WHATIS... ? a Cross Ratio? François Labourie

Quadruples of distinct lines are endowed with a unique invariant, the projective cross ratio: two U = ∂ M quadruples are equivalent under a linear transfor- ∞ Cx x mation if and only if they have the same cross ratio. The projective cross ratio turns out to characterise γ the geometry of the projective line. In projective z Cz A coordinates, the cross ratio is computed as a ratio involving four terms with some “crossing Ct symmetries”, hence its name. B t Consider four pairwise distinct lines (x, y, z, t) in the plane, all passing through the origin. There exist essentially unique coordinates so that x is Cy generated by (1, 0), y by (0, 1), z by (1, 1) and t y by (b, 1). Then, b := b(x, y, z, t) is the projective cross ratio of the four lines. The projective cross ratio satisfies a certain set R of functional rules. Cross ratio and distance. We single out two of these rules: a multiplicative cocycle rule (1) on the first and second variables and an additive rule (2): One can describe the real hyperbolic plane as (1) b(x, y, z, t) = b(x, w, z, t)b(w, y, z, t), a metric extension of the real projective line. In the complex line C, seen as an affine chart of (2) b(x, y, z, t) = 1 − b(t, y, z, x). the complex projective line, a circle is the set of Conversely, any set endowed with a function b of complex lines intersecting a given real plane and quadruples of points satisfying R can be realised as thus is in bijection with a real projective line. In a subset of the projective line so that the function b the Poincaré disk model, the hyperbolic plane is is the restriction of the projective cross ratio. This the disk bounded by the unit circle U, which we elementary though remarkable statement asserts identify with a real projective line. The geodesics that the projective cross ratio completely charac- of the hyperbolic plane are circles orthogonal to U. terises the projective line. Consequently, we may A pair of points in the real projective line U then define the projective cross ratio as a function on determines a unique geodesic. The cross ratio is quadruples satisfying some functional rules. related to the hyperbolic distance as follows. A The projective cross ratio has many descendants horosphere centred at a point x of the projective in algebraic geometry: invariants of configurations line is a circle in the Poincaré model tangent to U of planes, lines, or flags. We shall not pursue this at the point x. Let γ be the unique geodesic joining development here but rather restrict to real and two points A and B in the hyperbolic plane; let complex projective lines and the relationships of x and y be the end points at infinity of γ; let Cx the cross ratio with negatively curved manifolds, and Cy be the horospheres centred at x and y and hyperbolic dynamics, and Teichmüller theory. passing through A and B, respectively; let finally z and t be the centres of the two horospheres François Labourie is professor of mathematics at Uni- tangent to both Cx and Cy . Then the hyperbolic versité Paris-Sud, Orsay. His email address is francois. distance between A and B is the logarithm of the [email protected]. projective cross ratio of the four points x, y, z, t.

