Research on Schottky groups and Schottky spaces

肖特基群和肖特基空间研究

• 批准号: 10640158
• 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: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640158/
• 关键词: Schottky group; Schottky space; Jorgensen group; Jorgensen's inequality; Kleinian group; Teichmuller space; hyperbolic space; Hausdorff dimension; シヨツトキイ群; シヨツトキイ空間; 調和写像

We have studied the following three themes from 1998 to 1999.1. Classical Schottky space of real type.2. Jorgensen numbers.3. Jorgensen groups.1. Classical Schottky space of real type. We classified the classical Schottky space of real type of genus two into eight types. We represented the shape of their spaces and found generators for the Schottky modular groups acting on the above spaces. Furthermore we determine fundamental regions for the Schottky modular groups.2. Jorgensen numbers. We found the best lower bounds for all kinds of the classical Schottky spaces of real type considered in 1. We submited these results to J. Math. Soc. Japan.3. Jorgensen groups. A Jorgensen group is a non-elementary discrete group whose Jorgensen number is one. In 1998 we considered two one-parameter families of Jorgensen groups. We talked about these results at State University of New York at Stony Brook, Rutgers University at Newark and at New Brunswick in 1998. The results has been printed in Contemporary Mathematics (the proceeding of The Second Ahlfors-Bers Colloquium) in February, 2000. In 1999 we considered distribution of Jorgensen groups on the fiber over the unit circle. In particular, we studied new two one-paramter families of Jorgensen groups on it. We talked about the results at the international conference (ISAAC) in August, 1999. We invited Professor I. Kra at State University of New York at Stony Brook to talk about Schottky groups and the Schottky space in January, 2000. We are preparing to submit some results obtained between April, 1999 and February, 2000.

1998年至1999年我们研究了以下三个主题。1。实型经典肖特基空间2．乔根森数字3。乔根森组.1。实型经典肖特基空间。我们将属二实型的经典肖特基空间分为八种类型。我们表示了它们空间的形状，并找到了作用于上述空间的肖特基模块群的生成器。此外，我们还确定了肖特基模群的基本区域。2．乔根森数字。我们找到了 1 中考虑的各种实型经典肖特基空间的最佳下界。我们将这些结果提交给 J. Math。苏克。日本.3.乔根森团体。乔根森群是乔根森数为 1 的非初等离散群。 1998 年，我们考虑了 Jorgensen 群的两个单参数族。 1998 年，我们在纽约州立大学石溪分校、罗格斯大学纽瓦克分校和新不伦瑞克分校讨论了这些结果。结果已发表在 2000 年 2 月的《当代数学》（第二届 Ahlfors-Bers 研讨会论文集）上。 1999 年，我们考虑了 Jorgensen 群在单位圆上纤维上的分布。特别是，我们在其上研究了两个新的 Jorgensen 群单参数族。我们在1999年8月的国际会议（ISAAC）上谈到了这个结果。我们在2000年1月邀请了纽约州立大学石溪分校的I. Kra教授讲肖特基群和肖特基空间。我们准备提交1999年4月至2000年2月期间获得的一些结果。

项目成果

Marjatta Naatanen: "Arithmetic Fuchsian groups of signature(0;e_1,e_2,e_3,e_4)with 2 【less than or equal】e_1【less than or equal】e_2【less than or equal】e_3,e_4=∞"Contemporary Math. 240. 269-277 (1999)

Marjatta Naatanen：“签名算术 Fuchsian 群(0;e_1,e_2,e_3,e_4)与 2 【小于或等于】e_1【小于或等于】e_2【小于或等于】e_3,e_4=∞”当代数学.240.269-277（1999）

Marjatta Naatanen: "Weil-Petersson areas of the moduli space of tori"Results in Math.. 33. 120-133 (1998)

Marjatta Naatanen：“环面模空间的 Weil-Petersson 区域”数学结果.. 33. 120-133 (1998)

Hiroki Sato: "One-parameter families of extreme discrete groups for Jorgensen's inequality"Contemporary Math.. (2000)

Hiroki Sato：“Jorgensen 不等式的极端离散群的单参数族”当代数学.. (2000)

Marjatta Naatanen: "Arithmetic Fuchsian groups of signature(0;e_1,e_2,e_3,e_4)with2【less than or equal】e_1【less than or equal】e_2【less than or equal】e_3,e_4=∞"Contemporay Math.. 240. 269-277 (1999)

Marjatta Naatanen：“签名算术 Fuchsian 群(0;e_1,e_2,e_3,e_4)和 2【小于等于】e_1【小于等于】e_2【小于等于】e_3,e_4=∞”当代数学。 .240.269-277（1999）

Kazuo Akutagawa: "Kenmotsu-Bryant type representation formulas for constant mean curvature surfaces in H^3_1(-C^2) and S^3_1(C^2)" Ann.Global Anal.Geom.17. 49-75 (1999)

Kazuo Akutakawa：“H^3_1(-C^2) 和 S^3_1(C^2) 中恒定平均曲率曲面的 Kenmotsu-Bryant 型表示公式”Ann.Global Anal.Geom.17。