Research on Schottky groups and Schottky spaces

从1998年到1999年，我们研究了以下三个主题。实数型的经典肖特基空间约根森numbers.3。约根森groups.1。实型的经典肖特基空间。我们将2属实数型的经典Schottky空间划分为8种类型。我们表示了它们的空间形状，并找到了作用在上述空间上的肖特基模群的产生子。此外，我们还确定了Schottky模群的基本区域。约根森数字。我们找到了1中所考虑的各种实数型经典肖特基空间的最佳下界。我们将这些结果提交给了J. Math。Soc。Japan.3。约根森组。Jorgensen群是Jorgensen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、纽瓦克的罗格斯大学和新不伦瑞克分校讨论了这些结果。研究结果发表在2000年2月的《当代数学》（第二届Ahlfors-Bers学术讨论会的会刊）上。1999年，我们考虑在单位圆上的纤维上分布约根森群。特别地，我们在上面研究了两个新的单参数Jorgensen群族。我们在1999年8月的国际会议（ISAAC）上讨论了这些结果。2000年1月，我们邀请了纽约州立大学石溪分校的I. Kra教授来谈论肖特基群和肖特基空间。我们正准备提交1999年4月至2000年2月期间取得的一些结果。

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.