one side of the equivalence, and of Theorem 5.3, Lemma 4.3 and Guichard’s The- orem 2.4 [18] for the other side of the equivalence. 6The jet space J (T, R) Let J = J (T, R) be the space of one-jets of real-valued functions on the circle T. The structure of this section is as follows: 1,h h • In Paragraph 6.1, we describe the action of the group C (T) Diff (T) on J as well as the geometry of this latter space. 1,h • In Paragraph 6.2, we characterise geometrically the action of C (T) Diff (T) on J. • In Paragraph 6.3, we describe a homomorphism from PSL(2, R) to C (T) Diff (T) whose image acts faithfully and transitively on J. 6.1 Description of the jet space We describe in this section the geometric features of J that will be useful in the sequel, namely: • A structure of a principal R-bundle with connection given by – a projection δ onto T T, –an R-action given by a flow {ϕ }, – a connection form β whichisa contactform. • A foliation of J by affine leaves F . • A projection π onto T. 1,h h • An action of C (T) Diff (T) on J. 1,h h In Lemma 10.1, we shall characterise the action of C (T) Diff (T) using the prin- cipal bundle structure and the foliation F . We finally give in Paragraph 6.3.2 two alternate descriptions of J. CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 167 6.1.1 Projections We denote by j ( f ) the one-jet of the function f at the point x of T.We define the projections J → T, π : j ( f ) → x, J → T T, δ : j ( f ) → d f . We observethat each fibreof π carries an affine structure. In order to help the readers remember the heavy notation, we remark that δ is the projection that takes in account the “derivative” part. 1,h h 6.1.2 Action of C (T) Diff (T) 1,h h The group H(T) = C (T) Diff (T) acts by Hölder homeomorphisms on J 1,h 1 in the following way. Let (φ, h) be an element of C (T) Diff (T),where φ a C - diffeomorphism with Hölder derivatives of T and h is a C -function on T with Hölder derivatives. Let F = F(h,φ) be the homeomorphism of J given by 1 1 −1 F(h,φ) : j ( f ) → j ((h + f ) ◦ φ ). x φ(x) The homeomorphism F = F(h,φ) has the following properties: • F preserves the fibres of π,that is: π ◦ F = φ ◦ π . −1 ∞ • For every x in T, F restricted to π (x) is affine – in particular C –and the −1 derivatives of F| vary continuously on J. π (x) 1,h h Finally, the map (h,φ) → F(hφ) defines an action of C (T)Diff (T) on J by Hölder homeomorphisms. Alternatively, if we choose a coordinate θ on T and consider the identification J → T × R × R, ∂f j ( f ) → (θ, r, f ) = θ, , f (θ) , ∂θ then F(h,φ) is given by −1 ∂φ ∂h (25) (η, r, f ) → φ(η), (η) (η) + r , f + h(η) . ∂θ ∂θ 168 FRANÇOIS LABOURIE 6.1.3 Foliation Let F be the 1-dimensional foliation of J given by the fibres of the projection J → T × R, j ( f ) → (x, f (x)). Each leaf of F is included in a fibre of π and is an affine line with respect to the affine structure on the fibres of π . We observe that the leaf through z is identified (as an affine line) with T T for x = π(z). 1,h h This affine structure is invariant by the action of C (T) Diff (T) described above. 6.1.4 Canonical flow We define the canonical flow of J to be the flow 1 1 ϕ j ( f ) = j ( f + t), x x where we identify the real number t with the constant function that takes value t.The 1,h canonical flow commutes with the action of C (T) Diff (T) on J.Noticealsothat J/Z(H(T)) = T T, and that this identification turns δ : J → T T into a principal R-bundle. 6.1.5 Contact form We finally recall that J admits a contact form β. If we choose a coordinate θ on T and consider the identification J → T × R × R, ∂f j ( f ) → (θ, r, f ) = θ, , f (θ) , ∂θ then β = df − rdθ. Remarks 1. Note that a Legendrian curve for β which is locally a graph above T is the graph of one-jet of a function. 2. Moreover, the canonical flow ϕ preserves the 1-form β. t CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 169 3. Observe that β is a connection form for the principal R-bundle defined by δ, and that its curvature form is the canonical symplectic form of T T. 4. Here is another description of β. Recall that the Liouville form λ on the cotan- gent bundle p : T M → M is given by λ (u) =q|T p(u). q q In coordinates for T T,wehave, λ = rdθ.If f is the function on J given by 1 ∗ f ( j ( g)) = g(x),then β = df − δ λ. ∞ ∞ 6.2 A geometric characterisation of C (T) Diff (T) The following proposition shows that F , β and ϕ characterise the action of ∞ h ∞ 1,h C (T) Diff (T) on J. Later, in Lemma 10.1, we characterise C (T) Diff (T). Proposition 6.1. — Let ψ be a C -diffeomorphism of J. Assume that 1. ψ commutes with the flow ϕ , 2. ψ preserves the 1-form β, 3. ψ preserves the foliation F . Then ψ belongs to C (T) Diff (T). 6.2.1 A preliminary proposition We first prove an elementary remark Proposition 6.2. — Let α be a connection one-form on the R-bundle δ : J → T T. Assume that the curvature of α is the canonical symplectic form on T T. Then there exists a diffeo- morphism ξ of J which • commutes with the action of the canonical flow, • preserves the fibres of π and δ (that is send fibres to fibres), • is above a symplectic diffeomorphism of T T, • satisfies ξ β = α. Proof. — We choose a coordinate θ of T.Thus, T T is identified with T × R with coordinates (θ, r),and J with T × R × R with coordinates (θ, r, f ).Since α is a connection form, there exist functions α and α such that r θ α = df + α (r,θ)dr + α (r,θ)dθ. r θ Let γ = α − β = α (r,θ)dr + (α (r,θ) + r)dθ. r θ 170 FRANÇOIS LABOURIE Since the curvature of α is dθ ∧ dr, γ is closed. Therefore, there exist a function h on T T and a constant λ such that γ = dh + λdθ. A straightforward check now shows that the diffeomorphism ξ of J given by ξ(θ, r, f ) = (θ, r − λ, f + h(θ, r)). satisfies the condition of our proposition. 6.2.2 Proof of Proposition 6.1 Proof. — We use the notations and assumptions of Proposition 6.1. By Assump- tions (1) and (3), ψ preserves the fibres of π . There exists thus a C -diffeomorphism φ of T T such that π ◦ ψ = φ ◦ π. −1 Replacing ψ by ψ ◦ (0,φ ), we may as well assume that φ = id . By Assumption (1), ψ also preserves the fibres of δ. We choose a coordinate on T and use the identification J = T × R × R given in Paragraph 6.1.5. The previous discussion shows that ψ(θ, r, f ) = (θ, F(θ, r), H(θ, r, f )). Since ψ β = β,we obtainthat dH − Fdθ = df − rdθ. Hence ∂H ∂H ∂H = 1, F − = r, = 0. ∂f ∂θ ∂r Therefore, there exists a C -function g such that H = f + g(θ). It follows that ∂g ψ(θ, r, f ) = θ, r + , f + g(θ) . ∂θ This exactly means that ψ(θ, r, f ) = ( g, id ). j ( f ). ∞ ∞ In other words, ψ belongs to C (T) Diff (T). CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 171 6.3 PSL(2, R) and C (T) Diff (T) We consider the 3-manifold J = PSL(2, R).Wedenote by g the Killing form, which we consider as a biinvariant – with respect to both left and right actions of PSL(2, R) – Lorentz metric on J. Then: • Let ϕ = ∆ be the one-parameter group of diagonal matrices acting on the right on J. • Let F the orbit foliation by the right action of the one-parameter group of strictly upper triangular matrices. • Let β = i g,where X is the vector field generating ϕ . Alternatively, describing PSL(2, R) as the unit tangent bundle of the hyperbolic plane, we can identify ϕ with the geodesic flow, F with the horospherical foliation and β with the Liouville form. Then Proposition 6.3. — The one-form β is a contact form. Furthermore, there exists a C - diffeomorphism Ψ from J to J, that sends (ϕ , F,β) to (ϕ , F , β) respectively. Finally, this dif- t t feomorphism Ψ is unique up to left composition by an element of C (T) Diff (T). As an immediate application, we have Definition 6.4 (Standard representation). — Since the left action of PSL(2, R) preserves the flow ϕ , the foliation F and the 1-form β, combining Propositions 6.1 and 6.3, we obtain a group homomorphism PSL(2, R) → C (T) Diff (T), ι : −1 g → Ψ ◦ g ◦ Ψ , ∞ ∞ well defined up to conjugation by an element of C (T) Diff (T). The corresponding represen- ∞ ∞ tation from PSL(2, R) to C (T) Diff (T) is called standard. Remark. — The action of PSL(2, R) on RP T gives rise to an embedding of PSL(2, R) in Diff (T). The above standard representation is a nontrivial extension ∞ ∗ of this representation. Indeed, the natural lift of the action of Diff (T) on T T does not act transitively since it preserves the zero section. On the contrary, PSL(2, R) does act transitively through the standard representation. In Paragraph 6.3.2, we consider an alternate description of J which make the action of PSL(2, R) more canonical. 6.3.1 Proof of Proposition 6.3 Proof. — Itisimmediate tocheck that PSL(2, R) → PSL(2, R)/∆ = W, 172 FRANÇOIS LABOURIE is a principal R-bundle, whose connexion form is β and whose curvature is symplectic. It follows that β is a contact form. Let π be the projection W → RP = PSL(2, R)/B, where B is the 2-dimensional group of upper triangular matrices. Let ψ be a symplectic diffeomorphism from W to T T over a diffeomorphism φ from RP to T, that is so that π ◦ ψ = φ π.Let Ψ : J → J be a principal 0 0 0 0 −1 R-bundle equivalence over ψ .Let α = (ψ ) β.Let ξ be the symplectic diffeomor- −1 phism of J obtained by Proposition 6.2 applied to α = (ψ ) β. It follows Ψ = ξ ◦ Ψ has all the properties required. Finally, by Proposition 6.1, Ψ is well defined up to multiplication by an element of C (T) Diff (T). Since the right action of PSL(2, R) preserves F , β,and ϕ , we obtain a repre- sentation of PSL(2, R) well defined up to conjugation as follows ∞ ∞ PSL(2, R) → C (T) Diff (T), −1 g → Ψ ◦ g ◦ Ψ . 