Project Details
Projekt
Parabolics and Invariants
Subject Area
Mathematics
Term
from 2017 to 2020
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 338458954
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.
DFG Programme
Priority Programmes
Subproject of
SPP 2026:
Geometry at Infinity
