By Karel Dekimpe
Ranging from easy wisdom of nilpotent (Lie) teams, an algebraic conception of almost-Bieberbach teams, the basic 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 prove to generalize to the almost-Bieberbach teams. additionally, utilizing affine representations, particular cohomology computations might be conducted, or leading to a type of the almost-Bieberbach teams in low dimensions. the concept that of a polynomial constitution, an alternate for the affine constructions that usually fail, is brought.
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Additional resources for Almost-Bieberbach Groups: Affine and Polynomial Structures
Because F ( F i t t (F)) is a characteristic subgroup of Fitt (F) and so a normal subgroup of F, which implies that F ( F i t t (F)) = 1. We summarize this in the following theorem. 3 Let F be any polycyclic-by-finite group, then F is almost torsion free F is (almost-crystalIographic)-by-crystallographic. Moreover, in case F is almost torsion free, the intended almost-crystallographic subgroup can be taken as Fitt (F). In  one can find, as an exercise, the following property: If F is a solvable group, then C r ( F i t t (F)) C_ Fitt (F).
This implies that (mlm2) (ml 11 I2 0 2m2 ~ Aut And so r does not exist. This example shows that not all possible AC-groups were counted in the p r o o f of the theorem in . 4 is completely different. 2. Let us quickly recall some fundamentals of the theory of group extensions with non-abel]an kernel (see , ). Let N be a group. An extension with kernel N is a short exact sequence 1 ~ N ~ E ~ F ~ 1. This exact sequence induces a hom o m o r p h i s m r 9 F ~ Out N , which is called an abstract kernel.
Let N be a group. An extension with kernel N is a short exact sequence 1 ~ N ~ E ~ F ~ 1. This exact sequence induces a hom o m o r p h i s m r 9 F ~ Out N , which is called an abstract kernel. The extension usually is called compatible with r Given an abstract kernel r : F ~ Out N , the problem of studying all extensions compatible with r is well known in literature, and gave rise to the concept of non-abelian 2-cohomology sets . Let us write E x t r N ) for the set of equivalence classes of extensions compatible with r If this set is empty, one says that there is an obstruction to the algebraic realization of the abstract kernel.
Almost-Bieberbach Groups: Affine and Polynomial Structures by Karel Dekimpe