Complex hyperbolic and projective deformations of small Bianchi groups

Abstract

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