Picard模群与球面CR几何
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摘要
作为模群PSL(2,Z)在复双曲空间中的高维推广，Picard模群PU(2,1;Od)是一类最简单的复双曲算术格，其中Od是虚二次数域Q(i√d)中的代数整环，d是无平方因子的正整数。因为关于Picard模群的研究成果为发展复双曲离散群的一般性理论提供了十分重要的例子，所以Picard模群是复双几何理论的重要研究内容。
本论文的第一个工作是研究了Picard模群PU(2,1;Od)（d=2,3,7,11）的生成子。利用推广的连分式算法，我们找到了PU(2,1;O3)的一个新的有限生成子系。对于其他的Picard模群PU(2,1;Od)（d=2,7,11），我们通过首先找到固定无穷远点的稳定子群的一组生成子系，然后将其与Zhao的结果进行比较，从而获得PU(2,1;Od)（d=2,7,11）的一组新的生成子系。另外，对于高维的Picard模群，我们也通过利用使用连分式算法找到了PU(3,1;O3)一套生成子系。
根据Serre的定义，当一个有限生成群既不能分裂成非平凡的含有合并的乘积，又不具有到Z的满同态时，则称该群具有性质（FA）.考虑一个群是否具有性质（FA）是半单李群的一个重要问题。
本论文的第二个工作是研究了复双曲算术格Gauss-Picard模群PU(2,1;O1)和Eisenstein-Picard模群PU(2,1;O3)的姊妹群的性质（FA）. Eisenstein-Picard模群及其姊妹群是仅有的两个具有极小体积的带尖点的复双曲轨形的基本群，它们都是算术群。2007年，Stover证明了Eisenstein-Picard模群PU(2,1;O3)具有性质（FA），并问其他Picard模群PU(2,1;Od)是否也具有性质（FA）？我们证明了PU(2,1;O1)具有性质（FA）和Eisenstein-Picard模群PU(2,1; O3)的姊妹群具有性质（FA），这是对Stover提出的问题的部分肯定回答。该结果表明，作为仅有的两个具有极小体积的带尖点复双曲轨形的基本群，Eisenstein-Picard模群和Eisenstein-Picard模群的姊妹群都具有性质（FA）.
几何中一个很重要的问题就是研究空间拓扑性质和其在某种好的几何结构下的几何性质之间的联系。三维流形上的一个球面CR结构指的是一套S3中的坐标卡，其中的坐标变换是PU(2,1)中的元素在开集上的限制。众所周知，所有可以配置接触结构的三维流形很自然地可以配置CR结构。这个事实使得我们预想球面CR结构应该是研究三维流形的一个基本问题，因此，一个自然的问题就是哪些三维流形具有球面CR结构。
本论文的最后一个工作是研究了三维双曲流形八字结的补M=S3K的球面CR结构,其中K是八字结。通过粘合四面体的方法， Falbel构造了八字结的补的基本群π1(M)到PU(2,1)的三个互不共轭的表示ρi(i=1,2,3)。他详细研究了第一个表示ρ1，证明了该表示的离散性以及对应的球面CR结构的存在性，有意思的是该表示下的像群ρ1(π1(M))作用在复双曲空间上的极限集是整个边界S3.在本文中，我们主要研究了第二个表示ρ2，证明了该表示是离散的并且ρ2(π1(M))是Picard模群PU(2,1; O7)的无限次子群，证明了在该表示下存在一个分歧球面CR结构。我们还比较了这三个表示之间的异同：最大的相同之处是它们的像群ρi(π1(M))都是算术的，ρ1(π1(M))是Eisenstein-Picard模群PU(2,1; O3)的无限次子群，而ρ3(π1(M))和ρ2(π1(M))是共轭的，所以也可以看成是Picard模群PU(2,1;O7)的无限次子群；最大的不同之处则是ρ1(π1(M))含有一个无限次的曲面子群ρ1(F2)（F2是单洞环面的基本群，这是因为八字结的补是圆环上的单洞环面丛），而该曲面群在第二和第三个表示下的像群则是有限次（三次）的子群，该结果表明，八字结的补的基本群到PU(2,1)的三个表示中的第二个和第三个表示几乎是由其曲面子群决定的。
As the generalization of classic modular group PSL(2, Z) in high dimensionalcomplex hyperbolic space, Picard modular groups PU(2,1; Od) are the simplestcomplex hyperbolic arithmetic lattices, where Odis the ring of algebraic integersin the feld Q(id) and d is a square-free positive integer. Since studying thePicard modular groups will supply very useful examples for the research of thegeneral theory of complex hyperbolic discrete groups, it is an important subjectin the theory of complex hyperbolic geometry.
