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(id) , with d⩾ 1 squarefree) can be viewed as subgroups of SO (3 , 1) under the isomorphism PSL (2 , C) ≃ SO (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.