【作者】 王桦；

【导师】 王仙桃；

【作者基本信息】 湖南大学 ， 基础数学， 2004， 硕士

【摘要】 本文主要讨论了n维Mbius变换群的一些性质。具体安排如下： 第一章我们主要介绍所研究问题的一些背景，给出了我们得到的主要结果。 第二章介绍了有关Mbius变换的一些基本概念及性质，包括Clifford代数、Clifford矩阵、Mbius变换的分类等。 第三章利用二元生成Mbius群<f，g>纳稍豧，g的迹在C×C×C上定义了三元数组（β（f），β（g），γ（f，g））：讨论了二元生成Mbius子群<f，g>睦肷⑿浴⒊醯刃杂肴榈墓叵担玫搅思螮的一些性质，给出了集合D∩E₁的具体刻划及二元生成初等离散Mbius群所对应的三元数组的一些性质，并利用代数方法证明了集合E和D∪E均是C×C×C上的闭集。 Beardon A．F．等讨论了二维Mbius群与一维Mbius群的共轭关系，得到了一个充要条件及一些充分条件。在第四章中，我们继续讨论此问题，得到了二维Mbius子群与一维子群共轭的五个充分必要条件，从而推广了已有的相关结果。同时我们把所得结果推广到了高维情形中，建立了一条高维Mbius群与一维Mbius子群共轭的充分必要条件，所得结果推广了Maskit B．等的已有讨论。 在第五章中，我们首先推广了Hersonsky S．、Leutbecher A．、Shimizu H．、Ohtake H．等建立的关于含有严格抛物元素的高维Mbius群的不等式，得到了一个关于含有m阶严格抛物元素的高维Mbius子群的不等式，给出了这个不等式的两个应用。更多还原

【Abstract】 The main purpose of this thesis is to investigate properties of n-dimensional Mobius groups. This thesis is arranged as follows.Chapter 1 provides some background information about Mobius groups and statement of our main results.In Chapter 2, we introduce some basic concepts and properties of Mobius groups, including the Clifford algebra, Clifford matrices , classification of elements in M(Rn) etc.In Chapter 3, by using the traces of the generators / and g of two-generator Mobius subgroup (f, g), we define three-number sets (β(f),β(g),r(f, g)) in C× C× C and investigate the relationship among discreteness, elementariness and three-number sets. Some properties of the set E in C x C x C are given, and a precise description of the set D E1 and some properties of the three-number sets corresponding to two-generator elementary discrete Mobius subgroups are presented. Moreover, by using algebraic method, we prove that the sets E and D E are closed in C x C x C.Beardon A.F. etc discussed the conjugation between a subgroup in SL(2, C) and one in SL(2, R) and obtained a sufficient and necessary condition and some other sufficient conditions. In Chapter 4, we study this problem further and prove five necessary and sufficient conditions for a subgroup in SL(2, C) to be conjugate to one in SL(2,R). These generalize the known corresponding results. Also we generalize these discussions into the higer dimensional cases and one necessary and sufficient condition for a group in SL(2,Tn} to be conjugate to one in SL(2,R). Our result generalizes those obtained by Maskit B. etc.In Chapter 5, after by weakening the assumptions of the inequalities about high dimensional Mobius subgroups containing strictly parabolic element obtained by Hersonsky S., Leutbecher A., Ohtake H., Shimizu H. etc, we establish a new one. Finally, two applications of this inequality are given.更多还原