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Three-dimensional affine crystallographic groups. (English) Zbl 0571.57030

Summary: Those groups \(\Gamma\) which act properly discontinuously and affinely on \({\mathbb{R}}^ 3\) with compact fundamental domain are classified. First it is shown that such a group \(\Gamma\) contains a solvable subgroup of finite index, thus establishing a conjecture of Auslander in dimension three. Then unimodular simply transitive affine actions on \({\mathbb{R}}^ 3\) are classified; this leads to a classification of affine crystallographic groups acting on \({\mathbb{R}}^ 3\). A characterization of which abstract groups admit such an action is given; moreover it is proved that every isomorphism between virtually solvable affine crystallographic groups (respectively simply transitive affine groups) is induced by conjugation by a polynomial automorphism of the affine space. A characterization is given of which closed 3-manifolds can be represented as quotients of \({\mathbb{R}}^ 3\) by groups of affine transformations: a closed 3-manifold \(M\) admits a complete affine structure if and only if \(M\) has a finite covering homeomorphic (or homotopy-equivalent) to a 2-torus bundle over the circle.

MSC:

57S30 Discontinuous groups of transformations
20H15 Other geometric groups, including crystallographic groups
22E40 Discrete subgroups of Lie groups
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