Generators of a Picard modular group in two complex dimensions

Abstract

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.

The geometry of the Gauss–Picard modular group

“…In [8] we use a different method to give (essentially) the same generators for PU(2, 1; Z[i]). The advantage of the method used in [8] is that it gives a normal form for each element of the group.…”

“…The advantage of the method used in [8] is that it gives a normal form for each element of the group. This should be compared to the method given in Series [28] for producing normal forms for elements of PSL(2, Z).…”

Generators of the Eisenstein–picard Modular Group

“…In this section we extend the techniques of [2] to prove the following theorem. [5] The Eisenstein-Picard modular group 425 T 3.1.…”

“…In this section we recall some basic definitions and results from complex hyperbolic geometry which can be found, for example, in [2,[7][8][9][10].…”

“…Let C 2,1 denote the three-dimensional complex vector space C 3 equipped with the Hermitian form z, w = w * Jz = z 1 w 3 + z 2 w 2 + z 3 w 1 of signature (2,1). Here the matrix J is defined by…”