双曲空间上等距子群的离散性与四点对的模空间
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摘要
双曲几何和Klein群在低维拓扑,动力系统,黎曼几何等学科中有着重要的应用.Poincar(?),Fricke和Klein对Klein群理论的发展始于十九世纪末.1960后,随着拟共形理论的成熟,L.V.Ahlfors和L.Bers使得Klein群理论成为复分析中Teichm(u|¨)ller理论分支中一个活跃的领域.然后,在1980前后,W.P.Thurston革命性地发展了双曲几何和Klein群理论,使得双曲流形和Klein群理论吸引了众多拓扑学家的注意.现在,Klein群已经获得了巨大的发展.例如,L.V.Ahlfors提出了有限生成Klein群的零测度猜想;G.D.Mostow建立了高维有限体积双曲流形的刚性定理;D.P.Sullivan研究了Klein群作用在双曲空间边界上的动力行为;W.P.Thurston讨论了三维流形的分类及曲面叶状结构的分类.
在Poincar(?)等人发展Fuchs群和Klein群理论的同时,Picard也对复双曲Klein群进行了研究.尽管在Picard之后,又有G.Giraud和E.Cartan等人作了一些很重要的工作,但是复双曲几何的理论并没有像实双曲几何理论那样得到快速的发展.在S.Chen,L.Greenberg关于对称空间和G.D.Mostow关于复双曲空间的非算术格的构造的工作之后,复双曲离散子群的兴趣才被重新激发起来.许多著名数学家开始进行复双曲几何的研究,其中W.M.Goldman,R.Schwartz,J.R.Parker,E.Falbel等都在复Klein群方面做了许多非常重要的工作.
本文的主要目的是讨论复双曲空间和实双曲空间上等距离散子群的性质和复双曲空间及其边界上四点对的模空间,其中三个点在复双曲空间的边界上而另一点在复双曲空间空间中.我们主要得到了如下结果:
1.我们讨论了一个生成元为正则椭圆的二元生成复双曲群是离散群的必要条件,建立了相应的。J(?)rgensen不等式,当正则椭圆元素固定Lagrangian平面且具有实迹时,我们用此正则椭圆元素的迹及不动点刻画了J(?)rgensen不等式.
2.利用Klein-Maskist组合定理和椭圆元素所固定的复线或点之间的距离,我们得到了两个椭圆元素生成离散自由群的充分条件.类似的,我们也得到了两个抛物元素生成离散自由群的充分条件.
3.我们讨论了复双曲等距群n维子群的离散准则,利用恒等元素位于PU(1,n)中分别由斜驶元素和正则椭圆元素所构成的两个开集的边缘的事实及PU(1,n)中的n-维子群在PU(1,n)中要么离散,要么稠密的性质.我们得到了三个关于PU(1,n)中n-维子群的离散准则.
4.上个世纪九十年代,J.W.Anderson提出了两个具有相同轴集合的有限生成的n维Klein群是否共约的公开问题.我们构造了一个反例说明了此问题的回答一般是否定的,并给出了具有相同轴集合的n维Klein群共约的充分条件.
5.我们研究了一点位于复双曲空间内部,三点位于复双曲空间边界上四点对的模空间,其中这四点是不同的点.利用Cantan角不变量和复交比我们构造了这个实维数为6的具体模空间形式.
Hyperbolic geometry and Kleinian groups has many important applications in geometry and topology,physics,dynamical systems,Riemannian geometry.The development of the theory of Kleinian groups was started at the end of the nineteenth-century by Poincar(?),Fricke and Klein.After 1960,as theories of quasi-conformal mapping matured, L.V.Ahlfors and L.Bers brought Kleinian groups theory to active area of complex analysis as a branch of Teichm(u|¨)ller theory.Finally,in about 1980,W.P.Thurston brought a revolution to hyperbolic geometry and Kleinian groups,so that hyperbolic manifolds and Kleinian groups attracted the attention of many topologists.Now,A great deal of advances have been obtained in theory of Kleinian groups.For example,L.V.Ahlfors made Ahlfors' Measure Conjecture for finitely generated Kleinian groups.G.D.Mostow proved a rigidity theorem for hyperbolic n-manifolds of finite volume.D.P.Sullivan studied the dynamics of Kleinian groups acting on the boundary of hyperbolic space.W.P. Thurston provided the classification of geometric 3-manifolds and the foliations structure on surface.