6.3.2 Alternate description of J For the sake of completeness, we introduce two alternate descriptions of J.The second one – which we shall not use – has been suggested by the referee: • We have just seen that PSL(2, R) is an alternate description of J. • We also may consider the space E of half densities on T.Then J will be the 1/2 space of 1-jets of sections. The reader can work out the details and see that the natural action of PSL(2, R) which comes from its injection into Diff (T) coincides with the above action. 7 Anosov and ∞-Hitchin homomorphisms The purpose of this section is to define various types of homomorphisms from 1,h h π (Σ) to H(T) = C (T) Diff (T). 7.1 ∞-Fuchsian homomorphism and H-Fuchsian actions Definition 7.1 (Fuchsian homomorphism). — Let ρ be a Fuchsian homomorphism from π (Σ) to PSL(2, R). We say that the composition of ρ by the standard representation ι from 1 CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 173 PSL(2, R) to H(T) (cf. Definition 6.4) is ∞-Fuchsian,orinshort Fuchsian when there is no ambiguity. Definition 7.2 ( H-Fuchsian action on T). — We say an action of π (Σ) on T is H-Fuchsian if it is an action by C -diffeomorphisms with Hölder derivatives, which is Hölder conjugate to the action of a cocompact group of PSL(2, R) on RP . 7.2 Anosov homomorphisms We begin with a general definition. Let F be a foliation on a compact space, and d be a Riemannian distance along the leaves of F which comes from a leafwise continuous metric. In particular, d (x, y) < ∞ if and only if x and y are in the same leaf. Definition 7.3 (Contracting the leaves). — A flow ϕ contracts uniformly the leaves of a foliation F,if ϕ preserves F and if moreover ∀ >0, ∀α >0, ∃t : t ≥ t , d (x, y) ≤ α ⇒ d (ϕ (x), ϕ ( y)) ≤ . 0 0 F F t t This definition does not depend on the choice of d . We now use the notations of Section 6.1. Definition 7.4 (Anosov homomorphisms). — A homomorphism ρ from π (Σ) to H(T) is Anosov if: • the group ρ(π (Σ)) acts with a compact quotient on J = J (T, R), • the flow induced by the canonical flow ϕ on ρ(π (Σ)) \ J contracts uniformly the leaves t 1 of F (cf. definition above), • the induced action on T is H-Fuchsian. We denote the space of Anosov homomorphisms by Hom . Remarks 1. If ρ is Anosov, ρ(π (Σ)) usually only acts by homeomorphisms on J.How- ever, if we consider J as a C -filtered space (See the definitions in Para- graph 8.1.1), whose nested foliated structures are given by the fibres of the two projections δ and π,then ρ(π (Σ)) acts by C -filtered maps (i.e. smoothly along the leaves with continuous derivatives). Therefore, ρ(π (Σ))\ J has the ∞ ∞ structure of a C -lamination such that the flow induced by ϕ is C leaf- wise. 2. Our first examples of Anosov homomorphisms are ∞-Fuchsian homomor- phisms from π (Σ) in H(T). Indeed in this case, the canonical flow on 1 174 FRANÇOIS LABOURIE ρ(π (Σ)) \ J is conjugate to the geodesic flow for the corresponding hyper- bolic surface. Finally we define Definition 7.5 (∞-Hitchin homomorphisms). — An ∞-Hitchin homomorphism is a homomorphism from π (Σ) to H(T) which may be deformed into an ∞-Fuchsian homomorph- ism, through Anosov homomorphisms. In other words, the set Hom of Hitchin representations is the connected component of the set Hom of Anosov homomorphisms, containing the Fuchsian homomor- phisms. 8 Stability of ∞-Hitchin homomorphisms The set of homomorphisms from π (Σ) to a semi-simple Lie group G with Zariski dense images satisfies the following properties: • It is open in the set of all homomorphisms. • The group G/Z(G) acts properly on it, where Z(G) is the centre of G. We aim to prove that the set Hom of ∞-Hitchin homomorphisms from π (Σ) to H 1 H(T) enjoys thesameproperties. The canonical flow ϕ – considered as a subgroup of H(T) – is in the centre Z(H((T)) of H(T) and hence acts trivially on Hom . It follows that H(T)/Z(H(T)) acts on Hom . The mainresultof thissection is thefollowing: Theorem 8.1. — The set of Hitchin homomorphisms is open in the space of all homomorph- isms. We shall also see in the next section that the action by conjugation of H(T)/Z(H(T)) on Hom is proper and that the quotient is Hausdorff. In the proof of Theorem 8.1, we use the definitions and results obtained in Paragraph 8.1 which is independent of the rest of the article. It can be skipped at a first reading and is summarised as follows. Roughly speaking, a filtered space is equipped with a family of nested foliations which are transverse continuously, but posses a smooth structure along the leaves. A C -filtered function is continuous, smooth along the leaves and with continuous leafwise derivatives, and a C -filtered map between filtered spaces sends leaves to leaves and C -filtered functions to C -filtered functions. The theorem follows from the Stability lemma 8.10 which has its own interest and applications. CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 175 8.1 Filtered spaces and holonomy In this paragraph, logically independent from the rest of the article, we describe a notion of a topological space with “nested” laminations. This requires some defini- tions and notations. Let Z , ..., Z be a family of sets. Let 1 p Z = Z × ... × Z . 1 p For k< p,wenote p the projection Z → Z × ... × Z . 1 k th Let O be a subset of Z and x a point of O.The k -leave through x in O is (k) −1 O = O ∩ p { p (x)}. x k (k+1) (k) (1) We note that O ⊂ O .The higher dimensional leaf is O .If φ is a map from O x x x to a set V,we write (k) (k) φ = φ| . x O If V ⊂ W = W × ... × W ,we say φ is a filtered map if 1 k (k) (k) ∀k,φ O ⊂ V . x φ(x) 8.1.1 Laminations and filtered space In this section, we define a notion of filtered space for which it does make sense to say some maps are “smooth along leaves with derivatives varying continuously”. ∞ ∞ Definition 8.2 (C -filtered space). — A metric space P is C -filtered if the following conditions are satisfied: • There exists a covering of P by open sets U , called charts. i i i • There exist Hölder homeomorphisms ϕ , called coordinates,from U to V ×V ×...×V i i 1 2 p where for k> 1, V is an open set in a finite dimensional affine space. • Let −1 ϕ = ϕ ◦ (ϕ ) , ij i j −1 defined from W = ϕ (U ∩ U ) to W be the coordinate changes.Then ij i j ji – the coordinates changes are filtered map, (1) – the restriction (ϕ ) of the coordinate changes to the maximal leaf through x is a C - ij map whose derivatives depends continuously on x. 176 FRANÇOIS LABOURIE Remarks (k) 1. Due to the last assumption, for all k, (ϕ ) is a C -map whose derivatives ij depends continuously on x. 2. When p = 2,we speak of a laminated space. 3. We may want to specify the number of nested laminations, in which case we talk of p-filtered objects. 4. The Hölder hypothesis is somewhat irrelevant and could be replaced by other regularity assumptions, but it is needed in the applications. We now extend the definitions of the previous paragraph. Let P be a C -p-fil- tered space. Let k< p. th Definition 8.3 (Leaves). — The k -leaves of P are the equivalence classes of the equivalence relation generated by y R x ⇐⇒ p (ϕ ( y)) = p (ϕ (x)). k i k i Let P and P be C -p-filtered spaces. Let ϕ be coordinates on P and ϕ be coordinates on P. Definition 8.4 (Filtered maps and immersions). — Amap ψ from P to P is C -filtered if • ψ is Hölder, th th • ψ send k -leaves to k -leaves, (k) ∞ • (ϕ ◦ ψ ◦ ϕ ) are C and their derivatives vary continuously with x. j x A filtered immersion is a filtered map ψ whose leafwise tangent map is injective: in other −1 1 words, (ϕ ◦ ψ ◦ ϕ ) is an immersion. j x Observe that a filtered immersion is only an immersion along the leaves, there is no requirement about what happens transversely to the leaves. Definition 8.5 (Convergence of filtered maps). — Let {ψ } be a sequence of C -filtered n n∈N maps from P to P. The sequence C converges on every compact set if • it converges uniformly on every compact set, (k) • all the derivatives of (ϕ ◦ ψ ◦ ϕ ) converges uniformly on every compact set (as functions i n of x). Definition 8.6 (Affine leaves). — A P is C -filtered by affine leaves or carries a leafwise affine structure if furthermore ϕ | is an affine map. ij U x CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 177 8.1.2 Holonomy theorem We now prove the following result which generalises Ehresmann–Thurston holo- nomy theorem [10]. For P a p-filtered space, we denote by Fil(P) be the group of all bijections of P to itself which are all C -filtered leafwise immersions of P, equipped with the topology of C -convergence on compact sets. Theorem 8.7. — Let P be a p-filtered space. Let V be a connected p-filtered space. Let Γ ˜ ˜ be a discrete subgroup of Fil(V), such that V/Γ is a compact filtered space. Let U be the set of homomorphisms ρ from Γ to Fil(P) such that there exists a ρ-equivariant filtered immersion from V to P.Then U is open. Moreover, suppose ρ belongs to U.Let f be a ρ -equivariant filtered immersion from V 0 0 0 to P.If ρ is close enough to ρ , we may choose a ρ-equivariant filtered immersion f arbitrarily close to f on compact sets. We note again that we could have replaced the word “Hölder” by “continuous” from all the definitions and still have a valid theorem. Proof. —Let {U } be a finite covering of V/Γ such that: • The open sets U are charts on V/Γ. ˜ ˜ • The open sets U are trivialising open sets for the covering π : V → V/Γ. −1 i In other words, π (U ) is a disjoint union of open sets which are mapped homeomorphically to U by π . 