By Karel Dekimpe
Ranging from uncomplicated wisdom of nilpotent (Lie) teams, an algebraic thought of almost-Bieberbach teams, the elemental teams of infra-nilmanifolds, is built. those are a normal generalization of the well-known Bieberbach teams and lots of effects approximately usual Bieberbach teams end up to generalize to the almost-Bieberbach teams. additionally, utilizing affine representations, specific cohomology computations might be performed, or leading to a category of the almost-Bieberbach teams in low dimensions. the concept that of a polynomial constitution, another for the affine buildings that typically fail, is brought.
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Ranging from simple wisdom of nilpotent (Lie) teams, an algebraic concept of almost-Bieberbach teams, the basic teams of infra-nilmanifolds, is constructed. those are a average generalization of the well-known Bieberbach teams and plenty of effects approximately traditional Bieberbach teams end up to generalize to the almost-Bieberbach teams.
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Additional info for Almost-Bieberbach Groups: Affine and Polynomial Structures
Let Ad(W, R k) denote the set of continuous mappings of W into R k. In the same way as above we can make ~4(W, R k) into a (Gl(k, IR) • 7-/(W))-module, and there is an embedding M(W, Rk)~(GI(k,R) • n( W ) ) ~ n ( R k • W). Now, consider a group extension 1 ~ ~k ~ E ~ Q ~ 1 which induces an a u t o m o r p h i s m ~ : Q ~ Aut (Zk) via conjugation in E. o,~,o. A,4 >~(G x 7-l) such that the following diagram commutes 1 --~ 1 ~ ~k ~(w,R --~ k) -~ E ~x(v• ~ -~ Q r215 ~ 1 ~ 1 The resulting quotient space is also said to be a Seifert Fiber Space with typical fiber a (k-dimensional) torus.
In each of these restricted situations one now faces existence and uniquehess questions. In order to be able to solve these problems, we will first give a general t r e a t m e n t of the Seifert Fiber Space construction in the following section. 3 An algebraic description of the Seifert Fiber Space construction In this section we will first prove a general algebraic lemma and then apply this l e m m a to the Seifert Construction situation. B. Lee in . Let Q and Q1 be groups and suppose that there are two abelian groups Z and S such that Z is a Q-module, while S is a Ql-mOdule.
As is very well known, the group of affme transformations of R n can be seen as a subgroup of Gl(n + 1, R). The embedding of Aff(R ~) into Gl(n + 1, R) is given by m a p p i n g the affine transformation with linear part A 6 Gl(n, IR) and translational part a 6 R n (seen as column vector) onto the element (A a) 0 1 6 Gl(n + 1,]R). Moreover, i f w e i d e n t i f y a n e l e m e n t z 6 R~withtheelement ( x1 ) 6 [~n+l we can express the image of r under the affine transformation (A, a) 6 Aff(R n) by means of a m a t r i x multiplication.