Constructing faithful representations of finitely-generated torsion-free nilpotent groups
A finitely generated torsionfree nilpotent group \(G\) has a finite series, whose factors are central infinite cyclic groups. Every such group can be embedded into a group of upper unitriangular matrices with integer coefficients. Using the series the authors show how one obtains a consistent polycyclic presentation for \(G\) and then formulate an efficient algorithm for calculating a representation by unitriangular matrices. The inductive construction consists of an algorithm which extends a representation of this kind for a normal subgroup \(N\) of \(G\) to one for \(G\), where \(G/N\) is infinite cyclic. The central step is to describe the action of \(G\) on the dual space of the group ring \(\mathbb{Z} N\) by means of a suitable finite-dimensional faithful submodule. A table of examples is included which indicates the significant reduction in the embedding dimension this method achieves.
finitely generated torsionfree nilpotent groups
representations by unitriangular matrices
scientific article; zbMATH DE number 1753636