A complex hyperbolic structure for moduli of cubic surfaces

We show that the moduli space M of marked cubic surfaces is biholomorphic to (B⁴ − H)/Г, where B⁴ is complex hyperbolic four-space, Γ is a specific group generated by complex reflections, and H is the union of reflection hyperplanes for Γ. Thus M has a complex hyperbolic structure, i.e., an (incomplete) metric of constant negative holomorphic sectional curvature.     

Critical densities for random quotients of hyperbolic groups

We prove that in various natural models of a random quotient of a group, depending on a density parameter, for each hyperbolic group there is some critical density under which a random quotient is still hyperbolic with high probability, whereas above this critical value a random quotient is very probably trivial. We give explicit characterizations of these critical densities for the various models. To cite this article: Y. Ollivier, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Hyperbolic Structures on the Configuration Space of Six Points in the Projective Line

The oriented configuration space X⁺₆ of six points on the real projective line is a noncompact three-dimensional manifold which admits a unique complete hyperbolic structure of finite volume with ten cusps. On the other hand, it decomposes naturally into 120 cells each of which can be interpreted as the set of equiangular hexagons with unit area. Similar hyperbolic structures can be obtained by considering nonequiangular hexagons so that the standard hyperbolic structure on X⁺₆ is at the center of a five parameter family of hyperbolic structures of finite volume. This paper contributes to investigations of the properties of this family. In particular, we exhibit two real analytic maps from the set of prescribed angles of hexagons into R¹⁰ whose components are the traces of the monodromies at the ten cusps. We show that this map has maximal rank 5 at the center.     

Espace des modules des variétés hyperboliquement plongéesThe moduli space of hyperbolically imbedded manifolds

In this Note, we construct the moduli space of hyperbolically imbedded manifolds. We recall that the moduli space of compact hyperbolic manifolds has been constructed by Brody and Wright. To construct our moduli space, we use a general criterion to represent analytic functors by coarse moduli spaces due to Schumacher. The objects to deform are couples (X,D) where X is a compact manifold and D is a normal crossing divisor in X such that X?D is hyperbolically imbedded in X. This criterion is based on two ingredients: in our case, the first is the existence of semi-universal logarithmic deformation due to Kawamata. The second is a consequence of a theorem of stability of hyperbolically imbedded spaces through logarithmic deformations. We use the relative-distance of Kobayashi to simplify the proof. To cite this article: A. Khalfallah, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 237–242.     

The Laplace operator on a hyperbolic manifold I. Spectral and scattering theory

Using techniques of stationary scattering theory for the Schrödinger equation, we show absence of singular spectrum and obtain incoming and outgoing spectral representations for the Laplace-Beltrami operator on manifolds Mⁿ arising as the quotient of hyperbolic n-dimensional space by a geometrically finite, discrete group of hyperbolic isometries. We consider manifolds Mⁿ of infinite volume. In subsequent papers, we will use the techniques developed here to analytically continue Eisenstein series for a large class of discrete groups, including some groups with parabolic elements.     

Rank,rigidity of factors,and weak closure of measure-preserving <Emphasis Type="Bold">Z</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>-actions

Measure-preserving rank one actions of the groups Z ⁿ are considered. It is proved that if such an action is a non-finite expansion, then the corresponding quotient is a rigid action. It is also shown that in the case of a relatively weak mixing expansion the orthogonal projection to the quotient space belongs to the weak closure of the group of operators induced by the action considered.     

Real quadrics in C n , complex manifolds and convex polytopes

In this paper, we investigate the topology of a class of non-Kähler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics C ⁿ which are invariant with respect to the natural action of the real torus (S ¹) ⁿ onto C ⁿ . The quotient space is a simple convex polytope. The problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes. We prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich. We also resolve an associated wall-crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple convex polytope. Finally, as a nice consequence, we obtain that affine non-Kähler compact complex manifolds can have arbitrary amount of torsion in their homology groups, contrasting with the Kähler situation.     

Boundary amenability of relatively hyperbolic groups

Let K be a fine hyperbolic graph and Γ be a group acting on K with finite quotient. We prove that Γ is exact provided that all vertex stabilizers are exact. In particular, a relatively hyperbolic group is exact if all its peripheral groups are exact. We prove this by showing that the group Γ acts amenably on a compact topological space. We include some applications to the theories of group von Neumann algebras and of measurable orbit equivalence relations.     

On a J. C. C. Nitsche’s Type Inequality for the Hyperbolic Space H3

Let H ³ be the hyperbolic space identified with the unit ball B ³ = {x ∈ R ³ : |x| < 1} with the Poincaré metric d _h and let ??(x ₀, p, q) : = {x : p < d _h (x, x ₀) < q} ? H ³ be a hyperbolic annulus with the inner and outer radii 0 < p < q < ∞. We prove that if there exists a hyperbolic harmonic diffeomorphism between annuli ??(x ₀, a, b) and ?? (y ₀, α, β) in the hyperbolic space H ³, then β/α>1+ψ(a,b), where ψ is a positive function. In addition, for given two annuli in the hyperbolic space H ³, satisfying certain conditions, we construct radial harmonic mappings between them.     

