Complex hyperbolic & 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\sqrt{d}), with d >0 squarefree) can be viewed as 
subgroups of SO(3,1) under the isomorphism PSL(2,C) --> SO^0(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. 
This is joint work with Morwen Thistlethwaite.