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.

在2002至2003年期间，我们研究了以下四个主题。1.乔根森集团。2.皮卡德集团。3. Whitehead链接组4.经典Schottky空间与Jorgensen数1. Jorgensen群Jorgensen群是一个非初等的两生成元离散群，其Jorgensen数为1。Jorgensen群有抛物型和椭圆型两种类型。这里我们考虑抛物型。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）。我们获得了以下结果。(1)We在2002年发现了有限型的所有Jorgensen群和可数无限型的所有Jorgensen群;（2）在2003年发现了不可数无限型的所有Jorgensen群。因此，我们发现所有乔根森组的抛物型。结果（1）于2002年在北京国际数学家大会上发表，结果（2）于2003年在北京大学发表。我们构造了Picard群的一个新的基本区域，并利用这个基本区域找到了Picard群的两个生成元之间的八个关系。这一结果发表在2003年柏林的ISAAC大会上。证明了Whitehead环的Jorgensen数为2。因此，Whitehead链接不是Jorgensen群。我们在2002年国际拓扑会议上讨论了这个结果。4.经典Schottky空间和Jorgensen数。我们证明了存在一个经典的Schottky群，其Jorgensen数是一个给定的真实的数j[大于或等于]4。我们将在今年夏天的国际势理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）。

Hironori Kumura: "A note on the absence of eigenvzlues on negatively curved manifolds"Kyushu J. Math.. 56. 109-121 (2002)

Hironori Kumura：“关于负弯曲流形上不存在特征值的注释”Kyushu J. Math.. 56. 109-121 (2002)

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）。

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）。

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

芥川一夫：“相对山边不变量的积极性的障碍”数学。