Research on Jorgensen groups and Schottky spaces

约根森群和肖特基空间的研究

• 批准号: 14540170
• 负责人: SATO Hiroki
• 金额: $ 2.11万
• 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: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540170/
• 关键词: Jorgensen group; Jorgensen number; Jorgensen's inequality; Whitehead link group; Schottky space; Schottky group; Kleinian group; Uniformization of Riemann surface; ホワイトヘッドリンク; ピカール群

We have studied the following four themes from 2002 to 2003. 1.Jorgensen groups. 2.The Picard group. 3.The Whitehead link group. 4.Classical Schottky spaces and Jorgensen number.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 obtained the following. (1)We found all Jorgensen groups of finite type and all Jorgensen groups of countably infinite type in 2002, and (2)we found all Jorgensen groups of uncountably infinite type in 2003. Consequently we found all Jorgensen groups of parabolic type. The results (1) was talked at the International congress of Mathematicians in Beijing in 2002, and the result (2) was talked at Peking University in 2003.2.The Picard group. We constructed a new fundamental region for the Picard group and we found eight relations for two generators of the group by using the fundamental region. This result was published in the Proceedings of the ISAAC Congress in Berlin in 2003.3.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. We talked this result at the Internatonal Conference of Topology in 2002.4.Classical Schottky spaces and Jorgensen number. We showed that there exists a classical Schottky group whose Jorgensen number is a given real number j【greater than or equal】4. We will talk this result at the International Conference of Potential Theory this summer.

我们研究了2002年至2003年的以下四个主题。1。Jorgensen组。 2. Picard组。 3. Whitehead链接组。 4.古典Schottky空间和Jorgensen编号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 - 促销链类型和椭圆形类型，即Jorgensen组。在这里，我们考虑了抛物线类型。抛物线类型（有限类型，乡村无限类型和无限无限类型）的Jorgensen组有三种类型。我们获得了以下内容。 （1）我们在2002年发现了所有有限类型和所有乡村无限类型的Jorgensen群体，以及（2）我们在2003年发现了所有无限无限类型的Jorgensen群体。结果（1）在2002年的北京国际大会上在2002年的国际数学大会上进行了讨论，结果（2）在2002年聊天。我们为PICARD组建造了一个新的基本区域，并使用基本区域找到了该组的两个发电机的八个关系。该结果发表在2003年柏林的艾萨克大会会议录中。WhiteheadLink Group。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。我们在2002.4的国际拓扑会议上谈到了这一结果。古典肖特基空间和Jorgensen号码。我们表明，存在一个经典的肖特基群体，其jorgensen的数字是一个给定的实际数字j j【j【大于或等于】4。我们将在今年夏天的国际潜在理论会议上讨论这个结果。

项目成果

Hiroki Sato: "The Jorgensen number of the Whitehead link"RIMS koukyuroku, Kyoto Univ.. 1270. 77-83 (2002)

Hiroki Sato：“Whitehead 链接的 Jorgensen 数”RIMS koukyuroku，京都大学. 1270. 77-83 (2002)

Yusuke Okuyama: "Nevanlinna, Siegel, and Cremer"Indiana Univ.Math.J.. (to appear). (2004)

Yusuke Okuyama：“Nevanlinna、Siegel 和 Cremer”Indiana Univ.Math.J.（待出场）。

Toshihiro Nakanishi: "Complexification of lambda length as parameter for SL(2,C) representation space of punctured surface groups,"J.London Math.Soc.. (to appear). (2004)

Toshihiro Nakanishi：“将 lambda 长度作为穿孔表面群 SL(2,C) 表示空间的参数的复数”，J.London Math.Soc..（即将出现）。

Toshihiro Nakanishi: "Complexification of lambda length as parameter for SL(2,C) representation space of punctured surface groups"J.London Math.Soc.. (to appear). (2004)

Toshihiro Nakanishi：“作为穿孔表面群 SL(2,C) 表示空间参数的 lambda 长度的复数”J.London Math.Soc..（即将出现）。

Hiroki Sato: "Jorgensen groups and the Picard group"Proc. The Third ISAAC International Congress. 1. 249-258 (2002)

佐藤弘树：“乔根森群和皮卡德群”Proc。