WHAT IS a Cross Ratio?

Fran¸coisLabourie

Quadruples of distinct lines are endowed with thus is in bijection with a real projective line. In a unique invariant, the projective cross ratio: two the Poincar´edisk model, the hyperbolic plane is quadruples are equivalent under a linear trans- the disk bounded by the unit circle U which we formation if and only if they have the same cross identify with a real projective line. The geodesics ratio. The projective cross ratio turns out to of the hyperbolic plane are circles orthogonal to characterise the geometry of the projective line. U. A pair of points in the real projective line In projective coordinates, the cross ratio is com- U then determines a unique geodesic. The cross puted as a ratio involving four terms with some ratio is related to the hyperbolic distance as fol- ”crossing symmetries”, hence its name. lows. An horosphere centred at a point x of the Consider four pairwise distinct lines (x, y, z, t) projective line is a circle in the Poincar´emodel in the plane, all passing through the origin. There tangent to U at the point x. Let γ be the unique exist essentially unique coordinates so that x is geodesic joining two points A and B in the hy- generated by (1, 0), y by (0, 1), z by (1, 1) and t perbolic plane; let x and y be the end points at by (b, 1). Then, b := b(x, y, z, t) is the projective infinity of γ; let Cx and Cy be the horospheres cross ratio of the four lines. The projective cross centred at x and y and passing through A and ratio satisfies a certain set R of functional rules B respectively; let finally t and z the centres of We single out two of these rules: a multiplicative two horospheres tangent to each other as well as cocycle rule (1) on the first and second variables to Cx and Cy respectively. Then the hyperbolic and an additive rule (2) distance between A and B is the logarithm of the projective cross ratio of the four points y, z, x, t. (1) b(x, y, z, t) = b(x, w, z, t)b(w, y, z, t), This construction allows us to derive the hyper- (2) b(x, y, z, t) = 1 − b(t, y, z, x). bolic distance from the cross ratio and vice versa.

