Three-Dimensional Affine Crystallographic Groupd

Those groups r which act properly discontinuously and aillnely on II?’ with compact fundamental domain are classified. First it is shown that such a group f contains a solvable subgroup of finite index, thus establishing a conjecture of Auslander in dimension three. Then unimodular simply transitive alTine actions on IR’ are classified; this leads to a classification of atTine crystallographic groups acting on IR3. A characterization of which abstract groups admit such an action is given; moreover it is proved that every isomorphism between virtually solvable atline crystallographic groups (respectively simply transitive afline groups) is induced by conjugation by a polynomial automorphism of the affrne space. A characterization is given of which closed 3-manifolds can be represented as quotients of IR’ by groups of afftne transformations: a closed 3-manifold M admits a complete atline structure if and only if M has a finite covering homeomorphic (or homotopy-equivaient) to a 2-torus bundle over the circle.