Computational aspects of affine representations for torsion free nilpotent groups via the Seifert construction

Abstract

We study and describe faithful affine representations of a certain canonical form for torsion free, finitely generated nilpotent groups. The algebraic set-up used is that of the Seifert Fiber Space Construction conceived by Kyung Bai Lee and Frank Raymond, and can be considered as suitable for an iteration procedure. The representations obtained realise such a group as the fundamental group of a compact, complete affinely flat manifold. We investigate computational aspects for explicit constructions of these representations. An equivalent description for canonical form representations, in terms of matrices over polynomial rings, is presented. As these groups are uniform lattices in nilpotent Lie groups, it is interesting to draw the related picture on the Lie algebra level. An example (due to Dan Segal and Fritz Grunewald) is used to illustrate that the iteration aspect of this set-up should be well understood and will need a very careful treatment. Finally, we present canonical representations for several infinite families of groups, thus also giving positive evidence for the nilpotent case of a conjecture of Milnor's.