1 1 1 ˜ ˜ ˜ We choose an open set U ⊂ V,suchthat π is a homeomorphism from U to U . We make the following temporary definitions. A loop is a sequence of indexes i , ..., i such that i = i = 1 and 1 l l 1 i i j j+1 U ∩ U =∅. A loop defines uniquely a sequence of open sets U such that i i j j • π is a homeomorphism from U to U , i i j j+1 ˜ ˜ • U ∩ U =∅. 1 i ˜ ˜ A loop i , ..., i is trivialising if U = U . In general, we associate to a loop the 1 l 1 i ˜ ˜ element γ of Γ such that γ(U ) = U .The group Γ is the group of loops (with the product structure given by concatenation) modulo trivialising loops. This is just a way to choose a presentation of Γ adapted to the charts U . ij A cocycle is a finite sequence of g ={ g } of elements of G such that for every trivialising loop i , ..., i we have 1 p i i i i 1 2 p−1 p g ... g = 1. 178 FRANÇOIS LABOURIE Every cocycle g defines uniquely a homomorphism ρ from Γ to G = Fil(P),and furthermore the map g → ρ is open. Let g be a cocycle. A g-equivariant map f is a finite collection { f } such that • f is a filtered map from U to P, ij i j • f = g f on U ∩ U . j i It is easy to check that there is a one to one correspondence between ρ -equivariant filtered maps and g-equivariant maps. The following fact follows from the existence of partitions of unity: Let W , W , W be three open sets in V/Γ, such that 0 1 2 W ⊂ W ⊂ W ⊂ W . 0 1 1 2 Let h be a filtered map defined on W . Then there exists such that for any positive 2 0 smaller , for any filtered map h defined on W and -close to h on W then there exists h , 0 1 1 1 0 2-close to h on W , which coincides with h on W . 2 1 0 Let us now begin the proof. Let g be a cocycle associated to a covering U = (U ,..., U ).Let f be a g-equivariant immersion. Let now g be a cocycle arbitrarily 1 m close to g. We proceed by induction to build a g-equivariant map f defined on a smaller covering V = (V , ..., V ) and close to f . Our induction hypothesis is the following 1 m i i i i i • Let V be a collection of opens sets: V = (V ,..., V ), with V ⊂ U and 1 i−1 k i i (V ,..., V , U ,..., U ) is a covering. i m 1 i−1 • Let i i i f = f , ..., f , 1 i−1 i i i be a g-equivariant map defined on V = (V , ..., V ). 1 i−1 • Assume that f is close to f on V ,for all l<i. Let i i W = V ∪ ... ∪ V . 1 i−1 By the definition of a g-equivariant map, there exists a map h defined on W ∩ U i i i il such that h restricted to V ∩ U is equal to g f .The map h is close to f .Let Z i i i i i l l be a strictly smaller subset of W such that (Z , U , ..., U ) is still a covering. We use i i i m our preliminary observation to build f close to f on U and coinciding with h on i i i W ∩ U . We finally define i+1 i CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 179 i+1 • V = V ∩ Z , l l i+1 • V = U , i+1 • f = f for l< i. l l This completes the induction. In theend, weobtain a g-equivariant map f defined on slightly smaller open subsets of U ,and close to f . Therefore, it follows f is an immersion. The construction above proves also the last part of the statement about conti- nuity. 8.1.3 Completeness of affine structure along leaves Definition 8.8 (Leafwise complete). — The space V, C -filtered space by affine leaves, is leafwise complete if the universal cover of every leaf is isomorphic, in the affine category, to the affine space. We prove the following Lemma 8.9. — Let V be a compact space C -filtered by affine leaves. Let E be the vector bundle over V whose fibre at x is the tangent space at x of the leaf L . Assume that there exists a one-parameter group {ϕ } of homeomorphisms of V such that: t t∈R • For every leaf L , ϕ preserves the leaf L and acts as a one parameter group of translation x t x on L , generated by the vector field X. • The induced action of ϕ on the vector bundle E/R.X is uniformly contracting. Then V is leafwise complete. Proof. —For every x in V,let O ={u ∈ E | x + u ∈ L }, x x x O =∪ O . x∈V x We observe that O is an open subset of E invariant by ϕ . By hypothesis, we have L ⊂ O. Let L be the line bundle R.X.Since ϕ is contracting on F = E/L and V is compact, L admits a φ = ϕ invariant supplementary F . Let us recall the classical proof of this 1 0 fact. We choose a supplementary F to L.Then φ F is the graph of an element ω in 1 ∗ 1 K = F ⊗ L.Wenow identify F with F using the projection. Since the action of ϕ is 1 t uniformly contracting on F, the action of ϕ is uniformly contracting on K = F ⊗ L, −t hence exponentially contracting by compactness. It follows that the following element 180 FRANÇOIS LABOURIE of K = F ⊗ L is well defined: p=−∞ p ∗ α = (φ ) ω. p=−1 This section α satisfies the cohomological equation φ α − α = ω. This last equation exactly means that the graph F of α is φ invariant. Let u ∈ E.Write u = v + λX,with v ∈ F .For n large enough, φ (u) belongs to O.Since O is invariant by φ, we deduce that u ∈ O.Hence O = E and V is leafwise complete. 8.2 A stability lemma for Anosov homomorphisms 1,h h Let ρ be a homomorphism from π (Σ) to H(T) = C (T) Diff (T).We denote by ρ the associated representation (by projection) to Diff (T). The openness of the space of Anosov homomorphisms Hom is an immediate consequence of the following stability lemma. Moreover, corollaries of this lemma will enable us to associate to every ∞-Hitchin representation a cross ratio and a spectrum – see Paragraph 9.2.1 and 9.2.2. Lemma 8.10 (Stability lemma). — Let ρ be an Anosov homomorphism from π (Σ) to 0 1 H(T).Thenfor ρ close enough to ρ , there exists a Hölder homeomorphism Φ of J close to the identity which is a C -filtered immersion as well as its inverse intertwining ρ and ρ, that is −1 ∀γ ∈ π (Σ), ρ (γ) = Φ ◦ ρ(γ) ◦ Φ. 1 0 This lemma is proved in Paragraph 8.2.2. 8.2.1 Minimal action on the circle The following lemma is independent of the rest of the article. Lemma 8.11. — Let ρ and ρ be two homomorphisms from a group Γ to the group of 0 1 homeomorphisms of T. Suppose that every element of ρ (Γ) different from the identity has exactly two fixed points in T, one attractive and one repulsive. Suppose moreover that the fixed points of the nontrivial elements of ρ (Γ) are dense in T × T. Let f be a continuous map of nonzero degree from T to T intertwining ρ and ρ .Then f 0 1 is a homeomorphism. CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 181 Proof. —We first prove that f is injective. Let a and b be two distinct points of T such that f (a) = f (b) = c.Let I and J be the connected components of T\{a, b}. Since f has a nonzero degree, either T \{c}⊂ f (I) or T\{c}⊂ f ( J). Assume T\{c}⊂ f (I). By the density of fixed points in T × T, we can find an element γ in Γ such that the attractive fixed point γ of ρ (γ) belongs to I and the repulsive point γ belongs to J. We observe that for all n, n n T \ ρ (γ )(c) ⊂ f ρ (γ) (I) . 1 0 Therefore, there exist a positive and two sequences, {x } and{ y } ,suchthat n n∈N n n∈N n n • x belongs to ρ (γ) (I), y belongs to ρ (γ) (I), n 0 n 0 • d( f (x ), f ( y )) > . n n − n + Since γ belongs to J, the sequence {ρ (γ) (I)} converges uniformly to γ .Hence, lim d(x , y ) = 0. n n n→∞ The contradiction follows. We have proved that f is injective. Since f has nonzero degree, it is onto. Hence f is a homeomorphism. 8.2.2 Proof of Lemma 8.10 Proof. —Let ρ be an ∞-Hitchin homomorphism from π (Σ) to H(T).By 0 1 definition, P = ρ (π (Σ)) \ J is compact. The topological space J is a C -manifold 0 1 ∞ ∗ which is C -filtered by the fibres of the projection π : J → T and δ : J → T T. Observe that H(T) acts by C -filtered maps on J. It follows that the space P is also 0 ∞ C -manifold which is C -filtered by the images of the leaves of J. Let now ρ be a homomorphism of π (Σ) close enough to ρ .Let Γ= ρ (π (Σ)). 1 0 0 1 According to Theorem 8.7, we obtain a C -filtered immersion Φ from J to J,inter- twining ρ and ρ , and arbitrarily close to the identity on compact sets provided ρ is close enough to ρ . It remains to show that Φ is a homeomorphism. As a first step, we prove that ρ is H-fuchsian. Since Φ is a filtered map, there exists a Hölder map f from T to T close to the identity such that π ◦ Φ = f ◦ π. In particular, f intertwines ρ and ρ.Since ρ is Anosov, by definition ρ is an H- 0 0 Fuchsian representation, and thus satisfies the hypothesis of Lemma 8.11. The degree of f is nonzero since f is close to the identity. By Lemma 8.11, we obtain that f is a homeomorphism. Thus, ρ is also H-Fuchsian as we claimed. 182 FRANÇOIS LABOURIE The filtered equivariant immersion Φ induces an affine structure on the leaves of P, according to the definition of Paragraph 8.1.1. Due to the compactness of P,we deduce that there exists an action of a one-parameter group ψ on J such that, Φ ◦ ψ = ϕ ◦ Φ, t t where ϕ is the canonical flow. We observe that ψ preserves any given leaf and acts t t as a one-parameter group of translation on it. We recall that by definition the canonical flow contracts uniformly the leaves of F on the quotient ρ (π (Σ)) \ J. It follows that the same holds for ψ ,for Φ close 0 1 t enough to the identity. By Lemma (8.9), the affine structure induced by φ is leafwise complete. In par- −1 −1 ticular, for every x in T, Φ is an affine bijection from π {x} to π { f (x)}.Since f is a homeomorphism, we deduce that Φ itself is a homeomorphism, its inverse being also a filtered immersion. The result now follows. 