Complex Hyperbolic Geometry Research Articles

Orbifolds and orbibundles in complex hyperbolic geometry

We develop the theory of orbibundles from a geometrical viewpoint using diffeology. One of our goals is to present new tools allowing to calculate invariants of complex hyperbolic disc orbibundles over 2-orbifolds appearing in the geometry of 4-manifolds. These invariants are the Euler number of disc orbibundles and the Toledo invariant of PU(2,1)-representations of 2-orbifold groups.

Static wind loads on rigid roof model with complex hyperbolic geometry

Static wind loads on rigid roof model with complex hyperbolic geometry

Complex hyperbolic geometry and Hilbert spaces with complete Pick kernels

Suppose H is a finite dimensional reproducing kernel Hilbert space of functions on X. If H has the complete Pick property then there is an isometric map, Φ, from X, with the metric induced by H, into complex hyperbolic space, CHn, with its pseudohyperbolic metric. We investigate the relationships between the geometry of Φ(X) and the function theory of H and its multiplier algebra.

On a family of triangle groups in complex hyperbolic geometry

In this paper, we study the deformation of triangle groups of type (3,n,∞) determined by three R-circles R0, R1, R2 with only rotational symmetry, which generalizes the problem studied by Falbel and Parker in [4].

Counting and equidistribution in Heisenberg groups

We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$. We prove a Mertens' formula for the integer points over a quadratic imaginary number fields $K$ in the light cone of Hermitian forms, as well as an equidistribution theorem of the set of rational points over $K$ in Heisenberg groups. We give a counting formula for the cubic points over $K$ in the complex projective plane whose Galois conjugates are orthogonal and isotropic for a given Hermitian form over $K$, and a counting and equidistribution result for arithmetic chains in the Heisenberg group when their Cygan diameter tends to $0$.

Complex hyperbolic geometry of the figure-eight knot

We show that the figure eight knot complement admits a uniformizable spherical CR structure, i.e. it occurs as the manifold at infinity of a complex hyperbolic orbifold. The uniformization is unique provided we require the peripheral subgroups to have unipotent holonomy.

On the lack of density of Lipschitz mappings in Sobolev spaces with Heisenberg target

We study the question: When are Lipschitz mappings dense in the Sobolev space W 1 , p ( M , H n ) W^{1,p}(M,\mathbb {H}^n) ? Here M M denotes a compact Riemannian manifold with or without boundary, while H n \mathbb {H}^n denotes the n n th Heisenberg group equipped with a sub-Riemannian metric. We show that Lipschitz maps are dense in W 1 , p ( M , H n ) W^{1,p}(M,\mathbb {H}^n) for all 1 ≤ p > ∞ 1\le p>\infty if dim ⁡ M ≤ n \dim M \le n , but that Lipschitz maps are not dense in W 1 , p ( M , H n ) W^{1,p}(M,\mathbb {H}^n) if dim ⁡ M ≥ n + 1 \dim M \ge n+1 and n ≤ p > n + 1 n\le p>n+1 . The proofs rely on the construction of smooth horizontal embeddings of the sphere S n \mathbb {S}^n into H n \mathbb {H}^n . We provide two such constructions, one arising from complex hyperbolic geometry and the other arising from symplectic geometry. The nondensity assertion can be interpreted as nontriviality of the n n th Lipschitz homotopy group of H n \mathbb {H}^n . We initiate a study of Lipschitz homotopy groups for sub-Riemannian spaces.

Bounded cohomology and totally real subspaces in complex hyperbolic geometry

AbstractWe characterize representations of finitely generated discrete groups into (the connected component of) the isometry group of a complex hyperbolic space via the pullback of the bounded Kähler class.

Configurations of flags and representations of surface groups in complex hyperbolic geometry

In this work, we describe a set of coordinates on the PU(2,1)-representation variety of the fundamental group of an oriented punctured surface Σ with negative Euler characteristic. The main technical tool we use is a set of geometric invariants of a triple of flags in the complex hyperbolic plane $${\bf H^2_{\mathbb {C}}}$$ . We establish a bijection between a set of decorations of an ideal triangulation of Σ and a subset of the PU(2,1)-representation variety of π 1(Σ).

The Moduli Space of Points in the Boundary of Complex Hyperbolic Space

We consider the space \(\mathcal{M}(n,m)\) of ordered m-tuples of distinct points in the boundary of complex hyperbolic n-space, \(\mathbf{H}_{\mathbb{C}}^{n}\), up to its holomorphic isometry group PU(n,1). An important problem in complex hyperbolic geometry is to construct and describe the moduli space for \(\mathcal{M}(n,m)\). In particular, this is motivated by the study of the deformation space of complex hyperbolic groups generated by loxodromic elements. In the present paper, we give the complete solution to this problem.

The Moduli Space of Complex Geodesics in the Complex Hyperbolic Plane

We consider the space \(\mathcal{M}\) of ordered m-tuples of distinct complex geodesics in complex hyperbolic 2-space, \({\rm\bf H}_{\mathbb{C}}^{2}\), 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 \(\mathcal{M}\). 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.

