Research on Schottky spaces and Jorgensen groups

肖特基空间和乔根森群的研究

• 批准号: 12640168
• 负责人: SATO Hiroki
• 金额: $ 2.24万
• 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: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640168/
• 关键词: Schottky space; Schottky group; Jorgensen group; Jorgensen's inequalitu; Kleinian group; hyperbolic space; Picard group; Whitehead link; リーマン面

We have studied the following four themes from 2000 to 2001. 1. Jorgensen groups. 2. Jorgensen numbers of Classical Schottky spaces of real type. 3. The Picard group. 4. The Whitehead link.1. J0rgensen groups. A Jorgensen group is a discrete group whose Jorgensen number is one. First we considered two one-parameter families. The results appeared in Contemporary Mathematics in 2000. Furthermore we studied Jorgensen groups of parabolic type. We talked the results at the Meeting of AMS (UCLA, 2000) and at the International Coference of Complex Analysis (China, 2000). Recently we found almost all Jorgensen groups of parabolic type. We talked about the results at the Meeting of Discontinuous Groups at Shizuoka University (January, 2002 and the Geometry and Topology Seminar at University of Oregon in March, 2002.2. Jorgensen numbers of Classical Schottky space of real type. We found the best lower bounds for all kinds of the classical Schottky spaces of real type. The results appeared in J. Math. Soc. Japan in 2001.3. The Picard group. We have appointed out before that the Picard group is a two-generator group. This time we constructed a new fundamental region for the group and we found eight relations by using the fundamental region. We talked this result at the ISAAC Congress in Berlin in 2001. The result will appear in the Proceedings.4. The Whitehead link. We proved that the Jorgensen number of the Whitehead link is two. Therefore the Whitehead link is not a Jorgensen group. We talked the result at Kyoto University in 2001. We will talk this result at the internatonal conference in 2002.

从2000年到2001年，我们研究了以下四个主题。 1. Jorgensen群。 2.实型经典肖特基空间的乔根森数。 3. 皮卡德小组。 4. 怀特海链接。1．约根森团体。乔根森群是乔根森数为 1 的离散群。首先我们考虑两个单参数族。结果发表在 2000 年的《当代数学》上。此外，我们研究了抛物线型的乔根森群。我们在 AMS 会议（加州大学洛杉矶分校，2000 年）和国际复杂分析会议（中国，2000 年）上讨论了这些结果。最近我们发现几乎所有的约根森群都是抛物线型的。我们在静冈大学不连续群会议（2002年1月）和2002年3月俄勒冈大学几何与拓扑研讨会上讨论了结果。2.实型经典肖特基空间的约根森数。我们找到了最佳下界各种经典的实型肖特基空间，结果发表于 J. Math. Japan 2001.3。我们之前已经指定Picard群是一个二元群，这次我们为该群构建了一个新的基本域，并利用这个基本域找到了八个关系，我们在柏林的ISAAC大会上讨论了这个结果。 2001年，该结果将发表在论文集上。 4.怀特海链接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京都大学讨论了这个结果。 2001年。我们将在2002年的国际会议上谈论这个结果。

项目成果

Yoshihide Okumura: "Lifting problem and its application to Riemann surfaces"Proc. Eighth Inter. Conf. on Complex Analysis. 173-178 (2001)

Yoshihide Okumura：“提升问题及其在黎曼曲面上的应用”Proc。

Hiroki Sato: "Jorgensen's inequality for classical Schottky groups of real type, II"J. Math. Soc. Japan. 53. 791-811 (2001)

Hiroki Sato：“实型经典肖特基群的约根森不等式，II”J.

Hiroki Sato: "Jorgensen's inequality for classical Schottky groups of real type,II"J.Math.Soc.Japan. 53. 791-811 (2001)

Hiroki Sato：“实型经典肖特基群的约根森不等式，II”J.Math.Soc.Japan。

Kazuo Akutagawa: "An obstraction to the positivity of relative Yamabe invariants"Math. Z.. (to appear). (2002)

芥川一夫：“对相对山边不变量的积极性的阻碍”数学。

Yoshihide Okumura: "Lifting problem and its application to Riemann surfaces"Proc.Eighth International Conference on Complex Analysis. (2001)

Yoshihide Okumura：“提升问题及其在黎曼曲面上的应用”Proc.第八届国际复分析会议。