9 Cross ratios and properness of the action We prove in this section Theorem 9.1. — The action by conjugation of H(T)/Z(H(T)) on Hom is proper and the quotient is Hausdorff. This is a consequence of the more precise Theorem 9.10. In order to state and provethislasttheorem, we associate toevery ∞-Hitchin representation a cross ratio. This is done in the first two paragraphs of this section. 9.1 Corollaries of the Stability lemma 8.10 Since the space Hom is connected, we have Corollary 9.2. — Let ρ and ρ be two ∞-Hitchin homomorphisms. Then, there exists 0 1 a Hölder homeomorphism Φ of J which is a filtered immersion as well as its inverse which inter- −1 twines ρ and ρ , that is, for all γ in π (Σ)ρ (γ) = Φ ◦ ρ (γ) ◦ Φ. 0 1 1 0 1 2∗ 2 ∗ Let ∂ π (Σ) ={(x, y) ∈ ∂ π (Σ) | x = y}.Recall that H(T) acts on T T = ∞ 1 ∞ 1 J/Z(H(T)).We also prove Proposition 9.3. — Let ρ be an ∞-Hitchin homomorphism. Then, there exists a unique Hölder homeomorphism ∗ 2∗ Θ : T T → ∂ π (Σ) , ρ ∞ 1 CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 183 that intertwines • the action of ρ(π (Σ)) and the action of π (Σ), 1 1 ∗ 2∗ • the projection from T T onto T and the projection on the first factor from ∂ π (Σ) onto ∞ 1 ∂ π (Σ). ∞ 1 In other words: • For all γ in π (Σ), we have γ ◦ Θ = Θ ◦ ρ(γ). 1 ρ ρ • There exists a Hölder map f from T to ∂ π (Σ) such that { f (x)}× ∂ π (Σ) contains ∞ 1 ∞ 1 Θ (T T). Proof. — By Corollary 9.2, it suffices to show this result for the action of a Fuch- sian representation. In this case the statement is obvious. 9.2 Cross ratio, spectrum and Ghys deformations In this section, we associate a cross ratio to every ∞-Hitchin representation. We also define a natural class of deformations of these representations. Similar deforma- tions were introduced by E. Ghys in the context of the geodesic flow of hyperbolic surfaces in [13]. 9.2.1 Cross ratio associated to ∞-Hitchin representations Let (a, b, c) be a triple of distinct elements of ∂ π (Σ).Wedefine [b, c] to be ∞ 1 a the closure of the connected component of ∂ π (Σ) \{b, c} not containing a.Let ∞ 1 q = (x, y, z, t) be a quadruple of elements of ∂ π (Σ).We define qˆ to be following ∞ 1 2∗ closed curve, embedded in ∂ π (Σ) (Figure 2) ∞ 1 q = ({x}×[ y, t] ) ∪ ([x, z] ×{t}) ∪ ({z}×[ y, t] ) ∪ ([x, z] ×{ y}). x t z y FIG. 2. — The curve q ˆ 184 FRANÇOIS LABOURIE We choose the orientation on qˆ such that (x, y), (x, t), (z, t), (z, y) are cyclically or- dered. We finally define ε ∈{−1, 1} so that ε =−1, if (x, y, z, t) are cyclically ordered, ε = ε(σ)ε (σ(x),σ( y),σ(z),σ(t)) (x,y,z,t). Let ρ be an ∞-Hitchin representation. Let Θ be the homeomorphism from ∗ 2∗ T T to ∂ π (Σ) obtained in Proposition 9.3. Let λ = rdθ be the Liouville form on ∞ 1 T T (see Paragraph 6.1.5.(4) for a definition). Definition 9.4 (Associated cross ratio). — The associated cross ratio to the Hitchin rep- resentation ρ is B (x, y, z, t) = ε . exp λ . ρ q −1 Θ (qˆ) Remarks −1 1. The cross ratio is well defined and finite. Indeed Θ (q) is the union of four arcs in T T. Two of these arcs project injectively on T, the other two arcs project to a constant. In both cases, the integral of rdθ = λ is well defined and finite on such arcs. 2. The cross ratio just depends on the action of π (Σ) on T T. Here is another formu- lation of this observation. Let Ω (T) be the space of Hölder one-forms on T. h ∗ Note that Ω (T) Diff (T) naturally acts on T T.We also havea natural homomorphism 1,h h h h C (T) Diff (T) → Ω (T) Diff (T), d : ( f ,φ) → (df ,φ), whose kernel is the canonical flow. Therefore, two homomorphisms ρ and ρ such that d ◦ ρ = d ◦ ρ have the same associated cross ratio. In other 2 1 2 words, the cross ratio only depends on the representation as with values in h h Ω (T) Diff (T). 9.2.2 Spectrum Definition 9.5 (ρ-length). — Let ρ be an ∞-Hitchin representation. Let γ be an element in π (Σ).The ρ-length of γ – denoted by (γ) – is the positive number t such that 1 ρ ∃u ∈ J,ϕ (u) = ρ(γ)u. t CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 185 Remarks 1. In other words, (γ) is the length of the periodic orbit of ϕ in ρ(π (Σ))\ J ρ t 1 freely homotopic to γ.It is clear that (γ) just depends on the conjugacy class of γ . 2. The existence and uniqueness of such a positive number t –or of the previ- ously discussed closed orbit – follow from Corollary 9.2 and the description of standard representations. Definition 9.6 (Spectrum). — The marked spectrum of an ∞-Hitchin representation ρ is the map : γ → (γ). ρ ρ Definition 9.7 (Symmetric representation). — A representation is symmetric if (γ) = −1 (γ ). 9.2.3 Coherent representations and Ghys deformations Definition 9.8 (Coherent representation). — An ∞-Hitchin representation ρ is coherent if its marked spectrum coincides with the periods of the associated cross ratio. Remarks 1. Every coherent representation is symmetric. 2. Conversely, we prove in Proposition 10.12 that if is the period of the cross ratio associated to the representation ρ,then −1 (γ) = (γ) + (γ ) . B ρ ρ Consequently, every symmetric representation is coherent. 3. Theorems 13.1 and 12.1 provide numerous examples of coherent represen- tations. Definition 9.9 (Ghys deformation). — Let ρ be a Hitchin representation. Let ω be a non- 1 ω trivial element in H (Γ, R). We identify the centre of H(T) with the canonical flow ϕ .Let ρ be defined by ρ (γ) = ϕ .ρ(γ). ω(γ) ω ω We say that ρ is a Ghys deformation of ρ if the representation ρ is Hitchin. 186 FRANÇOIS LABOURIE Remarks 1. For ω small enough, ρ is a Ghys deformation. Indeed, for ω small enough, ρ is Hitchin by the Theorem 8.1. 2. We observe that ρ and ρ have the same associated cross ratio since they have the same action on T T. However, one immediately checks that (26) (γ) = (γ) + ω(γ). ρ ρ 3. The previous remarks provide many examples of representations with the same cross ratio but different marked spectra. 9.3 Action of H(T) on Hom We prove in this section that H(T)/Z(H(T)) acts properly on Hom .This is a consequence of the following result. Theorem 9.10. — The group H(T)/Z(H(T)) acts properly with a Hausdorff quotient on the set of ∞-Hitchin homomorphisms. Moreover, if two ∞-Hitchin homomorphisms ρ and ρ have the same associated cross ratio, 0 1 1 ω then there exists ω ∈ H (π (Σ)) such that ρ and ρ are conjugated by some element in H(T). 1 0 Consequently two representations with the same cross ratio and the same spectrum are equal. Remark. — Two representations with the same spectrum are not necessarily iden- tical (see remark after Theorem 13.1). h h 9.3.1 Characterisation of Ω (T) Diff (T) h h Let Ω (T) be the space of Hölder one-form on T. We observe first that Ω (T) Diff (T) acts naturally on T T and preserves the area. Conversely Lemma 9.11. — Let G be an area preserving Hölder homeomorphism of T T. Assume G is above a homeomorphism f of T.Then G belongs to Ω (T) Diff (T). Proof. — This lemma is classical for all cotangent spaces. We give the proof for completeness. We use the coordinates (r,θ) on T T.By hypothesis, G(r,θ) = ( g(r,θ), f (θ)). Since G preserves the area, we obtain f (θ ) g(r ,θ) 1 1 (27) (r − r )(θ − θ ) = dr dθ 0 1 0 1 f (θ ) g(r ,θ) 0 0 f (θ ) = ( g(r ,θ) − g(r ,θ))dθ. 0 1 f (θ ) 0 CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 187 Hence g(r,θ) is affine in r: g(r,θ) = ω(θ) + rβ(θ), where ω(θ) and β are Hölder. Since G is a homeomorphism, we observe that for all θ , β(θ) = 0 and f is an homeomorphism. By Equation (27), we obtain that −1 −1 f (θ ) − f (θ) = β(θ)dθ. −1 1,h −1 It follows that f is in C (T) with df = β.Since β never vanishes, f is actually h h is a diffeomorphism, and G belongs to Ω (T) Diff (T). 9.3.2 First step: Conjugation in Ω (T) Diff (T) We recall that H(T) acts on T T.We denoteby α → α˙ the projection from H(T) to Ω (T) Diff (T).Wefirstprove Proposition 9.12. — Let ρ and ρ be two ∞-Hitchin representations with the same as- 0 1 ρ ,ρ h 0 1 sociated cross ratio. Then there exists a unique element H = H in Ω (T) Diff (T) which intertwines ρ and ρ , that is ˙ ˙ 0 1 ∀γ ∈ π (Σ),∀y ∈ T T, H(ρ˙ (γ) · y) = ρ˙ (γ) · H( y). 1 0 1 Moreover, for any representation ρ, for any neighbourhood V of the identity in Ω (T) Diff (T), there exists a neighbourhood U of ρ, such that if two representations ρ and ρ in U 1 0 ρ ,ρ 0 1 have the same cross ratio, then H belongs to V. Proof. — Applying Proposition 9.3 twice, we obtain a unique Hölder homeo- morphism H of T T and a homeomorphism f of T such that H is above f and in- tertwines ρ˙ and ρ˙ . 