Spectral square means for period integrals of wave functions on real hyperbolic spaces

We study the period integrals of Laplace eigenfunctions on an arithmetic quotient X of the d-dimensional hyperbolic space along a fixed eigenfunction on an arithmetic quotient of (d−1)-dimensional hyperbolic space embedded in X. We introduce a certain counting function for period integrals and prove its asymptotic law.     

Harmonic manifolds with some specific volume densities

We show that a noncompact, complete, simply connected harmonic manifold (M ^d, g) with volume densityθ _m(r)=sinh^d-1 r is isometric to the real hyperbolic space and a noncompact, complete, simply connected Kähler harmonic manifold (M ^2d, g) with volume densityθ _m(r)=sinh^2d-1 r coshr is isometric to the complex hyperbolic space. A similar result is also proved for quaternionic Kähler manifolds. Using our methods we get an alternative proof, without appealing to the powerful Cheeger-Gromoll splitting theorem, of the fact that every Ricci flat harmonic manifold is flat. Finally a rigidity result for real hyperbolic space is presented.     

Nonarithmetic uniformization of some real moduli spaces

Some real moduli spaces can be presented as real hyperbolic space modulo a non-arithmetic group. The whole moduli space is made from some incommensurable arithmetic pieces, in the spirit of the construction of Gromov and Piatetski-Shapiro.     

Analyse harmonique des formes différentielles sur l'espace hyperbolique réel II. Transformation de Fourier sphérique et applications

We study the L² spherical Fourier transform associated with the bundle of differential forms over real hyperbolic spaces by using the Fourier-Jacobi transform on L² (R). Our results lead to the analytic Plancherel formula for the Fourier transform of differential forms, and to the exact expression for the heat kernel via the inversion of the Abel transform.     

Teichmller space of negatively curved metrics on complex hyperbolic manifolds is not contractible

We prove that for all n = 4k- 2 and k 2 there exists a closed smooth complex hyperbolic manifold M with real dimension n having non-trivial π1(T0(M)). T0(M) denotes the Teichm¨uller space of all negatively curved Riemannian metrics on M, which is the topological quotient of the space of all negatively curved metrics modulo the space of self-diffeomorphisms of M that are homotopic to the identity.     

Support of Virasoro unitarizing measuresSupport des mesures unitarisantes de l'algèbre de Virasoro

A unitarizing measure is a probability measure such that the associated L² space contains a closed subspace of holomorphic functionals on which the Virasoro algebra acts unitarily. It has been shown that the unitarizing property is equivalent to an a priori given formula of integration by parts, which has been computed explicitly. We show in this Note that unitarizing measures must be supported by the quotient of the homeomorphism group of the circle by the subgroup of Möbius transformations. To cite this article: H. Airault et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 621–626.     

The Index of a 1-Form on a Real Quotient Singularity

Let G be a finite Abelian group acting (linearly) on space ?ⁿ and, therefore, on its complexification ?ⁿ, and let W be the real part of the quotient ?ⁿ/G (in the general case, W ≠ ?ⁿ/G). The index of an analytic 1-form on the space W is expressed in terms of the signature of the residue bilinear form on the G-invariant part of the quotient of the space of germs of n-forms on (?ⁿ, 0) by the subspace of forms divisible by the 1-form under consideration.     

The Moduli Space of Complex Geodesics in the Complex Hyperbolic Plane

We consider the space M\mathcal{M} of ordered m-tuples of distinct complex geodesics in complex hyperbolic 2-space, H_\mathbbC²{\rm\bf H}_{\mathbb{C}}^{2}, up to its holomorphic isometry group PU(2,1). One of the important problems in complex hyperbolic geometry is to construct and describe the moduli space for M\mathcal{M}. This is motivated by the study of the deformation space of groups generated by reflections in complex geodesics. In the present paper, we give the complete solution to this problem.     

Homogeneous quaternionic Kähler structures of linear type

A classification of homogeneous quaternionic Kähler structures by real tensors is given and related to Fino's representation theoretic decomposition. A relationship between the modules whose dimension grows linearly and quaternionic hyperbolic space is found. To cite this article: M. Castrillón López et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Reduction theory over quadratic imaginary fields

Hurwitz developed a reduction theory for real binary quadratic forms of positive discriminant based on least-remainder continued fractions. For each quadratic imaginary field k, we develop a similar theory for complex binary quadratic forms of nonzero discriminant. This uses a Markov partition for the geodesic flow over the quotient of hyperbolic 3-space by the Bianchi group B_k. When k has a Euclidean algorithm, our theory is based on least-remainder continued fractions.