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.乔根森集团。2.皮卡德集团。3.白石链接组。4.经典肖特基空间与Jorgensen数。1.Jorgensen群。Jorgensen群是指Jorgensen数为1的非初等双生成元离散群。Jorgensen群有抛物型和椭圆型两种类型。这里我们考虑的是抛物线型。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)。我们获得了以下信息。(1)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)2003年我们发现了所有不可数无限型Jorgensen群。因此，我们发现所有的Jorgensen群都是抛物型的。结果(1)于2002年在北京举行的国际数学家大会上发表，结果(2)于2003年在北京大学发表。我们为Picard群构造了一个新的基本区域，并利用这个基本区域找到了群的两个生成元的八个关系。这一结果发表在2003年柏林艾萨克会议论文集。怀特黑德链接小组。我们证明了怀特黑德链的约根森数为2。因此，怀特黑德链接不是约根森集团。我们在2004年国际拓扑学会议上讨论了这一结果。经典肖特基空间和Jorgensen数。我们证明了存在一个经典的肖特基群，它的Jorgensen数是给定的实数j[大于或等于]4。我们将在今年夏天的位势理论国际会议上讨论这一结果。

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.