Presentations of groups acting discontinuously on direct products of hyperbolic spaces
Presentations of groups acting discontinuously on direct products of hyperbolic spaces
复制标题
对双曲空间的直积进行不连续作用的群的演示
作者：
E. Jespers;Ann Kiefer;Á. Río
刻画有限群G的整群环ZG的单位群U(ZG)的问题一直是人们关注的热点问题，给出这类群的表示是一个基本问题。在序的背景下，一个中心问题是描述O阶单位群在有理群代数QG的简单满象A中的表示。利用Poincare多面体定理的表示部分，Pita，del Rio和Ruiz提出了有限大族G的这种方法，随后Jespers，Pita，del Rio，Ruiz和Zalesskii描述了有限大族G的U(ZG)的结构。为了处理更多的群，我们想把Poincare方法推广到双曲空间的直积等距群的不连续子群。如果代数A有2次，则通过代数A的中心的Galois嵌入，我们将O阶约化范数1元的群视为这样的群，从而得到所述问题的解。这将在QG的2次单分支中给出阶数的单位群的表示，并特别地刻画不可约特征标度小于或等于2的每个群G的ZG的单位群。本文的目的是通过在Hilbert模群上执行这一方法来启动这一方法，即整数环上的二次投影线性群在有理数的实二次扩张中。这个群不连续地作用于两个二维双曲空间的直积。所构造的基本域类似于Fuchsian群或Kleinan群的Ford域。
The problem of describing the group of units U(ZG) of the integral group ring ZG of a finite group G has attracted a lot of attention and providing presentations for such groups is a fundamental problem. Within the context of orders, a central problem is to describe a presentation of the unit group of an order O in the simple epimorphic images A of the rational group algebra QG. Making use of the presentation part of Poincare’s Polyhedron Theorem, Pita, del Rio and Ruiz proposed such a method for a large family of finite groups G and consequently Jespers, Pita, del Rio, Ruiz and Zalesskii described the structure of U(ZG) for a large family of finite groups G. In order to handle many more groups, one would like to extend Poincare’s Method to discontinuous subgroups of the group of isometries of a direct product of hyperbolic spaces. If the algebra A has degree 2 then via the Galois embeddings of the centre of the algebra A one considers the group of reduced norm one elements of the order O as such a group and thus one would obtain a solution to the mentioned problem. This would provide presentations of the unit group of orders in the simple components of degree 2 of QG and in particular describe the unit group of ZG for every group G with irreducible character degrees less than or equal to 2. The aim of this paper is to initiate this approach by executing this method on the Hilbert modular group, i.e. the projective linear group of degree two over the ring of integers in a real quadratic extension of the rationals. This group acts discontinuously on a direct product of two hyperbolic spaces of dimension two. The fundamental domain constructed is an analogue of the Ford domain of a Fuchsian or a Kleinian group.
使用 Vorono 算法进行算术组计算
影响因子：
0.9
作者：
Oliver;R. Coulangeon;G. Nebe;Schönnenbeck;Sebastian
通讯作者：
Sebastian