On the moduli space of quadruples of points in the boundary of complex hyperbolic space

We consider the space \( \mathcal{M} \) of ordered quadruples of distinct points in the boundary of complex hyperbolic n-space, \( H_\mathbb{C}^n \), up to its holomorphic isometry group PU(n, 1): One of the important problems in complex hyperbolic geometry is to construct and describe a moduli space for \( \mathcal{M} \). For n = 2, this problem was considered by Falbel, Parker, and Platis. The main purpose of this paper is to construct a moduli space for \( \mathcal{M} \) for any dimension n ≥ 1.

A Measurable Cartan Theorem and Applications to Deformation Rigidity in Complex Hyperbolic Geometry

A Measurable Cartan Theorem and Applications to Deformation Rigidity in Complex Hyperbolic Geometry

Cartan angular invariant and deformations of rank 1 symmetric spaces

New geometric invariants in the quaternionic hyperbolic space and in the hyperbolic Cayley plane are introduced and studied. In these non-commutative and non-associative geometries they are a substitution for the Toledo invariant and the Cartan angular invariant well known in complex hyperbolic geometry. These new invariants are used for the investigation of quasi-Fuchsian deformations of quaternionic and octonionic hyperbolic manifolds. In particular, bendings are defined for such structures, which are the last two classes of locally symmetric structures of rank 1. Bibliography: 27 titles.

Open Sets of Maximal Dimension in Complex Hyperbolic Quasi-Fuchsian Space

Let pi(1), be the fundamental group of a closed surface Sigma of genus g > 1. One of the fundamental problems in complex hyperbolic geometry is to find all discrete, faithful, geometrically finite and purely loxodromic representations of pi(1) into SU(2, 1), (the triple cover of) the group of holomorphic isometries of H-C(2). In particular, given a discrete, faithful, geometrically finite and purely loxodromic representation rho(0) of pi(1), can we find an open neighbourhood of rho(0) comprising representations with these properties. We show that this is indeed the case when rho(0) preserves a totally real Lagrangian plane.

Asymptotic of complex hyperbolic geometry and L2-spectral analysis of Landau-like Hamiltonians

In this paper we show that the flat Hermitian complex geometry of Cn, n⩾1, is approximated by the complex hyperbolic geometry of the Bergman complex balls Bρn⊂Cn of radius ρ>0. Furthermore, it will be shown that some elements of the L2-spectral analysis, such as the spectrum, the L2-eigenprojector and the resolvent kernels, associated to the so-called Landau-like Hamiltonian HB,ρ on Bρn give rise to their analogous of the Landau-like Hamiltonian HB,∞ on Cn by letting ρ tend to infinity.

The moduli space of the modular group in complex hyperbolic geometry

We construct the space of discrete, faithful, type-preserving representations of the modular group into the isometry group of complex hyperbolic 2-space up to conjugacy. This is the first Fuchsian group for which the entire complex hyperbolic deformation space has been constructed. We also show how the ℂ-spheres of Falbel-Zocca are related to the ℝ-spheres (hybrid spheres) of Schwartz.

Rigidity and flexibility of triangle groups in complex hyperbolic geometry

We show that the Teichmüller space of the triangle groups of type ( p, q,∞) in the automorphism group of the two-dimensional complex hyperbolic space contains open sets of 0, 1 and two real dimensions. In particular, we identify the Teichmüller space near embeddings of the modular group preserving a complex geodesic.

Holomorphic foliations transverse to fibrations on hyperbolic manifolds

We regard some aspects of the following general question: Given a singular holomorphic foliation F on a complex manifold M. what can he said about F in the case that M M special geometry (e.g M is hyperbolic (of. [11,12] or in case the F leaves are embedded in a geometrically special way (e.g. F almost everywhere transverse to some fibration M → B with compact hyperbolic fibers:). We prove that the foliation exhibits meromorphic first integral under natural necessary conditions. We also study the classification of one foliations which are complete with respect to some holomorphic fibrations according to the genus of the fiber. The is applied to a remaining case in [1] and to poivnomial foliations in (See Section 4.1) Applications to systems of implicit ordinary differential equations without movable singularities are also given. The techniques and results that we obtain illustrate the interest of using featuring from Several Complex Variables and Hyperbolic Geometry in Holomorphie Dynamics and Complex Differential Equations. This paper is concieved to be accessible also to non-speciallists on singular holomorphic foliations and it contains paragraphs including some basic material on the subject.

Applications of Bruhat decompositions to complex hyperbolic geometry

The double coset space AΛ (n, ℂ) / U (n − 1, 1) is studied, where A consists of the diagonal matrices in GL (n, ℂ). This space naturally arises in the harmonic analysis on the hermitian symmetric space GL (n, ℂ) / U (n − 1, 1). It is shown here that these double cosets also represent a class of basic invariants related to complex hyperbolic geometry. An algebraic parametrization for the double cosets is given and it is shown how this may be used to conveniently compute the geometric invariants.