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1、上海交通大学 硕士学位论文 复双曲空间中的离散群及复双曲流形的体积的相关问题 姓名：付丽 申请学位级别：硕士 专业：基础数学 指导教师：乃兵 20090101 ? ? ? M obius? ? ?M obius? ?M obius? ? ? ?Kamiya?Parker?54?PU(2,1)? ?Kamiya? Parker?PU(2,1)?PU(n,1)? ?PU(n,1)? ? ?Bergman?0.2589?n=2? ?Kamiya?Parker?PU(2,1)? ? ?1.?g?PU(n,1)?g(q) = q? A U(n1)?g?AI? 1/4?h PU(n,1)? ?h(q) ?
2、= q,rh?h? 0(h(q),gh(q)0(h1(q),gh1(q) r2 h ?1 + ?1 4?A I? 2 ?2 ? ?2.?g?PU(n,1)?g : (,v,u) (A,v+ t,u),?A I? |2(A I) + it| 2 2?A I? ? , = 1 + ?1 4?A I? 2 ?G? i ?z1= (1,v1,u1)? ? Hn(C)?G? GI?g?z1?Bergman? ?GI?g?f? ?(z1,g(z1) (z1,f(z1)?z1?(z1,g(z1)/2 ?f G I? ? ?3.?g?PU(n,1)?g : (,v,u) (A,v + t,u),?A I? 2
3、/9?G?PU(n,1)?G? ?A?Hn(C)/G? ?Bergman?e 1.29551? 0.2589? ? ii Discrete Groups In Complex Hyperbolic Space And The Problems On The Volumes Of The Manifold ABSTRACT M obius group has been quite important in the complex analysis for over a hundred years, and it is always a main branch and studied by a l
4、ot of famous mathematicians, who also applied m obius groups to the complex hyperbolic manifold. The criterion of the discrete condition for the m obius group is one of the main subjects in the m obius group study, and also, it has great infl uence on the manifold and the algebra property of the dis
5、crete groups. Recently, the re- search of the discreteness of the m obius groups in the complex hyperbolic space attracts many mathematicians at home and abroad. One important result is given by Shigeyasu Kamiya and John R.Parker in PU(2,1) whose subgroup is generated by a screw parabolic motion and
6、 another m obius transformation. Based on the research by Shigeyasu Kamiya and John R.Parker, this paper extends the results in PU(2,1) to that in PU(n,1) and gives further research on the criterion of the discrete condition for groups of complex hyperbolic isome- tries one of whose generators is a
7、screw parabolic motion in PU(n,1). Then we use this result to give a sub-horospherical region precisely invariant under the stabliser of the fi xed point of the screw parabolic motion in G. Besides, we also show that, given a discrete subgroup of PU(n,1), if its subgroup stablising a point on the bo
8、undary of complex hyperbolic space consists only of screw parabolic motions (and the identity), the precisely invariant horoball or sub- horospherical region contains a ball of a uniform size with Bergman radius of 0.2589 which doesnt intersect any of its images. We get the three main results in the
9、 paper: Theroem 1. Let g be a positively oriented screw parabolic element of PU(n,1) fi xing q. Let A PU(n 1) denote the rotational part of g and suppose that ?A I? 1/4. Let h be any element of PU(n,1) not projectively fi xing qand let rhdenote the radius of the isometric sphere of h. If 0(B(),AB()0
10、(B1(),AB1() r2 B ?1 + ?1 4?A I? 2 ?2 then is not discrete. Theroem 2. Let g be a positively oriented screw parabolic map, g : (,v,u) iii (A,v + t,u),?A I? 2/9?where ?A I? |2(A I) + it| 2 2?A I? ? , = 1 + ?1 4?A I? 2 is precisely invariant under Gin G. Let z1= (1,v1,u1) be any point of Hn(C). As G is
11、 discrete, there is an element of GI with the shortest Bergman translation length at z1. That is, if f is any other element of G I, then (z1,g(z1) (z1,f(z1). This means that the open ball centered at z1with radius (z1,g(z1)/2 doest intersect any of its images under G I. In this condition, we can get
12、 the result as following: Theroem 3. Let g be a positively oriented scew parabolic map as above, g : (,v,u) (A,v + t,u),?A I? 1/t? ?B?rB t?t? ?G?Shimizu?q?M obius ?G?G?q?M obius ?U?G?H?B H,? B(U) = U,?B GH,B(U) ? U = ?PU(2,1)? ?1976?Troels Jorgensen?57?M obius? ?Jorgensen? ?B.57?f,g?PU(2,1;R)? |tr2(f) 4| + |tr(fgf1g1) 2| 1 ?f?g? ?I? Gaven J.Martin?12?Jorgensen?2?n? ?11?Dirichlet? ?Jorgensen? ?Waterman59?9? Cliff ord? ?Jorgensen?Shigeyasu Kamiya?John R.Parker? ?M obius? 1 John R.Parker23?PU(2,1)?PU(n,1)?Heisenberg? ?Shigeyasu Kamiya?Jorhn R.Parker?54