0 1 If ρ and ρ have the same associated cross ratio, then H preserves the area. By 0 1 Lemma 9.11, H belongs to Ω (T) Diff (T). The first part of the proposition now follows. The second part follows from Lemma 8.10. h h 9.3.3 Second step: From Ω (T) Diff (T) to H(T) We prove the following lemma Lemma 9.13. — Let φ : γ → φ be a representation from π (Σ) to Diff (T) with γ 1 nonzero Euler class. Let α be a continuous one-form. Let f : γ → f be a map from π (Σ) to the γ 1 188 FRANÇOIS LABOURIE 1 1 space C (T) of C -functions on T. Assume that φ ( f ) + f = f , η γ ηγ φ (α) − α = df . Then α is exact. Proof. —Let κ := α. We aim to prove that κ = 0.Wewrite T = R/Z.We choose a lift all our data to R,and use u˜ to describe a lift of u. In particular, there exists an element c in H (π (Σ), Z) such that ˜ ˜ ˜ ∀x ∈ R, c(γ, η) = φ (x) − φ ◦ φ (x) ∈ Z. γη γ η By definition, c is a representative of the Euler class of the representation φ.Let h ∈ C (R) such that α = dh. By definition ∀m ∈ Z, h(x + m) = h(x) + mκ. Observe now that for all γ ∗ ∗ ˜ ˜ ˜ ˜ d(φ (h) − h − f ) = φ α˜ − α˜ − d f = 0. γ γ γ γ Therefore, there exists η : π (Σ) → R, such that ˜ ˜ φ (h) − h = f + η(γ). γ γ Finally c(γ, η)κ ˜ ˜ = h ◦ φ − h ◦ φ ◦ φ γη γ η ∗ ∗ ∗ ˜ ˜ ˜ = φ (h) − φ φ (h) γη η γ ∗ ∗ ∗ ∗ ∗ ˜ ˜ ˜ ˜ ˜ = (φ (h) − h) − (φ φ (h) − φ (h)) − (φ (h) − h) γη η γ η η ˜ ˜ ˜ ˜ = f − φ f − f + η(γη) − η(γ) − η(η) γη η γ η = η(γη) − η(γ) − η(η). Hence κc is the coboundary of η. Since the Euler class is nonzero in cohomology by hypothesis, we obtain that κ = 0,hence α is exact. CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 189 9.3.4 Proof of Theorem 9.10 We first prove Proposition 9.14. — If two representations have the same cross ratio then, after a Ghys deformation, they are conjugated by an element of H(T). Proof. —Let ρ and ρ be two Hitchin representations with the same cross ratio. 0 1 Write i i ρ (γ) = f ,φ (γ) , i 1,h i h with f ∈ C (T) and φ (γ) ∈ Diff (T). By Proposition 9.12, the representations ρ˙ h h h h and ρ –with values in Ω (T) Diff (T) –are conjugated in Ω (T) Diff (T) by some element H = (α, F),where F ∈ Diff (T). 1,h In particular, after conjugating by (0, F) ∈ C (T) Diff (T) = H(T),we may assume that F is the identity and thus 0 1 φ = φ := φ . γ γ Since ρ is a representation, we have for i = 0, 1 ∗ i i i φ f + f = f . η η γ ηγ Since H = (α, id) intertwines ρ and ρ ,wealsohave ˙ ˙ 0 1 ∗ 1 0 φ α − α = df − df . γ γ γ 1 0 1,h Lemma 9.13 applied to f = f − f yields that α = dg for some function g in C (T). γ γ Conjugating by ( g, id), we may now as well assume that α = 0. It follows that there exists a homomorphism ω : π (Σ) → R, such that 1 0 f − f = ω(γ). γ γ This exactly means that ρ = ρ . This proves that two representations with the same cross ratio are conjugated after a Ghys deformation. Consequently two representations with the same cross ratio and spectrum are conjugated. Indeed if ρ and ρ havethesamespectrum and cross ratio, then by Formula (26), for all γ ∈ π (Σ), ω(γ) = 0.Hence ρ = ρ . 1 190 FRANÇOIS LABOURIE Finally, the proof of properness goes as follows. Suppose that we have a sequence of representations {ρ } converging to ρ , a sequence of elements {ψ } in H(T) n n∈N 0 n n∈N −1 such that {ψ ◦ ρ ◦ ψ } converges to ρ .Let d be the homomorphism H(T) → n n n∈N 1 h h Ω (T) Diff (T).Weaim to prove that {dψ } converges. n n∈N −1 Since ρ and ψ ◦ ρ ◦ ψ have the same spectrum and cross ratio, it follows n n n that ρ and ρ have thesamespectrum and cross ratio.Itfollows from thefirstpart 1 0 of the proof that there exists F such that −1 ρ = F ◦ ρ ◦ F. 0 1 We may therefore assume that ρ = ρ . 0 1 By the second part of Proposition 9.12, {d(ψ )} converges to the identity. The n n∈N properness is now proved. 1,h 10 A characterisation of C (T) Diff (T) We aim to characterise in a geometric way the action of group H(T) = 1,h C (T) Diff (T) generalising Proposition 6.3. Roughly speaking, we will describe it as a subgroup of the central extension of “exact symplectic homeomorphisms” of the Annulus (i.e. T T). The notion of exact symplectic homeomorphism does not make sense in an obvious way, but some homeomorphisms may be coined as “exact sym- plectic” as we shall see. We shall define in this section π-exact symplectomorphism of T T which are charac- terised by several equivalent properties (see Proposition 10.9). Using this notion, we can state our main result Lemma 10.1. — A Hölder homeomorphism of J belongs to H(T) if and only if 1. it preserves the foliation F , 2. it commutes with the canonical flow, 3. it is above a π-exact symplectic homeomorphism of T T. The aim of the following sections is • to recall basic facts about symplectic and Hamiltonian actions on the Annulus, • to extend this construction to a situation with less regularity, and in particular to give a “symplectic” interpretation of the action of H(T) on T T. 10.1 C -exact symplectomorphisms Let L be an orientable real line bundle over a manifold M, equipped with a con- nection ∇ whose curvature form ω is symplectic. CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 191 • For any curve γ joining x and y,let Hol(γ) : L → L , x y be the holonomy of ∇ along γ.If γ is a closed curve – that is x equals y – we identify GL(L ) with the multiplicative group R\{0} and consider Hol(γ) as a real number. • Let ν be a nonzero section of L.Let λ be the primitive of ω defined by ∇ ν = λ (X)ν. X ν If γ is a smooth closed curve, then Hol(γ) = e . Definition 10.2 (Exact symplectomorphisms). — A symplectic diffeomorphism of M is ∇ - exact if for any closed curve (28) Hol(σ) = Hol(φ(σ)). When the action of φ on H (M) is non trivial the notion actually depends on the choice of ∇ . We shall however usually say exact instead of ∇ -exact in order to simplify our exposition. The following proposition clarifies this last condition Proposition 10.3. — Let φ be a symplectic diffeomorphism. The following conditions are equivalent: • For any curve σ , Hol(σ) = Hol(φ(σ)). • For any non zero section ν of L, φ λ − λ is exact. ν ν • The action of φ lifts to a connection preserving action on L. ∗ ∗ Let now M = T T.Let σ : T → T T be the zero section considered as a curve in M.Then the map φ → Hol(φ(σ )), is a group homomorphism from the group of symplectomorphisms to the multiplicative real numbers whose kernel is the group of exact symplectomorphism. This definition is ad hoc: exact symplectomorphisms exist in a greater generality. 192 FRANÇOIS LABOURIE Proof. — We prove that for an exact symplectic diffeomorphism φ,the action lifts to an action on the line bundle L. We choose – on purpose – a more complicated proof. This proof however has the advantage of using very little regularity of φ,sothat we can reproduce it later in another context. We define a lift φ in the following way. We first choose a base point x in T T and η a curve joining x to φ(x ). For any point x in T T, we choose a curve γ joining 0 0 x to x, then we define −1 Ψ = Hol(φ(γ)) Hol(η) Hol(γ) : L → L . x x φ(x) Since for every closed curve σ , by definition of exactness Hol(σ) = Hol(φ(σ)), the linear map Ψ is independent of the choice of γ . Finally we define ˆ ˆ φ(x, u) = (φ(x), Ψ (u)). By construction, φ preserve the parallel transport along any curve γ,that is ˆ ˆ φ(Hol(γ)u) = Hol(φ(γ)).φ(u). This is what we wanted to prove. The other assertions of the proposition are obvi- ous. 10.2 π-symplectic homeomorphisms and π-curves The group Diff(T) of C -diffeomorphisms of the circle acts by area preserving ∗ 1 homeomorphisms on T T.Let Ω (T) be the vector space of continuous one-forms on T. This group also acts on T T by area preserving homeomorphisms in the fol- lowing way fdθ .(θ, t) = (θ, t + f (θ)). We observe that Diff(T) normalises this action. Definition 10.4 (π-symplectic homeomorphism). — The group of π-symplectic homeo- morphisms is π 1 Symp = Ω (T) Diff(T). Here is an immediate characterisation of π-symplectic homeomorphisms. The proof is identical to that of Lemma 9.11. CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 193 Proposition 10.5. — An area preserving homeomorphism of T T is π-symplectic if and only if it is above a diffeomorphism of T. In order to extend the notion of exact diffeomorphisms, using Proposition 10.3 as a definition, we need to have a class of less regular curves for which we can com- pute a holonomy. Definition 10.6 (π-curves and their holonomies). — A π-curve is a continuous curve c = ∗ 1 (c , c ) with values in T T such that c is C .We define θ r θ λ = c dc . r θ The holonomy of a closed π-curve is Hol(c) = e . We collect in the following proposition a few elementary facts. Proposition 10.7 1.The image of a π-curve by a π-symplectic homeomorphism is a π-curve, 2.Let φ = (α, ψ) be a π-symplectic homeomorphism. Let c be a closed π-curve which projects isomorphically on T,then Hol(φ(c)) = Hol(c)e . Definition 10.8 (π-exact symplectic homeomorphism). — The group of π-exact symplec- tomorphisms is π 1 Exact = (C (T)/R) Diff(T), where C (T)/R is identified with the space of exact continuous one-forms on T. By the above remarks, we deduce immediately, reproducing the proof of Prop- osition 10.3 Proposition 10.9 1.The group of π-exact symplectomorphisms is the group of π-symplectic homomorphisms φ such that for any closed π-curve c, Hol(c) = Hol(φ(c)). 194 FRANÇOIS LABOURIE 2.Let x ∈ T T. The action of any π-exact symplectomorphism φ lifts to an action of a homeomorphism φ on L such that for any π-curve c starting from x ,for any u ∈ L 0 x ˆ ˆ (29) φ(Hol(c)u) = Hol(φ(c)).φ(u). 3.Furthermore, φ is determined by Equation (29) up to a multiplicative constant. As a con- sequence, there exists an element (α, ψ) of C (T) Diff(T) such that the induced action of φ on J, identified with the frame bundle of L is (α, ψ). We observe that α is defined uniquely up to an additive constant. 4.Finally, if φ is Hölder, so is φ. From this we obtain immediately Lemma 10.1. 