In this thesis we frstly studied the generators of the Picard modular groupsPU(2,1; Od)(d=2,3,7,11). We obtained a new system of fnite generators ofPU(2,1; O3) by using the method of continued fraction algorithm. For the others,we frstly found a new fnite generator system of the stabilizer subgroup fxing thepoint at infnity, and then with Zhao’s results we obtained a new system of fnitegenerators of PU(2,1; Od)（d=2,7,11）. About the high dimensional Picardmodular groups, we found a system of fnite generators of the three dimensionalPicard modular group PU(3,1; O3) by using the continued fraction algorithm.
According to Serre’s theorem, we call a group has Property (FA) if it is fnitelygenerated and does not split into nontrivial free product with amalgamation anddoes not admit any homomorphism onto Z. Wether a group has Property (FA)is an important problem in the feld of semisimple Lie groups.
Secondly we studied the Property (FA) of the Gauss-Picard modular groupPU(2,1; O1) and the sister of the Eisenstein-Picard modular group PU(2,1; O3).Eisenstein-Picard modular group and its sister group are the only two fundamentalgroups of the arithmetic, cusped, complex hyperbolic orbifolds of minimal volume.In2007, Stover proved that Eisenstein-Picard modular group PU(2,1; O3), andhe asked wether the other Picard modular groups have Property (FA)? We provedthat the Gauss-Picard modular group has Property (FA), which is a partiallypositive answer for Stover’s question. We proved the sister of the Eisenstein-Picard modular group has Property (FA). It means that the Eisenstein-Picardmodular group and the sister of the Eisenstein-Picard modular group, as the onlytwo arithmetic fundamental groups of cusped, complex hyperbolic orbifolds ofminimal volume, have the same property of Property (FA).
In the theme of geometry, an important problem is to study the relationshipsof topology properties and geometric properties under a fne geometry structure of a space. A spherical CR structure on a3-dimensional manifold is a maximalsystem of coordinate charts in S3, and the transformation for the same domainis the restriction of an element in the group PU(2,1). It is well known that allof the3-dimensional manifolds can equipped with contact structure and naturallythe CR structure. This fact suspect that spherical CR structure should be a basicproblem in the study of3-dimensional manifolds, therefore, a natural question isto ask when a3-dimensional manifold has a spherical CR structure.
Lastly, we researched the spherical structure on the complement of fgure eightknot M=S3K, which is a3-dimensional hyperbolic manifold. Up to conju-gation, Falbel constructed three diferent representations ρi(i=1,2,3) from thefundamental group of the complement of fgure eight knot π1(M) to PU(2,1) bythe method of gluing two tetrahedra. He studied the frst representation ρ1moreprecisely, proved that the frst representation is discrete and there is a branchedspherical CR structure, more interesting, the limit set of the group ρ1(π1(M)) isthe whole boundary S3of the complex hyperbolic space. In this paper, we s-tudied the second representation ρ2and proved that it is discrete and the groupρ2(π1(M)) contained in the Picard modular group PU(2,1; O7) as an infnity in-dex subgroup, and there is a branched spherical CR structure associated to thisrepresentation. Besides, we compared the commonality and diferences of the threerepresentations. It is easy to see that the biggest commonality is that all of thegroups ρi(π1(M))(i=1,2,3) are arithmetic, since ρ1(π1(M)) is contained in theEisenstein-Picard modular group PU(2,1; O3) as an infnity index subgroup andρ3(π1(M)) is conjugated to ρ2(π1(M)) which means that it is also a subgroup ofinfnity index in the Picard modular group PU(2,1; O7). The biggest diferenceis that the ρ1(π1(M)) contains a surface subgroup ρ1(F2) of infnity index (F2isthe fundamental group of the once punctured torus since the complement of thefgure eight knot is the once punctured torus bundle over a circle), and ρi(F2) is afnite index (index three) subgroup ρi(π1(M)) when i=2,3, which means that thesecond and third representations are almost determined by the surface subgroupcontained in the fundamental group of the manifold.
