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.

我们研究了2004年至2006年的以下四个主题。1。Jorgensen组。 2。怀特黑德链接组。 3。jorgensen数字。 4。古典肖特基集团1。 Jorgensen组。 Jorgensen组是一个非元素的两期离散组，其Jorgensen编号为一个。有两种类型的抛物性类型和椭圆形类型，即Jorgensen组。在这里，我们考虑了抛物线类型。抛物线类型的Jorgensen组有三种类型（有限类型，无限类型和无限的无限类型）。我们在2004年和2006年发现了所有Jorgensen群体。结果是在日本数学学会，边缘，京都大学和芬兰约瑟乌的国际数学家大会上进行的。此外，它们是从大阪J. Math出版的。 Kodai数学。 J.和计算方法功能。理论2。 Whitehead链接组。我们证明了Whitehead链接的Jorgensen数量为两个。因此，怀特黑德链接不是一个约翰逊组。该结果发表在Boletin de Soc。垫。 Mex.3。 Jorgensen数字。我们表明，对于每个自然数字，每个实际数字都大于四个，那里都有一个经典的Schottky群体，其Jorgensen数字为数字4。古典Schottky团体。我们没有获得任何有意义的结果。我们正在计划以下内容：（1）研究由Jorgensen组（2）代表的3个模型的结构，以找到所有椭圆类型（3）组的Jorgensen组（3）组，以查找（经典）Schottky组和（4）的Jorgensen数量，以研究riemann schottalsical Schottky的统一化。

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

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

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

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