By Karel Dekimpe

Ranging from uncomplicated wisdom of nilpotent (Lie) teams, an algebraic concept of almost-Bieberbach teams, the basic teams of infra-nilmanifolds, is constructed. those are a normal generalization of the well-known Bieberbach teams and plenty of effects approximately usual Bieberbach teams end up to generalize to the almost-Bieberbach teams. additionally, utilizing affine representations, specific cohomology computations may be conducted, or leading to a type of the almost-Bieberbach teams in low dimensions. the concept that of a polynomial constitution, another for the affine constructions that typically fail, is brought.

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**Extra info for Almost-Bieberbach Groups: Affine and Polynomial Structures**

**Example text**

1 Two pairs (E, f ) and (E', f') as above are said to be equivalent if and only if E and E I are equivalent as group extensions of Z by Q via an isomorphism 0 (inducing the identity on Z and on Q) such that there exists an element s E S with fit? = # ( s ) f . Chapter 4: Canonical type representations 54 So if (E, f ) -,~ (E', f') via an isomorphism 0, then there exists an element s E S for which the following diagram commutes: E 0) E~ S~Q1 "-(s2 s~Ol Let us denote the set of equivalence classes of pairs (E, F ) by H(Q; Z, S).

Fin C_ E,~+I C_ . . be any ascending chain of finite normal subgroups of F. This chain is necessarilly finite, since every ascending chain of subgroups of r is finite (see [59]). Therefore we can choose a finite normal subgroup H o f t which is maximal among all finite normal subgroups. Now it is easy to see that H is unique, for if K was another such a normal subgroup, then H . K would contradict the maximality of H . 2 For a polycyclic-by-fmite group r , we will denote its maximal normal finite subgroup by F ( r ) .

If G denotes the Mal'cev completion of N , then the group extension induces a morphism (analogous and related to the abstract kernal of the extension) ~ a : F ~ Out (G) in the following way: For any ~ E F we choose an element z E E which maps onto ~. Then conjugation in E by x induces an a u t o m o r p h i s m of N . This a u t o m o r p h i s m lifts uniquely to an automorphism ~r($) of G. It is obvious that a ( 2 ) is unique up to inner automorphisms of N (and so of G). Therefore, the map qa: F ~ Out ( G ) : ~ ~ cr(~) Inn (G) is a well defined h o m o m o r p h i s m of groups.