Nikolay Gusevskii

Complex Hyperbolic Quasi-Fuchsian Groups and Toledo's Invariant

We consider discrete, faithful, type-preserving representations of the fundamental group of a punctured Riemann surface into PU(21), the holomorphic isometry group of complex hyperbolic space. Our main result is that there is a continuous family of such representations which interpolates between ℂ-Fuchsian representations and ℝ-Fuchsian representations. Moreover, these representations take every possible (real) value of the Toledo invariant. This contrasts with the case of closed surfaces where ℂ-Fuchsian and ℝ-Fuchsian representations lie in different components of the representation variety. In that case the Toledo invariant lies in a discrete set and indexes the components of the representation variety.

Complex Hyperbolic Structures on Disc Bundles over Surfaces

We study complex hyperbolic disc bundles over closed orientable surfaces that arise from discrete and faithful representations H_n->PU(2,1), where H_n is the fundamental group of the orbifold S^2(2,...,2) and thus contains a surface group as a subgroup of index 2 or 4. The results obtained provide the first complex hyperbolic disc bundles M->{\Sigma} that: admit both real and complex hyperbolic structures; satisfy the equality 2(\chi+e)=3\tau; satisfy the inequality \chi/2 PU(2,1) with fractional Toledo invariant; where {\chi} is the Euler characteristic of \Sigma, e denotes the Euler number of M, and {\tau} stands for the Toledo invariant of M. To get a satisfactory explanation of the equality 2(\chi+e)=3\tau, we conjecture that there exists a holomorphic section in all our examples. In order to reduce the amount of calculations, we systematically explore coordinate-free methods.

A note on trace fields of complex hyperbolic groups

We show that if

Representations of free Fuchsian groups in complex hyperbolic space

\Gamma

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

is an irreducible subgroup of

Complex hyperbolic Kleinian groups with limit set a wild knot

{\rm SU}(2,1)

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

, then

Complete minimal hypersurfaces in complex hyperbolic space

\Gamma

Complex Hyperbolic Structures on Disc Bundles over Surfaces. II. Example of a Trivial Bundle

contains a loxodromic element

Complex Hyperbolic Structures on Disc Bundles over Surfaces I. General Settings. A Series of Examples

A