A classification of $\mathbb{C}$-Fuchsian subgroups of Picard modular groups

Given an imaginary quadratic extension $K$ of $\mathbb{Q}$, we give a classification of the maximal nonelementary subgroups of the Picard modular group $\operatorname{PSU}_{1,2}(\mathcal{O}_K)$ preserving a complex geodesic in the complex hyperbolic plane $\mathbb{H}^2_\mathbb{C}$. Complementing work of Holzapfel, Chinburg-Stover and M\"oller-Toledo, we show that these maximal $\mathbb{C}$-Fuchsian subgroups are arithmetic, arising from a quaternion algebra $\Big(\!\begin{array}{c} D\,,D_K\\\hline\mathbb{Q}\end{array} \!\Big)$ for some explicit $D\in\mathbb{N}-\{0\}$ and $D_K$ the discriminant of $K$. We thus prove the existence of infinitely many orbits of $K$-arithmetic chains in the hypersphere of $\mathbb{P}_2(\mathbb{C})$.

GENERATORS OF THE EISENSTEIN–PICARD MODULAR GROUP IN THREE COMPLEX DIMENSIONS

Little is known about the generators system of the higher dimensional Picard modular groups. In this paper, we prove that the higher dimensional Eisenstein–Picard modular group PU(3, 1;ℤ[ω3]) in three complex dimensions can be generated by four given transformations.

GENERATORS OF THE EISENSTEIN–PICARD MODULAR GROUP

We prove that the Eisenstein–Picard modular group SU(2,1;ℤ[ω3]) can be generated by four given transformations.

PROPERTY (FA) OF THE GAUSS–PICARD MODULAR GROUP

In this note, we prove that the Gauss–Picard modular group PU(2,1;Θ1) has Property (FA). Our result gives a positive answer to a question by Stover [‘Property (FA) and lattices in SU(2,1)’, Internat. J. Algebra Comput.17 (2007), 1335–1347] for the group PU(2,1;Θ1).

A minimal volume arithmetic cusped complex hyperbolic orbifold

The sister of Eisenstein–Picard modular group is described explicitly in [10], whose quotient is a noncompact arithmetic complex hyperbolic 2-orbifold of minimal volume (see [16]). We give a construction of a fundamental domain for this group. A presentation of that lattice can be obtained from that construction, which relates to one of Mostow's lattices.