An explicit fundamental domain for the Picard modular group in two complex dimensions

Details An explicit fundamental domain for the Picard modular group in two complex dimensions An explicit fundamental domain for the Picard modular group in two complex dimensions

2005-09-29

arxiv.org/abs/math/0509708v1

Mathematics

Complex Variables

25 pages

Scientific paper

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 ${\bf CH}^{2}$. The complex hyperbolic space is a Hermitian symmetric space, its bounded realization is the unit ball in ${\bf C}^2$ 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.

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