本论文的第一个工作是研究了Picard模群PU(2,1;Od)（d=2,3,7,11）的生成子。利用推广的连分式算法，我们找到了PU(2,1;O3)的一个新的有限生成子系。对于其他的Picard模群PU(2,1;Od)（d=2,7,11），我们通过首先找到固定无穷远点的稳定子群的一组生成子系，然后将其与Zhao的结果进行比较，从而获得PU(2,1;Od)（d=2,7,11）的一组新的生成子系。另外，对于高维的Picard模群，我们也通过利用使用连分式算法找到了PU(3,1;O3)一套生成子系。
根据Serre的定义，当一个有限生成群既不能分裂成非平凡的含有合并的乘积，又不具有到Z的满同态时，则称该群具有性质（FA）.考虑一个群是否具有性质（FA）是半单李群的一个重要问题。
本论文的第二个工作是研究了复双曲算术格Gauss-Picard模群PU(2,1;O1)和Eisenstein-Picard模群PU(2,1;O3)的姊妹群的性质（FA）. Eisenstein-Picard模群及其姊妹群是仅有的两个具有极小体积的带尖点的复双曲轨形的基本群，它们都是算术群。2007年，Stover证明了Eisenstein-Picard模群PU(2,1;O3)具有性质（FA），并问其他Picard模群PU(2,1;Od)是否也具有性质（FA）？我们证明了PU(2,1;O1)具有性质（FA）和Eisenstein-Picard模群PU(2,1; O3)的姊妹群具有性质（FA），这是对Stover提出的问题的部分肯定回答。该结果表明，作为仅有的两个具有极小体积的带尖点复双曲轨形的基本群，Eisenstein-Picard模群和Eisenstein-Picard模群的姊妹群都具有性质（FA）.
几何中一个很重要的问题就是研究空间拓扑性质和其在某种好的几何结构下的几何性质之间的联系。三维流形上的一个球面CR结构指的是一套S3中的坐标卡，其中的坐标变换是PU(2,1)中的元素在开集上的限制。众所周知，所有可以配置接触结构的三维流形很自然地可以配置CR结构。这个事实使得我们预想球面CR结构应该是研究三维流形的一个基本问题，因此，一个自然的问题就是哪些三维流形具有球面CR结构。
本论文的最后一个工作是研究了三维双曲流形八字结的补M=S3K的球面CR结构,其中K是八字结。通过粘合四面体的方法， Falbel构造了八字结的补的基本群π1(M)到PU(2,1)的三个互不共轭的表示ρi(i=1,2,3)。他详细研究了第一个表示ρ1，证明了该表示的离散性以及对应的球面CR结构的存在性，有意思的是该表示下的像群ρ1(π1(M))作用在复双曲空间上的极限集是整个边界S3.在本文中，我们主要研究了第二个表示ρ2，证明了该表示是离散的并且ρ2(π1(M))是Picard模群PU(2,1; O7)的无限次子群，证明了在该表示下存在一个分歧球面CR结构。我们还比较了这三个表示之间的异同：最大的相同之处是它们的像群ρi(π1(M))都是算术的，ρ1(π1(M))是Eisenstein-Picard模群PU(2,1; O3)的无限次子群，而ρ3(π1(M))和ρ2(π1(M))是共轭的，所以也可以看成是Picard模群PU(2,1;O7)的无限次子群；最大的不同之处则是ρ1(π1(M))含有一个无限次的曲面子群ρ1(F2)（F2是单洞环面的基本群，这是因为八字结的补是圆环上的单洞环面丛），而该曲面群在第二和第三个表示下的像群则是有限次（三次）的子群，该结果表明，八字结的补的基本群到PU(2,1)的三个表示中的第二个和第三个表示几乎是由其曲面子群决定的。
As the generalization of classic modular group PSL(2, Z) in high dimensionalcomplex hyperbolic space, Picard modular groups PU(2,1; Od) are the simplestcomplex hyperbolic arithmetic lattices, where Odis the ring of algebraic integersin the feld Q(id) and d is a square-free positive integer. Since studying thePicard modular groups will supply very useful examples for the research of thegeneral theory of complex hyperbolic discrete groups, it is an important subjectin the theory of complex hyperbolic geometry.
