Parabolics and Invariants

群是数学所有领域的基本研究对象。群由一个物体的对称性组成。一种对称可以在另一种对称之后进行，这种操作通常被认为是乘法。在大多数群中，应用对称的顺序很重要（例如，平面反射后旋转与旋转后反射不同），但如果不是这样，则称为交换群。在任何群G上，交换性可以通过“阿贝尔化”来施加，从而导致可交换群a。在这个过程中，一些（通常是全部）信息丢失，而信息的丢失会反映在另一个群k上。这个项目涉及研究群的某些性质，它们在阿贝尔化方面的行为。一个例子是属性F_n （n是自然数），它反映了用有限数量的数据来描述一个群体的能力。给定某一类的群G，我们想同时计算G的所有子群H的性质F_n，这些子群在阿贝尔化下松散的信息不少于G。在许多已知的情况下，F_n类型的群可以由一个由完全不同性质产生的多面体参数化：对于任何群，通过在乘法上考虑加法运算可以得到更丰富的结构（群环）。这允许在群环上处理矩阵。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，然后对该元素进行阿贝尔化而得到的。第三个属性（更高代）与属性F_n有关。这里用于描述整个组的数据由子组组成，而不是由组元素组成。

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.