Conversely, any set endowed with a function b z of quadruples of points satisfying R can be re- x U = ∂ M alised as a subset of the projective line so that ∞ Cz Cx the function b is the restriction of the projective cross ratio. This elementary though remarkable statement asserts that the projective cross ratio γ A completely characterises the projective line. Con- C t t B sequently, we may define the projective cross ra- tio as a function on quadruples satisfying some functional rules. Cy The projective cross ratio has many descen- dants in algebraic geometry: invariants of config- y urations of planes, lines or flags. We shall not pursue this development here but rather restrict Conversely, the real projective line appears to real and complex projective lines and the rela- as the boundary at infinity of the hyperbolic tionships of the cross ratio with negatively curved plane. In the Poincar´edisk model, two oriented manifolds, hyperbolic dynamics and Teichm¨uller geodesics end up at the same point in U if they Theory. are asymptotic, that is, eventually remain at fi- One can describe the real hyperbolic plane as nite distance from each other. This permits the a metric extension of the real projective line. In extension of these ideas to Hadamard surfaces. the complex line C, seen as an affine chart of A two dimensional Riemannian manifold M is the complex projective line, a circle is the set of a Hadamard surface if it is simply connected, complex lines intersecting a given real plane and negatively curved and complete. This boundary 1 2 at infinity ∂ M of M is the set of equivalence of π1(S) in PSL(n, R), called the Hitchin com- classes of asymptotic∞ oriented geodesics. For the ponent, has been identifed by the author as the hyperbolic plane, the boundary at infinity is the space of cross ratios satisfying rules generalising projective line U.A horosphere is now the limit the additive rule (2). Finally, MS also contains of metric spheres passing through a given point the space of all negatively curved metrics on S. but whose centres go to infinity. The space MS is suspected to have an inter- Otal generalised the cross ratio to ∂ M by re- esting structure generalising the Poisson struc- versing the construction of the picture.∞ Starting ture on Hitchin components described by Gold- from four points in ∂ M, draw four horospheres man. Recall that a Poisson structure on a set Y and define the cross∞ ratio of x, y, z, t as the ex- is a Lie algebra structure on a set of functions on ponential of the distance between the two points Y , such that the Lie bracket satisfies a Leibniz A and B, with some sign convention. In general, rule with respect to multiplication. This notion the corresponding function satisfies all rules of R arises from classical mechanics and leads to quan- except the additive relation (2). Define a cross tum mechanics. By construction, every quadru- ratio as any function which satisfies this relaxed ple of points (x, y, z, t) of the boundary at infin- set of rules. Using these ideas, Otal proved that a ity defines a function on MS by b 7→ b(x, y, z, t). metric of negative curvature on a surface is char- These functions, when restricted to Fricke space, acterised by the length of closed geodesics. Bour- yield a natural class of functions whose Poisson don used similar cross ratios to define a coarse ge- brackets have been computed by Wolpert and ometry on the boundary at infinity of a general Penner. These functions were later quantised – negatively curved metric space. that is represented as operators on Hilbert spaces We now turn to dynamics and Teichm¨uller – by Chekhov, Fock, Penner. A more sophis- Theory. Assume that M is the universal cover of ticated construction led Fock and Goncharov to a closed surface S. Although ∂ M was defined quantise Hitchin components. from the metric geometry of M ∞, it only depends In other directions, cross ratios are instrumen- on π1(S). Therefore we denote it ∂ π1(S). The tal in generalising many properties of classical Te- boundary at infinity is homeomorphic∞ to a circle ichm¨ullertheory, such as McShane identities, to a and admits an action of π1(S). Thus: every neg- Higher Teichm¨uller-Thurston theory that is, the atively curved metric on S gives rise to a π1(S)- study of Hitchin components. invariant cross ratio on ∂ π1(S). The complex projective cross ratio strongly re- A cross ratio has a dynamical∞ interpretation. lates to hyperbolic 3-dimensional geometry. Two Consider the quotient of the space of triples of recent and beautiful examples are W. Neumann’s pairwise distinct points of ∂ π1(S) by the diago- study of Hilbert third problem in Hyperbolic ∞ nal action of π1(S). This quotient, which we de- Geometry, and the Quantum Hyperbolic Geome- note US, is compact. A cross ratio gives rise to try developed by Baseilhac, Bonahon, Benedetti, a one parameter group of transformations {φt} Kashaev etc. on US, defined by φt(x, y, z) = (x, y, w) where The simple functional rules satisfied by the t = b(x, y, z, w). The mutiplicative cocycle rule ubiquitous cross ratio, are flexible enough to de- (1) translates into φt+s = φt ◦ φs. This construc- scribe various geometric and dynamical situa- tion recovers the geodesic flow when the cross ra- tions. Yet they are rigid enough to carry impor- tio comes from a negatively curved metric. In tant information about dynamics, Poisson struc- general, there is an intimate relation between dy- tures and surface group representations. namics and the cross ratio. References What is the space MS of all cross ratios on ∂ π1(S)? The Fricke Space of hyperbolic [1] F. Labourie, Cross ratios, surface groups, PSL(n, R) structures∞ on S, usually identified with the Te- and diffeomorphisms of the circle, Publ. Math. IHES ichm¨ullerSpace T (S) of complex structures on (2007), no. 106, 139–213. S by the Uniformisation Theorem, naturally sits [2] J.-P. Otal, Sur la g´eometriesymplectique de l’espace des g´eod´esiques d’une vari´et´e `a courbure n´egative, in MS as the subset of projective cross ratios. Rev. Mat. Iberoamericana (1992), no 3, 441-456. Every such cross ratio identifies ∂ π1(S) with [3] W. D. Neumann, Hilbert’s 3rd problem and invariants the projective line and thus defines∞ a representa- of 3-manifolds, London Math. Soc. Lecture Note Ser. 1 (1998), 383–411 (electronic). tion of π1(S) in PSL(2, R) and a hyperbolic struc- ture on S. Similarly, a space of representations [4] W. Thurston, Three-dimensional geometry and topol- ogy, vol. 1, Princeton University Press (1997).