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Affine crystallographic groups. (English) Zbl 0854.57035
Bokut’, L. A. (ed.) et al., Third Siberian school on algebra and analysis. Proceedings of the third Siberian school, Irkutsk State University, Irkutsk, Russia, August 30-September 4, 1989. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 163, 165-170 (1995).
We study crystallographic groups of motions of an affine space in connection with the following old conjectures:
Conjecture A. A crystallographic group $$\Gamma$$ of motions of an affine space is almost solvable (i.e., contains a solvable subgroup of finite index).
We show that the following assertion is true: Theorem A. Let $$\Gamma$$ be a crystallographic group of motions of an affine space $$A$$ and let $$G^* \subset \text{Aff } V$$ be an algebraic subgroup containing $$\Gamma$$. If the real rank of any simple subgroup in $$G^*$$ does not exceed 1, then $$\Gamma$$ is almost simple.
The methods used in proving this theorem enable us to prove the following rigidity theorem for almost solvable crystallographic groups of affine motions.
Theorem B. Let $$\Gamma_1$$ and $$\Gamma_2$$ be two crystallographic almost solvable groups, $$\Gamma_1 \subset \text{Aff } V$$, $$\Gamma_2 \subset \text{Aff } V$$, and let $$G_1$$ and $$G_2$$ be their algebraic closures. Then any isomorphism $$\varphi : \Gamma_1 \to \Gamma_2$$ extends to a rational isomorphism $$\varphi : G_1 \to G_2$$.
For the entire collection see [Zbl 0816.00016].

MSC:
 57S30 Discontinuous groups of transformations 20H15 Other geometric groups, including crystallographic groups