We give a construction of a fundamental domain for PU(2, 1, Z[i]), that is the group of holomorphic isometries of complex hyperbolic space with coefficients in the Gaussian ring of integers Z[i]. We obtain from that construction a presentation of that lattice and relate it, in particular, to lattices constructed by Mostow.

Abstract. The goal of the article is to prove that four explicitly given transformations, two Heisenberg translations, a rotation and an involution generate the Picard modular group with Gaussian integers acting on the two dimensional complex hyperbolic space. The result answers positively a question raised by A. Kleinschmidt and D. Persson.

Abstract. Our main goal in this paper is to construct the first explicit fundamental domain of the Picard modular group acting on the complex hyperbolic space CH 2 . The complex hyperbolic space is a Hermitian symmetric space, its bounded realization is the unit ball in C 2 equipped with the Bergman metric. The Picard modular group is a discontinuous holomorphic automorphism subgroup of SU (2, 1) with Gaussian integer entries. This fundamental domain has finite volume, one cusp, explicitly given boundary surfaces and an interesting symmetry.

We compute explicitly the Bergman and Szego kernels for a class of pseudoconvex domains. The kernels are expressed in terms of Appell's multivariable hypergeometric functions. These explicit formulas are applied to investigate the asymptotic behavior of the Bergman kernel near some weakly pseudoconvex boundary points.

Our main goal is to analyze the geometric and spectral properties of the Picard modular group with Gaussian integer entries acting on the two-dimensional complex hyperbolic space.