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 built. those are a usual generalization of the well-known Bieberbach teams and plenty of effects approximately traditional Bieberbach teams end up to generalize to the almost-Bieberbach teams. in addition, utilizing affine representations, specific cohomology computations will be conducted, or leading to a category of the almost-Bieberbach teams in low dimensions. the idea that of a polynomial constitution, another for the affine constructions that usually fail, is brought.

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Then Fitt (F) is torsion free and maximal niIpotent in F F is almost torsion free. " First suppose that Fitt (F) is torsion free and maximal nilpotent in F. Consider H <3 F, a finite normal subgroup. Then H <~ Fitt ( F ) . H . H is an almost-crystallographic group (since Fitt (F) is maximal nilpotent) and so H has to be trivial. Conversely, suppose F is almost torsion free. It is easy to see that Fitt (F) has to be torsion free, since the elements of finite order of Fitt (F) 44 Chapter 3: Algebraic characterizations of A C-groups form a characteristic subgroup of Fitt (F).

So the desired q'~ E A cannot be found. Now suppose ~ is of finite order. Then the group generated by q and Fitt (F) is nilpotent if and only if p(q) E A ker(~ai), where the ~ai are i=1 as above. This intersection is a normal subgroup of Q and is finite, this means that q E Fitt (P). But now we use the fact that if F is almost torsion free, then Fitt (F) is almost-crystallographic. And so, Fitt (F) is maximal nllpotent in Fitt (F), implying we cannot find q outside Fitt (F). We summarize the results found so far in the following theorem.

Now we introduce quickly the basic building blocks for the representations we are interested in (we follow [44] and [14]): 2r K, IRk) = {continuous maps A: R K -+ IRk} 7-/(IRK) = {homeomorphisms h : R g ~ IRK}. A/I(~ K } ~ k ) , Vg E Gl(k,I~), Vh C ~~(RK). ~(]~K). 2 A s s u m e Y is polycyclic-by-finite with a torsion free filtration F . A representation p = po : F ~ H ( R K) will be called o f c a n o n i c a l t y p e with respect to F. 2) "where Chapter 4: Canonical type representations 50 9 A4>~(G • ~ ) stands for Ad(I~K'+~,IRk')>~(GI(ki, N) • ~ ( R K ' + ' ) ) , * j(z) : R Ki+' --+ Ii@i : z ~ z, Vz E ~kl and 9 Bi : F/Fi ~ GI(k/,Z) ~-~ GI(k/,IR) denotes the action of F/Fi induced on Z ki by conjugation in F / F / _ I .

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