Finite quotients of the Picard group and related hyperbolic tetrahedral and Bianchi groups
Finite quotients of the Picard group and related hyperbolic tetrahedral and Bianchi groups
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Picard 群及相关双曲四面体和 Bianchi 群的有限商
DOI：
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发表时间：
2001
期刊：
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通讯作者：
B. Zimmermann
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文献类型：
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作者：
L. Paoluzzi;B. Zimmermann
总结。-关于Picard群PSL(2，Z[I])的有限指标子群和有限商群有广泛的文献。本文的主要结果是对作为Picard群的有限商出现的所有线性分式群PSL(2，pm)进行分类。我们还对各种相关的双曲四面体群的线性分数型有限商进行了分类，它们使最小体积的尖点可定向双曲3-orboroles一致。此外，对于适当的PicardC，这些尖点四面体群是Bianci型的，即形式PSL(2，Z[ω])或PGL(2，Z[ω])。结果表明，这些四面体群的所有线性分式类型的有限商都是通过矩阵系数modp的约化得到的，而对于ω∈群，大多数商不是这样产生的(就像经典的模群PSL(2，Z)的情况一样)。从几何的角度出发，我们寻找的是具有覆盖群的双曲3-流形，其覆盖群同构于P-SL(2，Q)或P-GL(2，Q)，并按等距作用于极小体积的尖点双曲3-分支的正则覆盖。这些是具有最小体积的尖点双曲3-流形，它允许线性分式群的作用。我们还给出了具有大群作用的闭双曲3-流形的一些应用。由于上述群中所有阶数相对较小的有限商都是这种或密切相关的类型(类似于Riemann曲面上的Hurwitz作用)，所以线性分式群是第一类也是最重要的一类有限单群。
Summary. - There is an extensive literature on the finite index subgroups and the finite quotient groups of the Picard group P SL(2, Z[i]). The main result of the present paper is the classification of all linear fractional groups P SL(2, p m ) which occur as finite quotients of the Picard group. We classify also the finite quotients of linear fractional type of various related hyperbolic tetrahedral groups which uniformize the cusped orientable hyperbolic 3-orbifolds of minimal volumes. Also these cusped tetrahedral groups are of Bianchi type, that is of the form P SL(2, Z[ω]) or P GL(2, Z[ω]), for suitable ω ∈ C. It turns out that all finite quotients of linear fractional type of these tetrahedral groups are obtained by reduction of matrix coefficients modp whereas for the Picard group most quotients do not arise in this way (as in the case of the classical modular group P SL(2, Z)). From a geometric point of view, we are looking for hyperbolic 3-manifolds which are regular coverings, with covering groups isomorphic to P SL(2, q) or P GL(2, q) and acting by isometries, of the cusped hyperbolic 3-orbifolds of minimal volumes. So these are the cusped hyperbolic 3-manifolds of minimal volumes admitting actions of linear fractional groups. We also give some application to the construction of closed hyperbolic 3-manifolds with large group actions. We are concentrating in this work on quotients of linear fractional type because all finite quotients of relatively small order of the above groups are of this or closely related types (similar to the case of Hurwitz actions on Riemann surfaces), so the linear fractional groups are the first and most important class of finite simple groups to take into consideration.