10.3 Width We generalise the notion of width for a π-exact symplectomorphism φ in the following obvious way. Definition 10.10 (Action difference). — Let x and y two fixed points of φ.Let c be a π- curve joining x to y.If q is a curve, we denote by q the curve with the opposite orientation. The action difference of x and y is λ− λ c φ(c) ∆ (φ; x, y) = Hol(c ∪ φ(c)) = e . By Proposition 10.9, this quantity does not depend on c.Moreover, let φ be a lift of the action of φ on L. Then, for any fixed point z of φ, φ(z) is an element of GL(L ) –identified with R\{0} –and for any twofixed points x and y of φ ˆ ˆ ∆ (φ; x, y) = φ(x)/φ( y). Definition 10.11 (Action difference). — The width of φ is w(φ) = sup ∆ (φ; x, y). x,y fixed points This quantity is invariant by conjugation under π-symplectic homeomorphisms. We now relate it to the period of some cross ratio. Proposition 10.12. — Let ρ be an ∞-Hitchin homomorphism from π (Σ) to H(T).Let P be the projection P : H(T) → Exact . CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 195 Let ρ˙ = P ◦ ρ.Let be the spectrum of ρ (cf. Paragraph 9.2.1) and B be its associated cross ρ ρ ratio (cf. Paragraph 9.2.2) with periods .Then −1 log(w(ρ˙(γ))) = (γ) + (γ ) = 2 (γ). ρ ρ B Proof. —Let γ be a non trivial element of π (Σ).Let w be a point in J such that ϕ (w) = ρ(γ)(w). l (γ) The projection v = δ(w) is a fixed point of ρ( ˙ γ). Since L is a bundle associated to J, it follows that the associated action of ρ(γ) on L is a lift ρ( ˙ γ) of ρ( ˙ γ).Then (γ) ρ( ˙ γ)(v) = e . Since ρ( ˙ γ) has precisely two fixed points v and u,we obtain that −1 log(w(ρ(γ))) = (γ) + (γ ). ρ ρ We still have to identify the width with the period of the associated cross ratio. This requires some construction. Let q be the quadruple (γ , y,γ ,γ( y)) of points of + − ∂ π (Σ),where γ and γ are respectively the attractive and repulsive fixed points ∞ 1 + − + − of γ and y is any point different to γ and γ . According to Proposition 9.3, there exists a π (Σ)-equivariant map Θ from T T 2∗ ∗ to ∂ π (Σ) which identifies the fibres of T T → T to the first factor. Then, there ∞ 1 exists a π-curve C such that Θ(C) = q and by definition of the associated cross ratio B ,we have (30) 2 (γ) = 2 log(|B (γ , y,γ ,γ( y))|) = rdθ. B ρ + − ∗ + Let us describe C.Let t and s be the points in T T such that Θ (t) = (γ , y) and Θ(s) = (γ , y) respectively. We denote by [a, b] the arc joining a and b along that fibre, whenever a and b belong to the same fibre of T T. Using this notation, we can write C = c ∪[t, ρ( ˙ γ)(t)]∪ ρ( ˙ γ)(c) ∪[ρ(γ)(s), s], where c is a π-curve. Finally let c˜=[v, t]∪ c ∪[s, u], + − − + where v = Θ(γ ,γ ) and u = Θ(γ ,γ ) are the two fixed points of ρ(γ) –see Figure 3. 196 FRANÇOIS LABOURIE FIG. 3. — The π-curves C and c˜ We have log(w(ρ(γ))) = log(Hol(c ∪ ρ(γ)(c))) ˙ ˜ ˙ ˜ = log(Hol(c ∪[t, ρ(γ)(t)]∪ ρ(γ)(c) ∪[ρ(γ)(s), s])) ˙ ˙ = rdθ = 2 (γ). The assertion follows. 11 A conjugation theorem We use in this section the following setting and notation. 1. Let κ be a Hölder homeomorphism of T.Let D ={(s, t) ∈ T × T | κ(s) = t}. 2. Let p = ( p , p ) : M → D be a principal R-bundle over D equipped 1 2 κ κ with a connection ∇.Let ω be the curvature of ∇.Let f be such that ω = f (s, t)ds ∧ dt. We suppose that f is positive and Hölder. 3. For all s ∈ T,wedenote by c : t → (s, t) the curve in D whose first factor s κ is constant. 4. Let L be the one-dimensional foliation of M by the ∇ -horizontal sections of M along the curves c . Definition 11.1 (Anosov homomorphism). — A homomorphism ρ from π (Σ) to the group 0 1 1,h 1 Diff (M) of C -diffeomorphisms of M with Hölder derivatives is Anosov if 1. the action of ρ (π (Σ)) preserves ∇ , 0 1 2. the quotient ρ (π (Σ)) \ M is compact, 0 1 3.the R-action on ρ (π (Σ))\ M contracts uniformly the leaves of L (cf. Definition 7.3), 0 1 CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 197 1,h 4.there exist two H-Fuchsian homomorphisms ρ and ρ from π (Σ) to C (T), such that 1 2 1 −1 • ρ = κ ρ κ, 2 1 •∀γ ∈ π (Σ), u ∈ M, p(ρ (γ)u) = (ρ (γ)( p (u)), ρ (γ)( p (u))). 1 0 1 1 2 2 Our mainresultisthatevery Anosov representation on M is conjugated to an Anosov representation on J. We define more precisely what we mean by conjugation Definition 11.2 (Anosov conjugated). — An Anosov homomorphism ρ from π (Σ) to 0 1 1,h Diff (M) is Anosov conjugated to a homomorphism ρ from π (Σ) to H(T) by an homeo- morphism ψ from M to J,if: • The homeomorphism ψ intertwines the homomorphisms ρ and ρ , the foliations L and F as well as the R-actions on M and J. • There exists an area preserving homeomorphism ψ from D to T T such that δ ◦ ψ = ψ ◦ p. Remarks 1. The representation ρ with values in H(T) is Anosov in the sense of Defin- ition 7.4. 2. As a consequence of the next result, ψ is unique up to conjugation by elem- ents of H(T). 3. If ρ is Anosov conjugated to ρ and ρ ,then ρ and ρ are conjugated by 0 a b a b a π-symplectic homeomorphism. Theorem 11.3 (Conjugation theorem). — Every Anosov homomorphism ρ from π (Σ) to 0 1 1,h Diff (M) is Anosov conjugated to an Anosov homomorphism ρ from π (Σ) to H(T) by a homeo- morphism ψ. Moreover, ψ and ρ are unique up to conjugation by an element of H(T). Finally, if κ, ∇ depend continuously on a parameter, then we can choose ψ to depend con- tinuously on this parameter. To simplify the notations, we fix a trivialisation of M = R × D so that the sections U : t → (u, s, t) are horizontal along c .Let (s,u) s M → T, π : (u, s, t) → t, M → D , p : (u, s, t) → (s, t). We also observe that the foliation L is the foliation by the fibres of the projection (u, s, t) → (u, s). 198 FRANÇOIS LABOURIE 11.1 Contracting the leaves (bis) For later use, using the above notation, we prove the following easy result which allows us to restate Condition (3) of Theorem 11.3. 1,h Proposition 11.4. — Let ρ be a faithful homomorphism of π (Σ) to Diff (M) such 0 1 that ρ (π (Σ)) \ M is compact. Then the following conditions (2) and (2 ) are equivalent 0 1 (2) The action of R on ρ (π (Σ)) \ M contracts the leaves of L . 0 1 (2 ) Let {t} be a sequence of real numbers going to +∞.Let {γ} be a sequence of m∈N m∈N elements of π (Σ).Let (u, s, t) be an element of M such that {ρ (γ )(u + t , s, t)} , 0 m m m∈N converges to (u , s , t ) in M. Then for all w in T, with w = κ(s), the sequence 0 0 0 {ρ (γ )(s, w)} converges to (s , t ). 0 m m∈N 0 0 Proof. — To simplify the proof, we omit ρ .Let Π be the projection of M on π (Σ) \ M. To say that the R action on M contracts the leaves of L is to say that lim d(Π(u + h, s, t), Π(u + h, s, w)) = 0. h→∞ Assume Condition (2 ).Let {t} be a sequence of real numbers going to +∞.Let m∈N {γ} be a sequence of elements of π (Σ) such that after extracting a subsequence m∈N 1 –since π (Σ) \ M is compact – the sequence {γ (u + t , s, t)} , m m m∈N converges to (u , s , t ) in M. Then by hypothesis, 0 0 0 {γ (s, w)} , m m∈N converges to (s , t ). It follows that 0 0 {γ (u + t , s, w)} , m m m∈N converges to (u , s , w) in M.Hence 0 0 lim d(Π(u + h, s, t), Π(u + h, s, w)) = 0. h→∞ The converse relation is a similar yoga. CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 199 11.2 A conjugation lemma We prove a preliminary result. Let κ : T → T and D be as before. Let ds (respectively dt) be the Lebesgue probability measure on the first (respectively second) factor of T × T. Definition 11.5 (κ-infinite). — A positive continuous function f defined on D is κ-infinite if for all s, t such that κ(s) = t κ(s) t (31) f (s, u)du = f (s, u)du =∞. t κ(s) Lemma 11.6. — Assume that f is κ-infinite and Hölder. Then, there exists a Hölder homeomorphism ψ from D to T T such that • ψ ω = dθ ∧ dr,where ω = f (s, t)ds ∧ dt and where dθ ∧ dr is the canonical symplectic form of T T. • π ◦ ψ = π ,where π be the projection from D to the first T factor, D D κ • if c is a C -curve in D ,then ψ(c) is a π-curve. Furthermore if c is a closed curve then e = Hol(ψ(c)). 1,h • For any two elements γ , γ of Diff (T) such that γ = (γ ,γ ) preserves the 2-form 1 2 1 2 ω then −1 ψ ◦ γ ◦ ψ , is Hölder and is π-symplectic above γ . The homeomorphism ψ is unique up to right composition with a π-exact symplectomorphism. Finally ψ depends continuously on κ and f . Proof. — The uniqueness part of this statement follows from the characterisa- tion of π-exact symplectomorphisms given in Proposition 10.5. The proof of the ex- istence is completely explicit. Let g be a C -diffeomorphism of the circle T such that ∀s, g(s) = κ(s).Weconsider D → T T = T × R, ψ : (s, t) → s, f (s, u)du . g(s) It is immediate to check that • ψ ω = dθ ∧ dr, • π ◦ ψ = π . D 200 FRANÇOIS LABOURIE We also observe that ψ is a Hölder map and a homeomorphism. We now prove that −1 ψ is Hölder. We have −1 ψ (s, u) = (s,α(s, u)), where α(s,u) f (s, w)dw = u. g(s) It is enough to prove that α is Hölder. We work locally so that f is bounded from below by a positive constant k. Firstly, α(s,u) 1 1 |α(s, v) − α(s, u)|≤ f (s, w)dw = |u − v|. k k α(s,v) This proves that α is Hölder with respect to the second variable. Let us take care of the first variable. We have by definition of α,for all s and t α(s,u) α(t,u) f (s, w)dw = u = f (t, w)dw. g(s) g(t) Hence, α(t,u) g(s) α(t,u) (32) f (t, w)dw = f (t, w)dw + ( f (t, w) − f (s, w))dw. α(s,u) g(t) g(t) Sinceweworklocally, wemay assume that k ≤| f (t, w)|≤ K.Since f is a ζ -Hölder function for some ζ,we also have | f (t, w) − f (s, w)|≤ C|t − s| . Therefore, Equality (32) yields k|α(s, u) − α(t, u)|≤ K|t − s|+ C|t − s| . This finishes the proof that α is a Hölder function. By the construction of ψ,if c is a C -curve in D ,then ψ(c) is a π-curve. Note that λ(s, t) = f (s, u)du ds, g(s) is a primitive of ω. It follows that if c : s → (c (s), c (s)) is a C curve in D .Then 1 2 κ c (s) λ = f (s, u)c˙ (s)duds = log(Hol(ψ(c))). c T g(s) −1 By construction, ψ ◦ γ ◦ ψ is π-symplectic and Hölder. The continuity of ψ on κ and f follows from the construction. CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 201 11.3 Proof of Theorem 11.3 11.3.1 A preliminary lemma We prove Lemma 11.7. — Let ρ and ρ be two H-Fuchsian representations from π (Σ) in 1 2 1 Diff (T).Let κ be a Hölder homeomorphism of T such that κ ◦ ρ = ρ ◦ κ.Let f (s, t) 1 2 be a positive continuous function on D . Assume that, for all γ in π (Σ), ω = f (s, t)ds ∧ dt is invariant under the action of (ρ (γ), ρ (γ)).Then f is κ-infinite. 