Polycyclic-by-finite groups: from affine to polynomial structures

Summary

Introduction

This paper sketches the mathematics which arose from a question posed by John Milnor in 1977, concerning the existence of certain affinely flat manifolds. Although the original question is quite geometrical, several results have been obtained by translating Milnor's problem into the language of specific (discrete and Lie) group and Lie algebra representations. These representations are called affine structures. The examples, constructed in the beginning of the 90's, show that not all reasonable groups admit an affine structure, and gave a new boost to this research topic. Besides the investigation (without many results up till now) of the exact nature of the groups that (do not) allow an affine structure, we began to study what might replace the missing affine structures. Very recently we have shown that there is a notion of a polynomial structure, which can be seen as an alternative to the inadequate notion of an affine structure, and which exists on any polycyclic-by-finite group.

In this paper, we first give a historical survey of the most important results on affine structures and later we explain the ideas behind the origin and the existence of polynomial structures. The paper contains no new results, but the old results are put together in a nicer way and presented perhaps more clearly, without to many technical details.