论文标题
小比安奇组的复杂双曲和投影变形
Complex hyperbolic and projective deformations of small Bianchi groups
论文作者
论文摘要
bianchi组$ {\ rm bi}（d）= {\ rm psl}（2，\ mathcal {o} _d）<{\ rm psl}（\ rm psl}（2，\ c）$（其中$ \ nathcal {o \ nathcal {o} _d $表示$ \ q的$ \ sqrt $ \ sq rt $ \ s sqrt crt n of c（ \ geqslant 1 $ squareFree）可以被视为$ {\ rm so}（3,1）$的子组，在同构$ {\ rm psl}（\ rm psl}（2，\ c）\ c）\ simeq {\ simeq {\ rm so}^0（3,3,1）$。我们将这些组的变形研究成较大的谎言组$ {\ rm su}（3,1）$和$ {\ rm sl}（4，\ r）$，对于$ d $的小值。特别是我们表明，$ {\ rm bi}（3）$，它以$ {\ rm so}（3,1）$僵硬，承认为$ {\ rm su}（3,1）$和$ {\ rm sl}（bi rm sl}（4，\ r）$，$ n of in notemation $ {\ rm su}（3,1）$，$ $ {\ rm su}（3,1）$或$ {\ rm sl}（4，\ r）$是在$ {\ rm so}（3,1）$的内部连接的。我们还表明，$ {\ rm su}（3,1）$都没有变形既离散又忠实。
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.
