Duke Mathematical Journal

The geometry of the Eisenstein-Picard modular group

Elisha Falbel and John R. Parker

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The Eisenstein-Picard modular group PU(2,1;Z[ω]) is defined to be the subgroup of PU(2,1) whose entries lie in the ring Z[ω], where ω is a cube root of unity. This group acts isometrically and properly discontinuously on HC2, that is, on the unit ball in C2 with the Bergman metric. We construct a fundamental domain for the action of PU(2,1;Z[ω]) on HC2, which is a 4-simplex with one ideal vertex. As a consequence, we elicit a presentation of the group (see Theorem 5.9). This seems to be the simplest fundamental domain for a finite covolume subgroup of PU(2,1)

Article information

Duke Math. J., Volume 131, Number 2 (2006), 249-289.

First available in Project Euclid: 12 January 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]
Secondary: 11F60: Hecke-Petersson operators, differential operators (several variables)


Falbel, Elisha; Parker, John R. The geometry of the Eisenstein-Picard modular group. Duke Math. J. 131 (2006), no. 2, 249--289. doi:10.1215/S0012-7094-06-13123-X. https://projecteuclid.org/euclid.dmj/1137077885

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