Complex Hyperbolic Ideal Tetrahedral Groups

Tetrahedral groups are defined as groups of complex reflections on four planes, such that those planes form a tetrahedral configuration with vertices in the boundary of HC. We prove that the complex tetrahedral groups that are complexifications of real groups (i.e. subgroups of PO(3, 1)) do not admit discrete deformations. To achieve this, the structure of the subgroups that stabilize the vertices is studied and it is proven that they correspond to certain discrete representations of abstract triangle groups ∆ in the Heisenberg group. This representations are studied and complete results are given for the case of ∆ been of Euclidean or spherical type. We also show an explicit representation of a hyperbolic triangle group in H5.