Parabolics and Invariants

抛物线和不变量

• 批准号: 338458954
• 负责人: Professor Dr. Kai-Uwe Bux
• 金额: --
• 依托单位: Mathematisches Institut
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2019-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/338458954?language=en
• 关键词: Parabolics; Invariants

Groups are fundamental objects of study in all areas of mathematics. A group consists of symmetries of an object. One symmetry can be performed after another, an operation which is usually regarded as multiplication. In most groups the order in which symmetries are applied matters (for example a plane reflection followed by a rotation is not the same as the rotation followed by the reflection) but if it does not, the group is called commutative. On any group G commutativity can be imposed by "abelianizing" it, leading to a commutative group A. In the process some (often all) information is lost and the loss of information is reflected in another group K.This project is concerned with studying, for certain properties of groups, how they behave with respect to abelianization.One example are the properties F_n (n a natural number) that reflect how well a group can be described by finitely much data. Given a group G from a certain class, we want to simultaneously compute the properties F_n for all subgroups H of G which under abelianization loose no less information than G.In many known cases, the groups that are of type F_n can be parametrized by a polyhedron that arises from a completely different property: for any group one can obtain a richer structure (a group ring) by considering an addition operation on top of multiplication. This allows to work with matrices over the group ring. The polyhedra in question are obtained by first reducing certain matrices to a single group ring element and then abelianizing that element.The third property (higher generation) is related to the property F_n. Here the data used to describe the whole group consists of subgroups, rather than of group elements.

在数学的所有领域中，群都是基本的研究对象。群由对象的对称性组成。一种对称可以一个接一个地进行，这种运算通常被认为是乘法。在大多数群中，对称性的应用顺序很重要(例如，平面反射后的旋转与旋转后的反射不同)，但如果不是这样，群就被称为交换群。在任何群G上，交换性可以通过将其“化简”而被强加，从而得到一个交换群A。在这个过程中，一些(通常是全部)信息丢失，并且这种丢失的信息反映在另一个群K中。这个项目关注于研究群的某些性质，它们关于化简的行为。一个例子是性质F_n(n是自然数)，它反映了一个群可以用有限多的数据来刻画得多好。给定某一类的群G，我们想要同时计算G的所有子群H的性质F_n，这些子群在交换化下的信息量不少于G。在许多已知情况下，F_n类型的群可以由一个完全不同的性质产生的多面体来参数化：对于任何群，人们可以通过考虑乘法上的加法运算来获得更丰富的结构(群环)。这允许在群环上使用矩阵。所讨论的多面体是通过首先将某些矩阵简化为单个群环元素，然后将该元素简单化而得到的。第三个性质(更高的世代)与性质F_n有关。这里用于描述整个群的数据由子群组成，而不是由群元素组成。