Complex hyperbolic and projective deformations of small Bianchi groups

The Bianchi groups Bi(d) = PSL(2, O-d) < PSL(2, C) (where O-d denotes the ring of integers of Q(i root d), with d >= 1 squarefree) can be viewed as subgroups of SO(3, 1) under the isomorphism PSL(2, C) similar or equal to SO0(3, 1). We study the deformations of these groups into the larger Lie groups SU(3, 1) and SL(4, R) for small values of d. In particular we show that Bi(3), which is rigid in SO(3, 1), admits a 1-dimensional deformation space into SU(3, 1) and SL(4, R), whereas any deformation of Bi(1) into SU(3, 1) or SL(4, R) is conjugate to one inside SO(3, 1). We also show that none of the deformations into SU(3, 1) are both discrete and faithful.