Complex hyperbolic Kleinian groups were first studied by Picard at about the same time as Poincar(?) was developing the theory of Fuchsian and Kleinian groups.In spite of work by several other people,including G.Giraud and E.Cartan,the complex hyperbolic theory did not develop as rapidly as the real hyperbolic theory.Later,work of S.Chen and L.Greenberg on symmetric spaces and work of G.D.Mostow on the construction of nonarithmetic lattices led to a resurgence of interest in complex hyperbolic discrete groups. Many famous mathematicians began to investicate the complex hyperbolic geometry and obtained many important results.
The main purpose of the present thesis is to disscuss some properties of subgroups of isometries of complex hyperbolic space or real hyperbolic space and describe the moduli space of quadruples of points in complex hyperbolic space together with its boundary. We obtain the following main results.
Firstly,the authors obtained a necessary condition for two-generator discrete groups of PU(1,n),and provid a J(?)rgensen's inequality for non-elementary subgroups of isometries of complex hyperbolic space generated by two elements,one of which is regular elliptic element.We give a J(?)rgensen's inequality for subgroups containing regular elliptic with real trace and preserving a Lagrangian plane.
Secondly,by using some version of Klein-Maskit combination theorem and the dis- tance between the complex lines or points fixed by two elliptic elements,we explore the conditions for two elliptic elements and two parabolic elements in PU(1,2).to generate discrete free group.
Thirdly,we discuss three discreteness criterions of n-dimensional subgroup G of PU(1,n).This result generalize some discreteness criterions established by J.Gilman, S.Yang and A.Fang.The basic idea in proving our results is to note that the sets of loxodromic elements or regular elliptic elements are both open facing the identity element. A dense theorem of S.Chen and L.Greenberg which said that n—dimension subgroup of SU(1,n) is dense or discrete in SU(1,n) also play an important role in our proof.
Furthermore,In the late 90s of last century,J.W.Anderson asked whether two finitely generated Kleinian groups G_1,G_2 C Isom(H~n) with the same set of axes are commensurable.We give a very simple example to show the answer of Anderson's question is negative in general.We disscuss the conditons which imply two Kleinian groups are commensurable.
Finally,we study the moduli space of quadruples of points with three ideal points in boundary and one point in complex hyperbolic space.We constuct a moduli space of real dimension 6 by using Cartan angular invariant and generalized complex cross ratio.
在Poincar(?)等人发展Fuchs群和Klein群理论的同时,Picard也对复双曲Klein群进行了研究.尽管在Picard之后,又有G.Giraud和E.Cartan等人作了一些很重要的工作,但是复双曲几何的理论并没有像实双曲几何理论那样得到快速的发展.在S.Chen,L.Greenberg关于对称空间和G.D.Mostow关于复双曲空间的非算术格的构造的工作之后,复双曲离散子群的兴趣才被重新激发起来.许多著名数学家开始进行复双曲几何的研究,其中W.M.Goldman,R.Schwartz,J.R.Parker,E.Falbel等都在复Klein群方面做了许多非常重要的工作.
本文的主要目的是讨论复双曲空间和实双曲空间上等距离散子群的性质和复双曲空间及其边界上四点对的模空间,其中三个点在复双曲空间的边界上而另一点在复双曲空间空间中.我们主要得到了如下结果:
1.我们讨论了一个生成元为正则椭圆的二元生成复双曲群是离散群的必要条件,建立了相应的。J(?)rgensen不等式,当正则椭圆元素固定Lagrangian平面且具有实迹时,我们用此正则椭圆元素的迹及不动点刻画了J(?)rgensen不等式.
2.利用Klein-Maskist组合定理和椭圆元素所固定的复线或点之间的距离,我们得到了两个椭圆元素生成离散自由群的充分条件.类似的,我们也得到了两个抛物元素生成离散自由群的充分条件.
