Holomorphic dynamics, Painlevé VI equation and Character Varieties. Serge Cantat, Frank Loray To cite this version: Serge Cantat, Frank Loray. Holomorphic dynamics, Painlevé VI equation and Character Varieties.. Annales de l’Institut Fourier, Association des Annales de l’Institut Fourier, 2009, 59 (7), pp.2927-2978. hal-00186558v2 HAL Id: hal-00186558 https://hal.archives-ouvertes.fr/hal-00186558v2 Submitted on 5 Dec 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. HOLOMORPHIC DYNAMICS, PAINLEVE´ VI EQUATION AND CHARACTER VARIETIES. SERGE CANTAT, FRANK LORAY CONTENTS 1. Introduction 2 1.1. Character variety 3 1.2. Automorphismsandmodulargroups 4 1.3. Projective structures 6 1.4. Painlev´eVI equation 6 1.5. The Cayley cubic 7 1.6. Compactification and entropy 9 1.7. Bounded orbits 9 1.8. Dynamics, affine structures, and the irreducibility of PVI 11 1.9. Aknowledgement 11 2. The family of surfaces 12 2.1. The Cayley cubic 12 2.2. Mappingclassgroupaction 13 2.3. Twists 18 2.4. Character variety of the once-punctured torus 19 3. GeometryandAutomorphisms 20 3.1. Thetriangleatinfinityandautomorphisms 20 3.2. Consequencesandnotations 21 3.3. Elliptic,Parabolic,Hyperbolic 22 3.4. The Markov surface 24 3.5. An (almost) invariant area form 25 3.6. Singularities, fixed points, and an orbifold structure 25 4. BirationalExtensionandDynamics 27 4.1. Birational transformations of surfaces 27 4.2. Birational extension 28 4.3. Entropyofbirationaltransformations 31 5. Bounded Orbits 32 5.1. Dynamicsofparabolicelements 32 5.2. Bounded Orbits 34 hal-00186558, version 2 - 5 Dec 2007 6. Invariant geometric structures 40 Date: 2007. 1 2 SERGECANTAT,FRANKLORAY 6.1. Invariant curves, foliations and webs 40 6.2. Invariant Affine Structures 41 6.3. Proofoftheorem6.5;step1. 42 6.4. Proofoftheorem6.5;step2 44 7. IrreducibilityofPainlev´eVIEquation. 46 7.1. Phasespaceandspaceofinitialconditions 46 7.2. The Riemann-Hilbert correspondance and PVI-monodromy 46 7.3. Riccatisolutionsandsingularpoints 47 7.4. Algebraicsolutionsandperiodicorbits 47 7.5. Nishioka-Umemurairreducibility 47 7.6. Malgrangeirreducibility 48 7.7. Invariant geometric structures 48 7.8. Proof of theorem 7.1 49 7.9. Picard parameters of Painlev´eVI equation and the Cayley cubic 51 8. AppendixA 52 9. AppendixB 53 9.1. Painlev´eVI parameters (θα,θβ,θγ,θδ) and Okamoto symmetries 53 9.2. Reduciblerepresentationsversussingularities. 57 9.3. SU(2)-representationsversusboundedcomponents. 58 9.4. Ramified covers 64 References 67 1. INTRODUCTION This is the first part of a series of two papers (see [10]), the aim of which is to describe the dynamics of a polynomial action of the group (1.1) Γ∗ = M PGL(2,Z) M = Id mod(2) 2 { ∈ | } on the family of affine cubic surfaces (1.2) x2 + y2 + z2 + xyz = Ax + By +Cz + D, where A, B, C, and D are complex parameters. This dynamical system ap- pears in several different mathematical areas, like the monodromy of the sixth Painlev´edifferential equation, the geometry of hyperbolic threefolds, and the spectral properties of certain discrete Schr¨odinger operators. One of Γ our main goals here is to classify parameters (A,B,C,D) for which 2∗ pre- serves a holomorphic geometric structure, and to apply this classification to provide a galoisian proof of the irreducibility of the sixth Painlev´eequation. DYNAMICS, PAINLEVEVIANDCHARACTERVARIETIES.´ 3 S2 1.1. Character variety. Let 4 be the four punctured sphere. Its fundamen- tal group is isomorphic to a free group of rank 3; if α, β, γ and δ are the four loops which are depicted on figure 1.1, then 2 π1(S ) = α,β,γ,δ αβγδ = 1 . 