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Let G be a semisimple Lie group with no compact factors, K a maximal compact subgroup of G, and $\Gamma$ a lattice in G. We study automorphic forms for $\Gamma$ if G is of real rank one with some additional assumptions, using dynamical approach based on properties of the homogeneous flow on $\Gamma \backslash G$ and a Livshitz type theorem we prove for such a flow. In the Hermitian case G=SU(n,1) we construct relative Poincare series associated to closed geodesics on $\Gamma\backslash G/K$ for one-dimensional representations of K, and prove that they span the corresponding spaces of cusp forms.

Let G be a semisimple Lie group with no compact factors, K a maximal compact subgroup of G, and $\Gamma$ a lattice in G. We study automorphic forms for $\Gamma$ if G is of real rank one with some additional assumptions, using dynamical approach based on properties of the homogeneous flow on $\Gamma \backslash G$ and a Livshitz type theorem we prove for such a flow. In the Hermitian case G=SU(n,1) we construct relative Poincare series associated to closed geodesics on $\Gamma\backslash G/K$ for one-dimensional representations of K, and prove that they span the corresponding spaces of cusp forms.

Let G be a semisimple Lie group with no compact factors, K a maximal compact subgroup of G, and $\Gamma$ a lattice in G. We study automorphic forms for $\Gamma$ if G is of real rank one with some additional assumptions, using dynamical approach based on properties of the homogeneous flow on $\Gamma \backslash G$ and a Livshitz type theorem we prove for such a flow. In the Hermitian case G=SU(n,1) we construct relative Poincare series associated to closed geodesics on $\Gamma\backslash G/K$ for one-dimensional representations of K, and prove that they span the corresponding spaces of cusp forms.

In this paper we obtain some results of harmonic analysis on quantum complex hyperbolic spaces. We introduce a quantum analog for the Laplace-Beltrami operator and its radial part. The latter appear to be second order $q$-difference operator, whose eigenfunctions are related to the Al-Salam-Chihara polynomials. We prove a Plancherel type theorem for it.

Let B be the complex n-dimensional ball and X' be the toroidal compactification of a quotient B/G by a torsion free lattice G of SU(n,1). For an arbitrary G-rational boundary point p, denote by U(p) the commutant of the unipotent radical of the stabilizer of p in SU(n,1) and put G(U) for the subgroup of G, generated by the intersections of G with U(p) for all G-rational boundary points p. The present note establishes that the fundamental group of X' is isomorphic to the quotient G / G(U). As a consequence, the first integral homology group of X' turns to be a quotient of the first integral homology group of B/G by a finite group. The work shows that for any natural number N, there is a normal subgroup G(N) of G of finite index, such that the unramified covering of B/G by B/G(N), induced by the identity of the ball B extends to a covering of the corresponding toroidal compactifications with ramification index greater than N over the toroidal compactifying divisor of B/G(N). The argument exploits the residual finiteness of the lattices in SU(n,1). In the case of a complex dimension 2, the geometric genus of X' equals 1. If X' is not of general type, then the irregularity of X' does not exceed 2 and equals 2 only when X' is birational to an abelian surface. The torsion free surfaces X' of minimal volume are characterized by the Kodaira-Enriques classification types of their minimal models, as well as by lower and upper bounds on the number of the cusps.

We define a generalized Berezin transforms on line bundle over the complex hyperbolic space and we give it as a functions of the G-invariant laplacian on the line bundles.

We show that the complex hyperbolic metrics defined by Deligne-Mostow and Thurston on ${\mathcal{M}}_{0,n}$ are singular K\"ahler-Einstein metrics when ${\mathcal{M}}_{0,n}$ is embedded in the Deligne-Mumford-Knudsen compactification $\overline{\mathcal{M}}_{0,n}$. As a consequence, we obtain a formula computing the volumes of ${\mathcal{M}}_{0,n}$ with respect to these metrics using intersection of boundary divisors of $\overline{\mathcal{M}}_{0,n}$. In the case of rational weights, following an idea of Y. Kawamata, we show that these metrics actually represent the first Chern class of some line bundles on $\overline{\mathcal{M}}_{0,n}$, from which other formulas computing the same volumes are derived.

The paper initiates a systematic study of Moebius structures and Ptolemy spaces. We conjecture that every compact Ptolemy space with circles and many space inversions is Moebius equivalent to the boundary at infinity of a rank one symmetric space of noncompact type. We prove this conjecture for the class of complex hyperbolic spaces as our main result.

We introduce curvature-adapted foliations of complex hyperbolic space and study some of their properties. Generalized pseudo-Einstein hypersurfaces of complex hyperbolic space are classified. Analogous results for curvature-adapted hypersurfaces of quaternionic hyperbolic space are also obtained.

The higher-order superintegrability of the Tremblay-Turbiner-Winternitz system (related to the harmonic oscillator) is studied on the two-dimensional spherical and hiperbolic spaces, $S_\k^2$ ($\k>0$), and $H_{\k}^2$ ($\k 0$) and $H_{\k}^2$ ($\k

We classify all real hypersurfaces with three distinct constant principal curvatures in complex hyperbolic spaces of dimension greater than two.

We characterize the boundary at infinity of a complex hyperbolic space as a compact Ptolemy space that satisfies four incidence axioms.

We classify the semi-Riemannian submersions from a pseudo-hyperbolic space onto a Riemannian manifold under the assumption that the fibres are connected and totally geodesic. Also we obtain the classification of the semi-Riemannian submersions from a complex pseudo-hyperbolic space onto a Riemannian manifold under the assumption that the fibres are complex, connected and totally geodesic submanifolds.

Any Kaehler metric on the ball which is strongly asymptotic to complex hyperbolic space and whose scalar curvature is no less than the one of the complex hyperbolic space must be isometrically biholomorphic to it. This result has been known for some time in odd complex dimension and we provide here a proof in even dimension.

We construct examples of inhomogeneous isoparametric real hypersurfaces in complex hyperbolic spaces.

The space of marked n distinct points on the complex projective line up to projective transformations will be called a configuration space in this paper. There are two families of complex hyperbolic structures on the configuration space constructed by Deligne-Mostow and Thurston. We first confirm that these families are the same. Then in view of the deformation theory for real hyperbolic cone 3-manifolds, we review the families for small n.

We prove that, up to isometric congruence, there are exactly 2n+1 homogeneous polar foliations of the complex hyperbolic space. We also give an explicit description of each of these foliations.

We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal curvatures of the homogeneous hypersurfaces together with their multiplicities.