By Karel Dekimpe

Ranging from simple wisdom of nilpotent (Lie) teams, an algebraic concept of almost-Bieberbach teams, the elemental teams of infra-nilmanifolds, is built. those are a typical generalization of the well-known Bieberbach teams and lots of effects approximately traditional Bieberbach teams end up to generalize to the almost-Bieberbach teams. in addition, utilizing affine representations, particular cohomology computations may be performed, or leading to a class of the almost-Bieberbach teams in low dimensions. the idea that of a polynomial constitution, an alternate for the affine buildings that typically fail, is brought.

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**Example text**

P r o o f : Assume that the induced abstract kernel is not injective. This means that there exists an element in E \ N and an element no E N such that Vn E N : x n x - i = nonno 1. 4, is not injective. 1, N is not maximal nilpotent inE. 3 If N is abelian, then it is well known that the converse of the s t a t e m e n t is also true. However, in general, as the following example shows, this is not the case. e. r : F -+ Out N might be injective, without N being maximal nilpotent. 4 Consider the following torsion free virtually 2-step nilpotent group.

Z in stead of i(z), which is justified by So, fs = (z + ~ ( a ) , j ( a ) ) . It is -+ S >~Q1 is a m o r p h i s m of groups. T h e first thing to show is that the choice of lift ~ is u n i m p o r t a n t . So consider another lift of A, say A. T h e n we define g : Q -+ Z : c ~ ~ ) , ( a ) - ~(a). T h e fact t h a t g(a) takes images in Z follows from the fact that A and are b o t h lifts of the same )~. Using this g, we introduce a m a p O: E ( i ) -~ E ( ~ ) : ( z , a ) ~ ( z - g(a), a). Some elementary computations show that 0 is an isomorphism of groups inducing the identity on b o t h Z and Q and such that =s oe.

0 1 1 " Chapter 4: Canonical type representations 48 So, an affine representation of a group E can be seen as a matrix representation and already many computer programs can deal very well with (formal) matrices. From the geometrical point of view, we will describe in this chapter some of the latest developements concerning the conjecture of John Milnor ([52]) stating that any torsion free polycyclic-by-finite group occurs as the fundamental group of a compact, complete affinely fiat manifold.