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.

我们从2002年到2003年研究了以下四个主题。 1. 乔根森群 2. 皮卡德群 3. 怀特海德群 4. 经典肖特基空间和乔根森数。 1. 乔根森群是一个非乔根森群。 -Jorgensen 数为 1 的初等二元离散群 这里我们考虑两种类型：抛物线型和椭圆型。抛物型乔根森群有三种类型（有限型、可数无限型和不可数无限型）： (1)我们找到了所有有限型乔根森群和所有可数无限型乔根森群。 2002 年，我们发现了所有不可数无限类型的 Jorgensen 群。我们发现了所有抛物型类型的 Jorgensen 群。结果（1）在国际会议上进行了讨论。 2002年北京数学家大会，结果(2)于2003年在北京大学发表。2.皮卡德群 我们为皮卡德群构造了一个新的基本区域，并利用该群的两个生成元找到了八种关系。这个结果发表在2003年柏林ISAAC会议记录上。 3.怀特海链接群我们证明了怀特海链接的乔根森数是二。因此Whitehead链不是Jorgensen群。我们在2002年的国际拓扑学会议上讨论了这个结果。 经典肖特基空间和Jorgensen数我们证明存在一个经典肖特基群，其Jorgensen数是给定的实数j【更大。大于或等于】4.我们将在今年夏天的国际势论会议上讨论这个结果。

项目成果

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）。

Hiroki Sato: "Jorgensen groups and the Picard group"Proceedings of The Third ISAAC International Congress. 1. 149-158 (2003)

Hiroki Sato：“Jorgensen 群和 Picard 群”第三届 ISAAC 国际大会论文集。

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。

Kazuo Akutagawa: "An obstruction to the positivity of relative Yamabe invariant"Math.Z. 243. 85-98 (2003)

Kazuo Akutakawa：“相对 Yamabe 不变量的积极性的障碍”Math.Z。

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

奥山佑介：“Nevanlinna、Siegel 和 Cremer”Indiana Univ.Math.J. 登场。