Research on Jorgensen groups and Schottky spaces

从2002年到2003年，我们研究了以下四个主题。1.约根森组。2.皮卡德集团。3.Whitehead链接组。4.经典肖特基空间和约根森1号。约根森组。Jorgensen群是Jorgensen数为1的非初等双生离散群。Jorgensen群有抛物线型和椭圆型两种类型。这里我们考虑抛物线型。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）。我们得到了以下结果。(1)在2002年我们发现了所有有限型Jorgensen群和所有可数无限型Jorgensen群；(2)在2003年我们发现了所有不可数无限型Jorgensen群。因此，我们找到了所有抛物型Jorgensen群。结果(1)在2002年北京国际数学家大会上被讨论，结果(2)于2003年在北京大学被讨论。皮卡德集团。我们为Picard群构造了一个新的基域，并利用这个基域找到了群的两个生成元之间的8个关系。这一结果于2003年3月发表在柏林艾萨克大会的会议记录上。Whitehead链接组。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。我们在2002年的国际拓扑学会议上讨论了这个结果。经典肖特基空间和约根森数。我们证明了存在一个经典Schottky群，其Jorgensen数是给定实数j【大于等于】4。我们将在今年夏天的国际势理论会议上讨论这个结果。

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.