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.

2002年到2003年我们研究了以下四个主题。 1.Jorgensen群。 2.皮卡德群。 3.怀特海链接群。 4.经典肖特基空间和约根森数。1.约根森群。乔根森群是乔根森数为 1 的非初等二元离散群。 Jorgensen 群有抛物线型和椭圆型两种类型。这里我们考虑的是抛物线型。抛物型乔根森群有三种类型（有限型、可数无穷型和不可数无穷型）。我们得到了以下内容。 (1)我们在2002年找到了所有有限类型的Jorgensen群和所有可数无限类型的Jorgensen群，并且(2)我们在2003年找到了所有不可数无限类型的Jorgensen群。因此我们找到了所有抛物型类型的Jorgensen群。结果(1)是2002年在北京国际数学家大会上发表的，结果(2)是2003年在北京大学发表的。2.皮卡德小组。我们为皮卡德群构建了一个新的基本区域，并利用该基本区域找到了该群的两个生成元的八种关系。这一结果发表在2003年柏林ISAAC大会论文集上。3.怀特海德链接组。我们证明了怀特海链接的乔根森数是二。因此怀特海链接不是乔根森群。我们在2002年4月的国际拓扑会议上谈到了这个结果。经典肖特基空间和约根森数。我们证明存在一个经典肖特基群，其乔根森数是给定的实数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。