Parabolics and Invariants

群是所有数学领域的基本研究对象。一个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反射后的旋转不同），但如果不是这样，则该群称为交换群。在任何群G上，交换性都可以通过“阿贝尔化”来实现，从而得到交换群A。在这个过程中，一些（通常是所有）信息丢失，而信息的丢失反映在另一个群K中。本项目关注的是研究群的某些性质如何关于阿贝尔化表现。一个例子是性质F_n（n是自然数），它反映了一个群可以用多少数据来描述。给定一个来自某类的群G，我们希望同时计算G的所有子群H的性质F_n，这些子群H在阿贝尔化下释放的信息不比G少。在许多已知的情况下，F_n型群可以由一个多面体参数化，该多面体由一个完全不同的性质产生：对于任何群，通过在乘法之上考虑加法运算，可以获得更丰富的结构（群环）。这允许在群环上处理矩阵。第三个性质（高代）与性质F_n有关。这里用来描述整个群的数据是由子群组成的，而不是由群元素组成的。

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.