Complex hyperbolic and projective deformations of small Bianchi groups

The Bianchi groups $\mathrm{B}\mathrm{i}(d)=\mathrm{P}\mathrm{S}\mathrm{L}(2,{\mathcal{O}}_d)<\mathrm{P}\mathrm{S}\mathrm{L}(2,\text{C})$ (where ${\mathcal{O}}_d$ denotes the ring of integers of $\text{Q}(i\sqrt{d})$, with $d⩾1$ squarefree) can be viewed as subgroups of $\mathrm{S}\mathrm{O}(3,1)$ under the isomorphism $\mathrm{P}\mathrm{S}\mathrm{L}(2,\text{C})\simeq {\mathrm{S}\mathrm{O}}^0(3,1)$. We study the deformations of these groups into the larger Lie groups $\mathrm{S}\mathrm{U}(3,1)$ and $\mathrm{S}\mathrm{L}(4,\mathbb{R})$ for small values of $d$. In particular we show that $\mathrm{B}\mathrm{i}(3)$, which is rigid in $\mathrm{S}\mathrm{O}(3,1)$, admits a 1-dimensional deformation space into $\mathrm{S}\mathrm{U}(3,1)$ and $\mathrm{S}\mathrm{L}(4,\mathbb{R})$, whereas any deformation of $\mathrm{B}\mathrm{i}(1)$ into $\mathrm{S}\mathrm{U}(3,1)$ or $\mathrm{S}\mathrm{L}(4,\mathbb{R})$ is conjugate to one inside $\mathrm{S}\mathrm{O}(3,1)$. We also show that none of the deformations into $\mathrm{S}\mathrm{U}(3,1)$ are both discrete and faithful.