# Almost-Bieberbach Groups: Affine and Polynomial Structures by Karel Dekimpe

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 common 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 may be conducted, or leading to a class of the almost-Bieberbach teams in low dimensions. the idea that of a polynomial constitution, another for the affine buildings that typically fail, is brought.

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This section is devoted to the investigation of such torsion elements. 1 Let 1 --~ Z k ~ E ---+F ~ 1 be any central extension, where F is a finite group. Then r ( E ) is a characteristic subgroup of E . P r o o f : Once we know t h a t r ( E ) is a group, it follows a u t o m a t i c a l l y t h a t it is characteristic. 1) for some 2-cocycle c : F x F ~ Z k. Since ~k is a vector space, the inclusion map i : Zk ~ Rk induces a trivial map i. : H2(F, Zk) -+ H2(F, R k) = 0 on the cohomology level.

R e m a r k that #(s) o fX(0, a) = (s + X(c~) - ~ s,j(v~)) while f~, o 0 = ( - g ( ~ ) + ~'(~1, j(~)) which implies that s + A(~) - ~ s = - g ( ~ ) + ~'(a) ~ ,V - )~ = 5~ with ~ = p(s). Therefore, seen as elements of HI(Q, ~ ) , {A} = {A'}, which was to be shown. ~2 is surjective: Suppose ( E , f ) is a pair satisfying the necessary conditions. E is an extension of Q by Z and so there exists a 2-cocyle c : Q x Q ~ Z such that E can be seen as the set Z x Q and where the multiplication is given by Vz, y E Z;Vct, fl E Q : (z,c~)(y,/3) = (z +~ y + c(a,~),a~).

1), we should show that there are only finitely m a n y conjugacy classes of finite subgroups in Out N. But Aut N is an arithmetic group ([59]) and I n n N ~ N / Z ( N ) is a torsion free, finitely generated nilpotent group. This implies that the subgroups K of Aut N such that [K : Inn N] < oo, lie in finitely m a n y conjugacy classes in Aut N ([59]). By dividing out Inn N, we obtain our result. e. those occurring as holonomy group for crystallographic groups. 6 At the end of this chapter, I would also like to mention the work of F.