Research on Jorgensen groups and classical Schottky groups

约根森群和经典肖特基群的研究

• 批准号: 16540147
• 负责人: SATO Hiroki
• 金额: $ 2.37万
• 依托单位: Shizuoka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540147/
• 关键词: Jorgensen group; Jorgensen number; classical Schottky group; Schottky group; Kleinian group; Schottky space; Jorgensen's inequality; uniformization of a Riemann surface; 三角群; 3次元双曲多様体

We have studied the following four themes from 2004 to 2006. 1. Jorgensen groups. 2. The Whitehead link group. 3. Jorgensen numbers. 4. Classical Schottky groups.1. Jorgensen groups. A Jorgensen group is a non-elementary two-generator discrete group whose Jorgensen number is one. There are two types-parabolic type and elliptic type-for Jorgensen groups. Here we considered of parabolic type. There are three types for Jorgensen groups of parabolic type (finite type, countably infinite type and uncountably infinite type). We found all Jorgensen groups during 2004, and 2006. The results were talked at the Mathematical Society of Japan, the RIMS, Kyoto university and the International congress of Mathematicians in Joensuu, Finland. Furthermore, they were published from Osaka J. Math. Kodai Math. J. and Comput Methods Funct. Theory.2. The Whitehead link group. We proved that the Jorgensen number of the Whitehead link is two. Therefore the Whitehead link is not a Jorgensen group. This result was published at Boletin de Soc. Mat. Mex.3. Jorgensen numbers. We showed that for every natural number and every real number greater than four there exists a classical Schottky group whose Jorgensen number is the number.4. Classical Schottky groups. We did not obtain any meaningful results.We are planning the following : (1) To study structures of 3-manifolds represented by Jorgensen groups (2) To find all Jorgensen groups of elliptic type (3) To find Jorgensen numbers of (classical) Schottky groups and (4) To study a uniformization of a Riemann surfaces by classical Schottky groups.

从2004年到2006年，我们研究了以下四个主题。1. 约根森组。2. Whitehead链接组。3. 约根森数字。4. 古典肖特基组。约根森组。Jorgensen群是Jorgensen数为1的非初等双生离散群。Jorgensen群有抛物线型和椭圆型两种类型。这里我们考虑抛物线型。抛物型Jorgensen群有三种类型（有限型、可数无限型和不可数无限型）。我们在2004年和2006年找到了所有乔根森群。研究结果在日本数学学会、RIMS、京都大学和芬兰约恩苏举行的国际数学家大会上发表。此外，他们发表于大阪J.数学。恒大数学。J.和计算方法函数。Theory.2。Whitehead链接组。我们证明了怀特黑德链的乔根森数是2。因此怀特黑德家族不是乔根森家族。这一结果发表在《社会科学公报》上。Mex.3垫。约根森数字。我们证明了对于每一个自然数和每一个大于4的实数，都存在一个Jorgensen数为4的经典肖特基群。古典肖特基乐团。我们没有得到任何有意义的结果。我们计划：(1)研究由Jorgensen群表示的3-流形的结构(2)找到所有椭圆型Jorgensen群(3)找到（经典）Schottky群的Jorgensen数(4)研究经典Schottky群对Riemann曲面的均匀化。

项目成果

Jorgensen groups of parabolic type III (uncountably infinite case)

抛物线类型 III 的约根森群（不可数无限情况）

• 发表时间: 2005
• 期刊: Kodai Math. J. 28 2
• 作者: H.Sato;C.L;M.Oichi
• 通讯作者: M.Oichi

Jorgensen groups of parabolic type I (finite case)

I 型抛物线乔根森群（有限情况）

• 发表时间: 2005
• 期刊: Comput. Methods Funct. (Theory 5) 2
• 作者: H.Sato;C.L;M.Oichi
• 通讯作者: M.Oichi

Recent development of cohomological dimension theory Existence and applicaion of Edwards-Walsh resolutions

上同调维数论的最新发展爱德华兹-沃尔什决议的存在和应用

• 发表时间: 2004
• 期刊: Sugaku Expositions 17・2
• 作者: Hiroki Sato;C.Li;M.Oichi;Hiroki Sato;Akira Koyama
• 通讯作者: Akira Koyama

Jorgensen groups of parabolic type II (Countably infinite case)

抛物型 II 型乔根森群（可数无限情况）

• 发表时间: 2004
• 期刊: Osaka.J.Math. 41
• 作者: Hiroki Sato;C.Li;M.Oichi
• 通讯作者: M.Oichi

Complexification of lambda length as parameter for SL(2,C) representation space of punctured surface groups

将 lambda 长度复作为刺孔表面群 SL(2,C) 表示空间的参数

• 发表时间: 2004
• 期刊: London Math.Soc. 70
• 作者: T.Nakanishi;M.Naatanen
• 通讯作者: M.Naatanen