3.我们讨论了复双曲等距群n维子群的离散准则,利用恒等元素位于PU(1,n)中分别由斜驶元素和正则椭圆元素所构成的两个开集的边缘的事实及PU(1,n)中的n-维子群在PU(1,n)中要么离散,要么稠密的性质.我们得到了三个关于PU(1,n)中n-维子群的离散准则.
4.上个世纪九十年代,J.W.Anderson提出了两个具有相同轴集合的有限生成的n维Klein群是否共约的公开问题.我们构造了一个反例说明了此问题的回答一般是否定的,并给出了具有相同轴集合的n维Klein群共约的充分条件.
5.我们研究了一点位于复双曲空间内部,三点位于复双曲空间边界上四点对的模空间,其中这四点是不同的点.利用Cantan角不变量和复交比我们构造了这个实维数为6的具体模空间形式.
Hyperbolic geometry and Kleinian groups has many important applications in geometry and topology,physics,dynamical systems,Riemannian geometry.The development of the theory of Kleinian groups was started at the end of the nineteenth-century by Poincar(?),Fricke and Klein.After 1960,as theories of quasi-conformal mapping matured, L.V.Ahlfors and L.Bers brought Kleinian groups theory to active area of complex analysis as a branch of Teichm(u|¨)ller theory.Finally,in about 1980,W.P.Thurston brought a revolution to hyperbolic geometry and Kleinian groups,so that hyperbolic manifolds and Kleinian groups attracted the attention of many topologists.Now,A great deal of advances have been obtained in theory of Kleinian groups.For example,L.V.Ahlfors made Ahlfors' Measure Conjecture for finitely generated Kleinian groups.G.D.Mostow proved a rigidity theorem for hyperbolic n-manifolds of finite volume.D.P.Sullivan studied the dynamics of Kleinian groups acting on the boundary of hyperbolic space.W.P. Thurston provided the classification of geometric 3-manifolds and the foliations structure on surface.
Complex hyperbolic Kleinian groups were first studied by Picard at about the same time as Poincar(?) was developing the theory of Fuchsian and Kleinian groups.In spite of work by several other people,including G.Giraud and E.Cartan,the complex hyperbolic theory did not develop as rapidly as the real hyperbolic theory.Later,work of S.Chen and L.Greenberg on symmetric spaces and work of G.D.Mostow on the construction of nonarithmetic lattices led to a resurgence of interest in complex hyperbolic discrete groups. Many famous mathematicians began to investicate the complex hyperbolic geometry and obtained many important results.
The main purpose of the present thesis is to disscuss some properties of subgroups of isometries of complex hyperbolic space or real hyperbolic space and describe the moduli space of quadruples of points in complex hyperbolic space together with its boundary. We obtain the following main results.
Firstly,the authors obtained a necessary condition for two-generator discrete groups of PU(1,n),and provid a J(?)rgensen's inequality for non-elementary subgroups of isometries of complex hyperbolic space generated by two elements,one of which is regular elliptic element.We give a J(?)rgensen's inequality for subgroups containing regular elliptic with real trace and preserving a Lagrangian plane.
Secondly,by using some version of Klein-Maskit combination theorem and the dis- tance between the complex lines or points fixed by two elliptic elements,we explore the conditions for two elliptic elements and two parabolic elements in PU(1,2).to generate discrete free group.
Thirdly,we discuss three discreteness criterions of n-dimensional subgroup G of PU(1,n).This result generalize some discreteness criterions established by J.Gilman, S.Yang and A.Fang.The basic idea in proving our results is to note that the sets of loxodromic elements or regular elliptic elements are both open facing the identity element. A dense theorem of S.Chen and L.Greenberg which said that n—dimension subgroup of SU(1,n) is dense or discrete in SU(1,n) also play an important role in our proof.
Furthermore,In the late 90s of last century,J.W.Anderson asked whether two finitely generated Kleinian groups G_1,G_2 C Isom(H~n) with the same set of axes are commensurable.We give a very simple example to show the answer of Anderson's question is negative in general.We disscuss the conditons which imply two Kleinian groups are commensurable.
Finally,we study the moduli space of quadruples of points with three ideal points in boundary and one point in complex hyperbolic space.We constuct a moduli space of real dimension 6 by using Cartan angular invariant and generalized complex cross ratio.
引文
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