【作者】 杨芳；

【导师】 蒋月评；

【作者基本信息】 湖南大学 ， 数学， 2024， 博士

【摘要】 复双曲格是复双曲空间的全纯等距群PU(n,1)中体积有限的离散子群,与复几何、代数几何、数论、低维拓扑等有紧密的联系.关于复双曲格的研究,一个非常重要的问题是构造复双曲非算术格.目前,我们只知道在PU(2,1)和PU(3,1)中的有限个复双曲非算术格.当n≥4时,PU(n,1)中是否存在复双曲非算术格仍然是一个公开问题.最近,Deraux,Parker和Paupert构造了 PU(2,1)中一族新的复双曲非算术格.本文的一个工作是研究这些新的复双曲非算术格的C-Fuchs子群的算术性.n维Picard模群PU(n,1;Od)是PU(n,1)中矩阵元素属于Od中的子群,其中Od是有理数域上的虚二次扩域Q((?))的代数整环.Picard模群是一类重要的复双曲算术格.本文的另一个工作是研究Picard模群的表示和Fuchs子群.本文的主要工作总结如下:1.研究了 PU(2,1)中一些非算术格的C-Fuchs子群的算术性.通过选取Cayley变换,将这些C-Fuchs子群变换为SL(2,R)中相应的Fuchs群,从而得到这些C-Fuchs子群的算术性.我们的结果是对Wells在其博士论文中提出的一个关于复双曲非算术格的C-Fuchs子群算术性的问题的肯定回答.2.研究了二维Eisenstein-Picard模群PU(2,1;O3)的一个无限指数子群K的Fuchs子群.从K中的一个抛物元素出发,找到了该抛物元素固定的实平面,从而获得了K的一个R-Fuchs子群.显然,这也是二维Eisenstein-Picard模群的一个R-Fuchs子群.用类似的方法,我们也得到了二维Eisenstein-Picard模群的一个C-Fuchs 子群.3.研究了三维Gauss-Picard模群PU(3,1;O1)的一个表示,其中O1=Z[i].利用Mark和Paupert在研究二维Picard模群的群表示时提出的方法,首先得到了PU(3,1;O1)中固定无穷远点的稳定子群的表示.然后根据该稳定子群的一个粗糙基本域,证明了 PU(3,1;O1)的覆盖深度至多为4,并且找到了该粗糙基本域中深度至多为4的Z[i]-有理点,同时给出了 PU(3,1;O1)的表示.更多还原

【Abstract】 Complex hyperbolic lattices are discrete subgroups of finite volume in the holomorphic isometry group PU(n,1)of complex hyperbolic spaces.They are closely connected with complex geometry,algebraic geometry,number theory,low dimensional topology,and so on.On the study of complex hyperbolic lattices,a very important problem is to construct complex hyperbolic non-arithmetic lattices.At present,we only know finite complex hyperbolic non-arithmetic lattices in PU(2,1)and PU(3,1).For n≥4,whether there exist complex hyperbolic non-arithmetic lattices remains an open question.Recently,Deraux,Parker and Paupert produced a family of new complex hyperbolic non-arithmetic lattices in PU(2,1).One contribution of this thesis is to study the arithmeticity of C-Fuchs subgroups of these new non-arithmetic lattices.The n-dimensional Picard modular groups PU(n,1;Od)are subgroups in PU(n,1)with entries in Od,where Od is the ring of algebraic integers of the imaginary quadratic extension Q((?))over the rational number field.Picard modular groups are a class of the significant complex hyperbolic arithmetic lattices.Another contribution of the thesis is to investigate the presentations and Fuchs subgroups of Picard modular groups.The main results of this thesis can be summarized as follows.We firstly investigated the arithmeticity of C-Fuchs subgroups of some non-arithmetic lattices.By choosing Cayley transformations,we transformed these C-Fuchs subgroups into the corresponding Fuchs groups in SL(2,R),and thus obtained the arithmeticity of these CFuchs subgroups.Our result gave a positive answer to a question proposed by Wells in his doctoral thesis in terms of the arithmeticity of C-Fuchs subgroups of complex hyperbolic non-arithmetic lattices.Secondly,we studied the Fuchs subgroups of a subgroup K of infinite index of the twodimensional Eisenstein-Picard modular group PU(2,1;O3).We began with a pure parabolic element in K,found a real plane fixed by the pure parabolic element,and thus had a R-Fuchs subgroup of K.Obviously,it is also a R-Fuchs subgroup of the two-dimensional EisensteinPicard modular group.Applying a similar approach,we also got a C-Fuchs subgroup of the two-dimensional Eisenstein-Picard modular group.Lastly,we explored a presentation for the three-dimensional Gauss-Picard modular group PU(3,1;O1),where O1=Z[i].Using the method presented by Mark and Paupert in the study of presentations for the two-dimensional Picard modular group,we had a presentation for the stabilizer subgroup that fixes the point at infinity in PU(3,1(O1).Then by a coarse fundamental domain for the stabilizer subgroup,we demonstrated that the covering depth of PU(3,1;O1)is at most 4,found the Z[i]-rational points whose depth is at most 4 in this coarse fundamental domain and gave a presentation for PU(3,1;O1).更多还原