1234 Notices of the AMS Volume 55, Number 10 This construction allows us to derive the hyperbolic cross ratio identifies ∂∞π1(S) with the projective distance from the cross ratio and vice versa. line and thus defines a representation of π1(S) Conversely, the real projective line appears as in PSL(2, R) and a hyperbolic structure on S. the boundary at infinity of the hyperbolic plane. In Similarly, a space of representations of π1(S) in the Poincaré disk model, two oriented geodesics PSL(n, R), called the Hitchin component, has been end up at the same point in U if they are asymptotic, identifed by the author as the space of cross ratios that is, they eventually remain at finite distance satisfying rules generalising the additive rule (2). from each other. This permits the extension of these Finally, MS also contains the space of all negatively ideas to Hadamard surfaces. A two-dimensional curved metrics on S. Riemannian manifold M is a Hadamard surface The space MS is suspected to have an interest- if it is simply connected, negatively curved, and ing structure generalising the Poisson structure on complete. This boundary at infinity ∂∞M of M is the Hitchin components described by Goldman. Recall set of equivalence classes of asymptotic oriented that a Poisson structure on a set Y is a Lie algebra geodesics. For the hyperbolic plane, the boundary structure on a set of functions on Y , such that the at infinity is the projective line U.A horosphere is Lie bracket satisfies a Leibniz rule with respect to now the limit of metric spheres passing through a multiplication. This notion arises from classical given point but whose centres go to infinity. mechanics and leads to quantum mechanics. By Otal generalised the cross ratio to ∂∞M by construction, every quadruple of points (x, y, z, t) reversing the construction of the picture. Starting of the boundary at infinity defines a function on from four points in ∂∞M, draw four horospheres MS by b , b(x, y, z, t). These functions, when and define the cross ratio of x, y, z, t as the expo- restricted to Fricke space, yield a natural class of nential of the distance between the two points functions whose Poisson brackets have been com- A and B, with some sign convention. In general, puted by Wolpert and Penner. These functions were the corresponding function satisfies all rules of R later quantised—that is represented as operators except the additive relation (2). Define a cross ratio on Hilbert spaces—by Chekhov, Fock, and Penner. as any function which satisfies this relaxed set of A more sophisticated construction led Fock and rules. Using these ideas, Otal proved that a metric Goncharov to quantise Hitchin components. of negative curvature on a surface is characterised In other directions, cross ratios are instrumen- by the length of closed geodesics. Bourdon used tal in generalising many properties of classical similar cross ratios to define a coarse geometry Teichmüller theory, such as McShane identities, to on the boundary at infinity of a general negatively a higher Teichmüller-Thurston theory, that is, the curved metric space. study of Hitchin components. We now turn to dynamics and Teichmüller theo- The complex projective cross ratio strongly ry. Assume that M is the universal cover of a closed relates to hyperbolic 3-dimensional geometry. Two surface S. Although ∂∞M was defined from the recent and beautiful examples are W. Neumann’s metric geometry of M, it depends only on π1(S). study of Hilbert’s third problem in hyperbolic Therefore we denote it ∂∞π1(S). The boundary at geometry, and the Quantum Hyperbolic Geome- infinity is homeomorphic to a circle and admits try developed by Baseilhac, Bonahon, Benedetti, an action of π1(S). Thus: every negatively curved Kashaev, etc. metric on S gives rise to a π1(S)-invariant cross The simple functional rules satisfied by the ratio on ∂∞π1(S). ubiquitous cross ratio are flexible enough to de- A cross ratio has a dynamical interpretation. scribe various geometric and dynamical situations. Consider the quotient of the space of triples of Yet they are rigid enough to carry important infor- pairwise distinct points of ∂∞π1(S) by the diag- mation about dynamics, Poisson structures, and onal action of π1(S). This quotient, which we surface group representations. denote US, is compact. A cross ratio gives rise to a one-parameter group of transformations {φt } Further Reading on US, defined by φt (x, y, z) = (x, y, w) where [1] F. Labourie, Cross ratios, surface groups, PSL(n, R) t = b(x, y, z, w). The mutiplicative cocycle rule (1) and diffeomorphisms of the circle, Publ. Math. IHES translates into φt+s = φt ◦ φs . This construction (2007), no. 106, 139–213. recovers the geodesic flow when the cross ratio [2] J.-P. Otal, Sur la géometrie symplectique de l’espace comes from a negatively curved metric. In general, des géodésiques d’une variété à courbure négative, Rev. Mat. Iberoamericana (1992), no 3, 441-456. there is an intimate relation between dynamics and [3] W. D. Neumann, Hilbert’s 3rd problem and invariants the cross ratio. of 3-manifolds, The Epstein Birthday Schrift, 383–411 What is the space MS of all cross ratios on (electronic), Geom. Topol. Monogr., 1, Geom. Topol. Publ., ∂∞π1(S)? The Fricke space of hyperbolic structures 1998. on S, usually identified with the Teichmüller [4] W. Thurston, Three-Dimensional Geometry and space T (S) of complex structures on S by the Topology, vol. 1, Princeton University Press (1997). Uniformisation Theorem, naturally sits in MS as the subset of projective cross ratios. Every such

November 2008 Notices of the AMS 1235