1 2 Proof. — For the sake of simplicity, we write ρ (γ) = γ .We first observe that theinvarianceof ω yields that for all γ in π (Σ) 1 2 dγ dγ 1 2 (33) f (s, t) = (s) (t) f (γ (s), γ (t)). ds dt For any (s, t) in D , we may find a sequence {γ } as well as (s , t ) in D κ n n∈N 0 0 κ such that (34) lim γ (s) = s , n→∞ (35) lim γ (t) = t , n→∞ dγ (36) lim (s) =+∞. n→∞ ds Relation (33) yields t t 1 2 dγ dγ n n 1 2 f (s, u)du = (s) (u) f γ (s), γ (u) du n n ds du κ(s) κ(s) 1 t 2 dγ dγ n n 1 2 = (s) (u) f γ (s), γ (u) du n n ds du κ(s) γ (t) dγ n 1 = (s) f γ (s), u du ds 2 γ (κ(s)) γ (t) dγ n 1 = (s) f γ (s), u du. ds 1 κ(γ (s)) From Assertions (34) and (35), we deduce that γ (t) t lim f γ (s), u du = f (s , u)du > 0. n→∞ κ(γ (s)) κ(s ) n 202 FRANÇOIS LABOURIE Hence Assertion (36) shows that f (s, u)du =∞. κ(s) A similar argument yields κ(s) f (s, u)du =∞. 11.3.2 Proof of Theorem 11.3 Proof. — Recall our hypothesis and notation. Let ρ and ρ be two H-Fuchsian 1 2 representations. Let κ be a Hölder homeomorphism of T such that κ ◦ ρ = ρ ◦ κ. 1 2 Let p : M = R × D → D be a principal R-bundle over D equipped with a con- κ κ κ nection ∇ . Assume the curvature ω of ∇ is such that ω = f (s, t)ds ∧ dt with f positive and Hölder. Let M → T, π : (u, s, t) → t, M → D , p : (u, s, t) → (s, t). Assume that π (Σ) acts on M by C -diffeomorphisms with Hölder derivatives. Assume that this action preserves ∇,and that • π (Σ) \ M is compact, • the action of R on π (Σ) \ M contracts uniformly the fibres of π , 1 1 • we have p(γ u) = (ρ (γ)( p(u)), ρ (γ)( p(u))). 1 2 We want to prove there exists an R-commuting Hölder homeomorphism ψ from M to J over a homeomorphism ψ from D to T T, a representation ρ from π (Σ) to κ 1 H(T) element of Hom ,such that ˆ ˆ ψ ◦ γ = ρ(γ) ◦ ψ. By Lemma 11.7, f is κ-infinite. Hence, by Lemma 11.6 there exists a Hölder homeo- morphism ψ from D to T T unique up to right composition with a π-exact symplec- tic homeomorphism, such that 1. ψ ω = dθ · dr, 2. π ◦ ψ = π , D CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 203 3. if γ , γ are two C diffeomorphisms with Hölder derivatives of the circle 1 2 such that γ = (γ ,γ ) preserves the 2-form ω then 1 2 −1 ψ ◦ γ ◦ ψ , is Hölder and π-symplectic above γ , 4. if c is a C -curve in D,then ψ(c) is a π-curve and Hol (c) = Hol(ψ(c)), where Hol (c) = λ. Let γ be an element of π (Σ).Since γ acts on M preserving the connection ∇ , it follows that µ(γ) = (ρ (γ), ρ (γ)) acts on D in such a way that for all C -curves 1 2 κ Hol (c) = Hol (µ(γ)c). ∇ ∇ We now show that (37) Hol (c) = Hol (µ(γ)c). ω ω We choose a trivialisation of L. In this trivialisation ∇= D + λ + α,where D is the trivial connection and α is a closed form. Then Hol (c) = Hol (c).e . ∇ ω To prove Equality (37), it suffices to show (38) α = α. f (γ)(c) c But ρ (γ) preserves the orientation of T, hence is homotopic to the identity by a fam- ily of mapping f . It follows that µ(γ) is also homotopic to the identity through the −1 family ( f ,κf κ ).This implies that µ(γ) acts trivially on the homology. Hence Equal- t t ity (38). −1 It follows from (3) and (4) that g(γ) = ψ ◦ µ(γ) ◦ ψ is a π-exact symplecto- morphism. Finally, we describe the construction of a map ψ from M to J above ψ,com- muting with R action, “preserving the holonomy” well defined up right composition by an element of R. Let fix an element y of the fibre of p above x in D and an 0 0 κ element z of thefibreof δ above ψ(x ).Let y be an element of the fibre of p above 0 0 some point x of D .Let c be path joining x to x. We observe that there exists ζ in R κ 0 such that y = ζ + Hol (c)y . ∇ 0 We define ψ( y) = ζ + Hol(ψ(c))z . Since for any closed curve Hol (c) = Hol(ψ(c)), it follows that ψ( y) is independent of thechoiceof c. By construction, ψ satisfies the required conditions. The uniqueness statement follows from Lemma 10.1. The continuity statement follows by the corresponding con- tinuity statement of Lemma 11.6 and the construction. 204 FRANÇOIS LABOURIE 12 Negatively curved metrics We first use our conjugation Theorem 11.3 to prove the space Rep contains an interesting space. Theorem 12.1. — Let M be the space of negatively curved metrics on the surface Σ iden- tified up to diffeomorphisms isotopic to the identity. Then, there exists a continuous injective map ψ : M → Rep . Moreover, for every g, ψ( g) is coherent. Furthermore, for any γ in π (Σ) (γ) = (γ). g ψ( g) Here (γ) is the length of the closed geodesic for g freely homotopic to γ,and is the ψ( g)- g ψ( g) length of γ . We first recall some facts about the geodesic flow and the boundary at infinity of negatively curved manifolds. 12.1 The boundary at infinity and the geodesic flow Let Σ be a compact surface equipped with a negatively curved metric. Let Σ be ˜ ˜ its universal cover. Let UΣ (respectively UΣ) be the unitary tangent bundle of Σ (re- spectively Σ). Let ϕ be the geodesic flow on these bundles. Let ∂ Σ be the boundary t ∞ at infinity of Σ. We collect in the following proposition classical facts: Proposition 12.2 ˜ ˜ 1. The boundaries at infinity of π (Σ) and Σ coincide ∂ Σ = ∂ π (Σ). 1 ∞ ∞ 1 2. ∂ Σ has a C -structure depending on the choice of the metric such that the action of 1,h π (Σ) on it is by C -diffeomorphisms. The action of π (Σ) is Hölder conjugate to 1 1 a Fuchsian one. 1,h 2∗ 3.Let G be the space of geodesics of Σ.Then G is C -diffeomorphic to (∂ π (Σ)) . ∞ 1 4. UΣ → G is a principal R-bundle (with the action of the geodesic flow). Furthermore the Liouville form is a connection form for this bundle which is invariant under π (Σ),and its curvature is symplectic. We recall in the next proposition the identification of the unitary tangent bundle with a suitable subset of the cube of the boundary at infinity. Let 3+ 3 ∂ π (Σ) ={oriented (x, y, z) ∈ (∂ π (Σ)) | x = y = z = x}. ∞ 1 ∞ 1 For any (x, y) let 3+ L ={(w, x, y) ∈ (∂ π (Σ)) }. (x,y) ∞ 1 CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 205 Proposition 12.3. — There exists a homeomorphism f of ∂ π (Σ) with UΣ such that ∞ 1 −1 if ψ = f ◦ ϕ ◦ f then ψ (L ) = L . t t t (x,y) (x,y) 3+ Moreover, for any sequence of real number {t} going to infinity, let (x, y, z) ∈ ∂ π (Σ) , m∈N ∞ 1 let {γ} be a sequence of elements of π (Σ), such that m∈N 1 {γ ◦ ψ (z, x, y)} converges to (z , x , y ). m t n∈N 0 0 0 3+ Then, for any w, v such that (w, x, v) ∈ ∂ π (Σ) , there exists v such that ∞ 1 0 {γ ◦ ψ (w, x, v)} converges to (v , x , y ). m t n∈N 0 0 0 Proof. — We first explain the construction of the map f .Let (z, x, y) ∈ 3+ ∂ π (Σ) .Let c the geodesic in Σ going from x to y.Let c(t ) be the image of the ∞ 1 0 projection of z on c, that is the unique minimum on c of the horospherical function as- sociated to z.Weset f (z, x, y) = c(t ). Then, the first property of f is obvious, and the second one is a classical consequence of negative curvature, namely that two geodesics with the same endpoints at infinity go exponentially closer and closer. 12.2 Proof of Theorem 12.1 Using Proposition 11.4, Propositions 12.2 and 12.3 yield that the hypotheses of Theorem 11.3 are satisfied. Therefore, we obtain a continuous map ψ from M to Hom /H(T).For a hy- perbolic metric, we obtain precisely the associated ∞-Fuchsian representation. Since the space of negatively curved metrics on a compact surface is connected, all the rep- resentations are actually in Rep . Finally if ψ( g ) = ψ( g ), then the two metrics have the same length spectrum 0 1 and therefore are isometric by Otal’s Theorem [27]. This proves injectivity. The con- tinuity follows from the continuity statement of Theorem 11.3. 13 Hitchin component We now prove that Rep contains all these Hitchin components. Theorem 13.1. — There exists a continuous injective map ψ : Rep (π (Σ), PSL(n, R)) → Rep , H H such that, if ρ ∈ Rep (π (Σ), PSL(n, R)) then, for any γ in π (Σ), we have 1 1 (39) (γ) = w (γ), ψ(ρ) ρ where (γ) is the ψ(ρ)-length of γ,and w (γ) is the width of γ with respect to ρ. ψ(ρ) ρ 206 FRANÇOIS LABOURIE Moreover, the cross ratios associated to ρ and ψ(ρ) coincide, and the representation ψ(ρ) is coherent (cf. Definition 9.8). Remarks. — It follows in particular that if ρ belongs to Rep (π (Σ), PSL(n, R)), ∗ ∗ and if ρ is the contragredient representation, then ψ(ρ) and ψ(ρ ) have the same spectrum although they are different representations. 