4 h | i pβ pγ β γ δ α pα pδ FIGURE 1. The four punctured sphere. S2 π S2 Let Rep( 4) be the set of representations of 1( 4) into SL(2,C). Such a representation ρ is uniquely determined by the 3 matrices ρ(α), ρ(β), ρ γ S2 and ( ), so that Rep( 4) can be identified with the affine algebraic vari- ety (SL(2,C))3. Let us associate the 7 following traces to any element ρ of S2 Rep( 4): a = tr(ρ(α)) ; b = tr(ρ(β)) ; c = tr(ρ(γ)) ; d = tr(ρ(δ)) x = tr(ρ(αβ)) ; y = tr(ρ(βγ)) ; z = tr(ρ(γα)). The polynomial map χ : Rep(S2) C7 defined by 4 → (1.3) χ(ρ)=(a,b,c,d,x,y,z) is invariant under conjugation, by which we mean that χ(ρ′) = χ(ρ) if ρ′ is conjugate to ρ by an element of SL(2,C). Moreover, S2 (1) the algebra of polynomial functions on Rep( 4) which are invariant under conjugation is generated by the components of χ; 4 SERGECANTAT,FRANKLORAY (2) the components of χ satisfy the quartic equation (1.4) x2 + y2 + z2 + xyz = Ax + By +Cz + D, in which the variables A, B, C, and D are given by A = ab + cd, B = bc + ad, C = ac + bd, (1.5) and D = 4 a2 b2 c2 d2 abcd. − − − − − S2 S2 (3) the algebraic quotient Rep( 4)//SL(2,C) of Rep( 4) by the action of SL(2,C) by conjugation is isomorphic to the six-dimensional quar- tic hypersurface of C7 defined by equation (1.4). S2 χ S2 The affine algebraic variety Rep( 4)//SL(2,C) will be denoted ( 4) and S2 called the character variety of 4. For each choice of four complex parame- ters A, B, C, and D, S(A,B,C,D) (or S if there is no obvious possible confusion) will denote the cubic surface of C3 defined by the equation (1.4). The family of these surfaces S(A,B,C,D) will be denoted Fam. Remark 1.1. The map C4 C4;(a,b,c,d) (A,B,C,D) defined by (1.5) is a non Galois ramified cover→ of degree 24.7→ Fibers are studied in Appendix B. It is important to notice that a point m S will give rise to repre- ∈ (A,B,C,D) sentations of very different nature depending on the choice of (a,b,c,d) in the fiber, e.g. reducible or irreducible, finite or infinite image. Remark 1.2. As we shall see in section 2.4, if we replace the four puntured sphere by the once puntured torus, the character variety is naturally fibered by the family of cubic surfaces S(0,0,0,D. 1.2. Automorphisms and modular groups. The group of automorphisms π S2 S2 Φ ρ ρ Φ 1 Aut( 1( 4)) acts on Rep( 4) by composition: ( , ) − . Since inner χ S2 7→ ◦ automorphisms act trivially on ( 4), we get a morphism from the group of π S2 outer automorphisms Out( 1( 4)) into the group of polynomial diffeomor- χ S2 phisms of ( 4): 2 2 Out(π1(S )) Aut[χ(S )] (1.6) 4 → 4 Φ fΦ 7→ such that fΦ(χ(ρ)) = χ(ρ Φ 1) for any representation ρ. ◦ − π S2 The group Out( 1( 4)) is isomorphic to the extended mapping class group S2 S2 MCG∗( 4), i.e. to the group of isotopy classes of homeomorphisms of 4, that preserve or reverse the orientation. It contains a copy of PGL(2,Z) which is obtained as follows. Let T2 = R2/Z2 be a torus of dimension 2 and σ be the involution of T2 defined by σ(x,y)=( x, y). The fixed point set of σ is the 2-torsion subgroup H T2, isomorphic− to− Z/2Z Z/2Z: ⊂ × (1.7) H = (0,0); (0,1/2); (1/2,0); (1/2,1/2) . { } DYNAMICS, PAINLEVEVIANDCHARACTERVARIETIES.´ 5 The quotient T2/σ is homeomorphic to the sphere, S2, and the quotient map π : T2 T2/σ = S2 has four ramification points, corresponding to the four fixed points→ of σ. The affine group GL(2,Z) ⋉ H acts linearly on T2 and commutes with σ. This yields an action of PGL(2,Z) ⋉ H on the sphere S2, that permutes ramification points of π. Taking these four ramification points S2 as the punctures of 4, we get a morphism 2 (1.8) PGL(2,Z) ⋉ H MCG∗(S ), → 4 which, in fact, is an isomorphism (see [5], section 4.4). The image of PGL(2,Z) is the stabilizer of π(0,0), freely permuting the three other points. χ S2 As a consequence, PGL(2,Z) acts by polynomial transformations on ( 4). The image of H permutes the 4 punctures by products of disjoint transpo- χ S2 sitions and acts trivially on ( 4), so that the action of the whole mapping S2 χ S2 class group MCG∗( 4) on ( 4) actually reduces to that of PGL(2,Z) (see section 2.2). Γ Let 2∗ be the subgroup of PGL(2,Z) whose elements coincide with the identity modulo 2. This group coincides with the (image in PGL(2,Z) of σ Γ S2 the) stabilizer of the fixed points of , so that 2∗ acts on 4 and fixes its four Γ χ S2 punctures. Consequently, 2∗ acts polynomially on ( 4) and preserves the fibers of the projection (a,b,c,d,x,y,z) (a,b,c,d). 7→ From this we obtain, for any choice of four complex parameters (A,B,C,D), Γ a morphism from 2∗ to the group of polynomial diffeomorphisms of the ` surface S(A,B,C,D).