Complex hyperbolic and projective deformations of small Bianchi groups

TL;DR

This paper investigates how small Bianchi groups can be deformed into larger Lie groups, revealing specific cases of rigidity and flexibility, and analyzing the nature of these deformations in terms of discreteness and faithfulness.

Contribution

It demonstrates that certain Bianchi groups admit non-trivial deformations into ${ m SU}(3,1)$ and ${ m SL}(4, )$, providing new insights into their deformation spaces and rigidity properties.

Findings

${ m Bi}(3)$ has a 1-dimensional deformation space into ${ m SU}(3,1)$ and ${ m SL}(4, )$.

${ m Bi}(1)$ deformations into these groups are conjugate to those inside ${ m SO}(3,1)$.

None of the deformations into ${ m SU}(3,1)$ are both discrete and faithful.

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 $d \geqslant 1$ squarefree) can be viewed as subgroups of ${\rm SO}(3,1)$ 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 ${\rm SU}(3,1)$ and ${\rm SL}(4,\R)$ for small values of $d$. In particular we show that ${\rm Bi}(3)$, which is rigid in ${\rm SO}(3,1)$, admits a 1-dimensional deformation space into ${\rm SU}(3,1)$ and ${\rm SL}(4,\R)$, whereas any deformation of ${\rm Bi}(1)$ into ${\rm SU}(3,1)$ or ${\rm SL}(4,\R)$ is conjugate to one inside ${\rm SO}(3,1)$. We also show that none of the deformations into ${\rm SU}(3,1)$ are both discrete and faithful.