13.1 Hyperconvex curves We recall a proposition obtained in Section 2.2.1 Proposition 13.2. — Let ρ be a hyperconvex representation. Let 1 n−1 ξ = (ξ , ..., ξ ), be the limit curve of ρ.Then, there exist 1 n • two C embeddings with Hölder derivatives, η and η ,of T in respectively P(R ) and 1 2 ∗n P(R ), • two representations ρ and ρ from π (Σ) in Diff (T), the group of C -diffeomorphisms 1 2 1 of T with Hölder derivatives, • a Hölder homeomorphism κ of T, such that 1 n−1 1. η (T) = ξ (∂ π (Σ)) and η (T) = ξ (∂ π (Σ)), 1 ∞ 1 2 ∞ 1 2.the map η is ρ equivariant, i i 3.if κ(s) = t, then the sum η (s) + η (t) is direct, 1 2 4. κ intertwines ρ and ρ . 1 2 We shall use we following the continuous map −1 1 −1 n−1 (40) η = η ◦ ξ ,η ◦ ξ , 1 2 2∗ from ∂ π (Σ) to D . ∞ 1 κ 13.2 Proof of Theorem 13.1 We use in this section the independent results proved in the Section 4.4. Let ρ be a hyperconvex representation. Let η , η and κ as in Proposition 2.7. 1 2 Let η = (η ,η ).Let D be defined as usual. We observe that π (Σ) acts by ρ˙ = 1 2 κ 1 (ρ ,ρ ) on D . By Proposition 2.7, η is a ρ-equivariant C map from D to 1 2 κ κ 2∗ n ∗n ⊥ P(n) = P(R ) × P(R )\{(D, P)|D ⊂ P }. CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 207 2∗ In Section 4.7, we show there exists a principal R-bundle L on P(n) equipped with an action of PSL(n, R) and an invariant connection whose curvature is a symplectic form Ω . We pull back this structure on D : • Let M be the induced bundle by η.Note that M is equipped with an action of π (Σ) induced from the PSL(n, R) action on L. • Let ϕ be the flow of the induced R-action on M. • Let L be the foliation of M by horizontal sections along the curves c : t → (s, t) in D . We observe the following + − Proposition 13.3. — Let γ be an element of π (Σ).Let x = η(γ ,γ ) be a fixed point of γ in D .Let λ and λ be the largest and smallest eigenvalues (in absolute values) of ρ(γ). κ max min Then the action of γ on the fibre M of M above x is given by the translation by max log . min Proof. — This follows from the last point of Proposition 4.7. 13.2.1 Proof By Propositions 5.4 and 5.5, the 2-form ω = η Ω is non degenerate and Hölder. To complete the proof of Theorem 13.1, we prove • the quotient π (Σ) \ M is compact (in Proposition 13.4), • the action of R on M contracts the leaves of L (in Proposition 13.6). We now explain how these properties imply the theorem: by Theorem 11.3, there exists a homeomorphism ψ from M to J, an area preserving homeomorphism ψ from D to T T such that • ψ commutes with the R-action, • δ ◦ ψ = ψ ◦ p, • ψ sends L to F , and a representation ρ from π (Σ) to H(T) such that ˆ ˆ ψ ◦ γ = ρ(γ) ◦ ψ. In particular, ρ belongs to Hom . Letusprove that ρ actually belongs to Hom . By the definition of the Hitchin component and the fact that we can choose ψ to depend continuously on our param- eters, it suffices to verify this assertion whenever ρ is an n-Fuchsian representation. In 208 FRANÇOIS LABOURIE this case, the action of π (Σ) extends to a transitive action of PSL(2, R) and is by definition in Hom . The statement (39) about the spectrum follows from Proposition 13.3 which im- plies the spectrum is symmetric. Therefore, by definition, ψ(ρ) is symmetric or, equiv- alently, coherent. Finally, by construction and Proposition 4.7, the two cross ratios co- incide. By Theorem 9.13, an ∞-Hitchin representation which is symmetric is deter- mined by its cross ratio. It follows that ψ is injective. Finally the continuity statement follows from the continuity statement of The- orem 11.3. 13.2.2 Compact quotient Let M be the R-bundle over D defined by M = η L. Inducing the connection form L, M is equipped with a connection whose curvature form is ω.Furthermore π (Σ) acts on M, by the pull back of the action of PSL(n, R) on L.We now prove Proposition 13.4. — The quotient π (Σ) \ M is compact. Recall that π (Σ) acts with a compact quotient on 3+ ∂ π (Σ) = ∞ 1 {oriented triples (x, y, z) ∈ (∂ π (Σ)) , x = y, y = z, x = z}. ∞ 1 We first prove: Proposition 13.5. — There exists a continuous onto π (Σ)-equivariant proper map l from 3+ 2∗ ∂ π (Σ) to M.Moreover, themap l is above the map η = (η , η ) from ∂ π (Σ) to D . ˙ ˙ ˙ ∞ 1 1 2 ∞ 1 κ 3+ 3+ Proof. —Let (z, x, y) be an element of T = ∂ π (Σ) . The two transverse ∞ 1 flags ξ(x) and ξ( y) define a decomposition R = L (x, y) ⊕ L (x, y) ⊕ ... ⊕ L (x, y), 1 2 n such that ξ (x) = L (x, y) and n−1 (41) ξ ( y) = L (x, y) ⊕ ... ⊕ L (x, y). 2 n Let u be a nonzero element of ξ (z).Let u be the projection of u on L (x, y).By i i hyperconvexity, u = 0. We choose u, up to sign, so that |u ∧ ... ∧ u |= 1. 1 n CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 209 n−1 ⊥ Finally we choose f in ξ ( y) so that f , u = 1.The pair (u , f ) is well defined up 1 1 to sign, and hence defines a unique element l(z, x, y) of the fibre of M above η( ˙ x, y). Our first assertion is that l: 3+ T → M, (z, x, y) → l(z, x, y), is proper. 3+ Let {(z , x , y )} be a sequence of elements of T such that m m m m∈N {l(z , x , y ) = (u , f )} , m m m m m m∈N converges to (u , f ) with f , u = 1.In particular, {(x , y )} converges to (x , y ), 0 0 0 0 m m m∈N 0 0 with x = y . We may assume after extracting a subsequence that {z } converges 0 0 m m∈N to z .To prove l is proper, it suffices to show that x = z and y = z . 0 0 0 0 0 Suppose that this is not the case and let us first assume that z = x .Let π be 0 0 m 1 1 n−1 the projection of ξ (z ) on ξ (x ) along ξ ( y ).We observe that π converges to the m m m m 1 1 1 1 identity from ξ (z ) = ξ (x ) to ξ (x ).Let v ∈ ξ (z ) such that π (v ) = u .Since 0 0 0 m m m m m {u } converges to a nonzero element u , {v } converges to u . As a consequence, m m∈N 0 m m∈N 0 all the projections {v } of v on L (x , y ) converge to zero for i>1.Hence, m∈N m i m m 1 n 1 = v ∧ ... ∧ v → 0, m m and the contradiction. n n−1 ⊥ Suppose now that z = y . Using the volume form of R ,weidentify ξ ( y) 0 0 with L (x, y) ∧ ... ∧ L (x, y). 2 n 2 n We use the same notations as in the previous paragraph. Then v ∧ ...∧ v is identified m m with f . It follows that 2 n (42) v ∧ ... ∧ v → f . m m By hyperconvexity 1 p p+1 ξ (z ) ⊕ ξ ( y ) → ξ ( y ). m m 0 We also have, ξ ( y) = L (x, y) ⊕ ... ⊕ L (x, y). n−p+1 n 210 FRANÇOIS LABOURIE 1 1 Since ξ (z ) → ξ ( y ), it follows that for all k m 0 "v " → 0. "v " In particular 1 n−1 "v " → 0. 2 n "v ∧ ... ∧ v " m m Thanks to Assertion (42), we finally obtain that "v "→ 0.Hence 1 n 1 ="v ∧ ... ∧ v " m m 1 2 n ≤"v ""v ∧ ... ∧ v "→ 0, m m m 3+ and the contradiction. Therefore (z , x , y ) ∈ T and l is proper. 0 0 0 It remains to prove that l is onto. Note first that for every (x, y), 3+ L = l(T ) ∩ M , (x,y) η( ˙ x,y) is an interval, being the image of an interval. If (γ ,γ ) is a fixed point of γ,then + − L is invariant by ρ(γ) that acts as a translation on M . It follows that (γ ,γ ) (γ ,γ ) + − + − L = M . (γ ,γ ) (γ ,γ ) + − + − 2 3+ Since the set of fixed points of elements of π (Σ) is dense in ∂ π (Σ) ,and l(T ) is 1 ∞ 1 3+ closed by properness, we conclude that l(T ) = M and that l is onto. As a corollary, we prove Proposition 13.5 3+ Proof of Proposition 13.5. —Since π (Σ) acts properly on T and l is proper and onto, the action of π (Σ) on M is proper. Indeed for any compact K in M −1 −1 {γ ∈ π (Σ) | γ(K) ∩ K =∅} ⊂ {γ ∈ π (Σ) | γ(l (K)) ∩ l (K) =∅}. 1 1 Hence, {γ ∈ π (Σ) | γ(K) ∩ K =∅} −1 −1 ≤ {γ ∈ π (Σ) | γ(l (K)) ∩ l (K) =∅} < ∞. Since π (Σ) has no torsion elements and acts properly, it follows that the ac- tion of π (Σ) on M is free. The space π (Σ) \ M is therefore a topological manifold. 1 1 CROSS RATIOS, SURFACE GROUPS, PSL(n, R) AND DIFFEOMORPHISMS OF THE CIRCLE 211 3+ Finally, the quotient map of l from π (Σ) \ T (which is compact) being onto, it follows π (Σ) \ M is compact. 13.2.3 Contracting the leaves We notice that M is topologically a trivial bundle. Let L be the foliation of M by horizontal sections along the curves c : t → (s, t) in D . We choose a π (Σ)- s κ 1 invariant metric on M.Weprove now: Proposition 13.6. — The flow ϕ contracts the leaves of L on M. Proof. — In the proof of Proposition 13.5, we exhibited a proper onto continuous 3+ π (Σ)-equivariant map l from ∂ π (Σ) to M. This map is such that 1 ∞ 1 (43) l(z, x, y) ∈ M . η( ˙ x,y) By Proposition 12.3, the choice of a hyperbolic metric on Σ gives rises to a flow 3+ ψ by proper homeomorphisms on ∂ π (Σ) such that t ∞ 1 3+ 1. For all t ∈ R,for all (z, x, y) ∈ ∂ π (Σ) , there exists w such that ∞ 1 ψ (z, x, y) = (w, x, y). 2. For any sequence {t} of real numbers going to infinity, let (z, x, y) ∈ m∈N 3+ ∂ π (Σ) and let {γ} be a sequence of elements of π (Σ),suchthat ∞ 1 m∈N 1 {γ ◦ ψ (z, x, y)} converges to (z , x , y ). m t n∈N 0 0 0 3+ Then for any w, v such that (w, x, v) ∈ ∂ π (Σ) , there exists v such that ∞ 1 0 {γ ◦ ψ (w, x, v)} converges to (w , x , y ). m t n∈N 0 0 0 Since Σ is compact and since l and the flows are proper, there exist positive constants a and B such that ∀u,∀T,∃t ∈]T/a − b, Ta + b[ such that l(ψ (u)) = ϕ (l(u)). t T Therefore, the following assertion holds: Let {t} be a sequence of real numbers going to infinity, let u ∈ M,let {γ} m∈N m∈N be a sequence of elements of π (Σ),such that {γ ◦ ϕ (u)} converges to u ∈ M , m t n∈N 0 η(x ,y ) m ˙ 0 0 212 FRANÇOIS LABOURIE then for any w, s such that w ∈ M , there exists v such that η( ˙ x,s) 0 {γ ◦ ϕ (w)} converges to v ∈ M . m t n∈N 0 η(x ,y ) m 0 0 By Proposition 11.4, the action contracts of φ contracts the leaves of L . REFERENCES 1. D. 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Journal
Publications mathématiques de l'IHÉS – Springer Journals
Published: Dec 18, 2007