In this thesis we frstly studied the generators of the Picard modular groupsPU(2,1; Od)(d=2,3,7,11). We obtained a new system of fnite generators ofPU(2,1; O3) by using the method of continued fraction algorithm. For the others,we frstly found a new fnite generator system of the stabilizer subgroup fxing thepoint at infnity, and then with Zhao’s results we obtained a new system of fnitegenerators of PU(2,1; Od)（d=2,7,11）. About the high dimensional Picardmodular groups, we found a system of fnite generators of the three dimensionalPicard modular group PU(3,1; O3) by using the continued fraction algorithm.
According to Serre’s theorem, we call a group has Property (FA) if it is fnitelygenerated and does not split into nontrivial free product with amalgamation anddoes not admit any homomorphism onto Z. Wether a group has Property (FA)is an important problem in the feld of semisimple Lie groups.
Secondly we studied the Property (FA) of the Gauss-Picard modular groupPU(2,1; O1) and the sister of the Eisenstein-Picard modular group PU(2,1; O3).Eisenstein-Picard modular group and its sister group are the only two fundamentalgroups of the arithmetic, cusped, complex hyperbolic orbifolds of minimal volume.In2007, Stover proved that Eisenstein-Picard modular group PU(2,1; O3), andhe asked wether the other Picard modular groups have Property (FA)? We provedthat the Gauss-Picard modular group has Property (FA), which is a partiallypositive answer for Stover’s question. We proved the sister of the Eisenstein-Picard modular group has Property (FA). It means that the Eisenstein-Picardmodular group and the sister of the Eisenstein-Picard modular group, as the onlytwo arithmetic fundamental groups of cusped, complex hyperbolic orbifolds ofminimal volume, have the same property of Property (FA).
In the theme of geometry, an important problem is to study the relationshipsof topology properties and geometric properties under a fne geometry structure of a space. A spherical CR structure on a3-dimensional manifold is a maximalsystem of coordinate charts in S3, and the transformation for the same domainis the restriction of an element in the group PU(2,1). It is well known that allof the3-dimensional manifolds can equipped with contact structure and naturallythe CR structure. This fact suspect that spherical CR structure should be a basicproblem in the study of3-dimensional manifolds, therefore, a natural question isto ask when a3-dimensional manifold has a spherical CR structure.
Lastly, we researched the spherical structure on the complement of fgure eightknot M=S3K, which is a3-dimensional hyperbolic manifold. Up to conju-gation, Falbel constructed three diferent representations ρi(i=1,2,3) from thefundamental group of the complement of fgure eight knot π1(M) to PU(2,1) bythe method of gluing two tetrahedra. He studied the frst representation ρ1moreprecisely, proved that the frst representation is discrete and there is a branchedspherical CR structure, more interesting, the limit set of the group ρ1(π1(M)) isthe whole boundary S3of the complex hyperbolic space. In this paper, we s-tudied the second representation ρ2and proved that it is discrete and the groupρ2(π1(M)) contained in the Picard modular group PU(2,1; O7) as an infnity in-dex subgroup, and there is a branched spherical CR structure associated to thisrepresentation. Besides, we compared the commonality and diferences of the threerepresentations. It is easy to see that the biggest commonality is that all of thegroups ρi(π1(M))(i=1,2,3) are arithmetic, since ρ1(π1(M)) is contained in theEisenstein-Picard modular group PU(2,1; O3) as an infnity index subgroup andρ3(π1(M)) is conjugated to ρ2(π1(M)) which means that it is also a subgroup ofinfnity index in the Picard modular group PU(2,1; O7). The biggest diferenceis that the ρ1(π1(M)) contains a surface subgroup ρ1(F2) of infnity index (F2isthe fundamental group of the once punctured torus since the complement of thefgure eight knot is the once punctured torus bundle over a circle), and ρi(F2) is afnite index (index three) subgroup ρi(π1(M)) when i=2,3, which means that thesecond and third representations are almost determined by the surface subgroupcontained in the fundamental group of the manifold.
引文
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[94] Dan Y. Elliptic points of the Picard modular group. Monatsh. Math.,2009,156(4):391–396