Research on Schottky spaces and Jorgensen groups

从2000年到2001年，我们研究了以下四个主题。1. 约根森组。2. 实型经典肖特基空间的Jorgensen数。3. 皮卡德集团。4. 怀特黑德的联系。J0rgensen组。乔根森群是一个乔根森数为1的离散群。首先我们考虑了两个单参数族。研究结果发表在2000年的《当代数学》杂志上。进一步研究了抛物型Jorgensen群。我们在AMS会议（UCLA, 2000）和国际复合体分析会议（中国，2000）上讨论了结果。最近我们发现了几乎所有的抛物型Jorgensen群。我们在静冈大学的不连续群会议（2002年1月）和俄勒冈大学的几何与拓扑研讨会（2002年3月）上讨论了这些结果。实型经典肖特基空间的Jorgensen数。我们找到了各种实型经典肖特基空间的最佳下界。研究结果发表在《数学》杂志上。Soc。2003年的日本。皮卡德集团。我们之前说过皮卡德群是一个双发电机群。这次我们为群构造了一个新的基域，并利用基域找到了8个关系。我们在2001年柏林的艾萨克大会上讨论了这个结果。结果将在会议记录中公布。Whitehead的联系。我们证明了怀特黑德链的乔根森数是2。因此怀特黑德家族不是乔根森家族。2001年我们在京都大学讨论了这个结果。我们将在2002年的国际会议上讨论这一结果。

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.