alexa Three-dimensional affine crystallographic groups


Journal of Generalized Lie Theory and Applications

Author(s): David Fried, William M Goldman

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Those groups Γ which act properly discontinuously and affinely on R3 with compact fundamental domain are classified. First it is shown that such a group Γ contains a solvable subgroup of finite index, thus establishing a conjecture of Auslander in dimension three. Then unimodular simply transitive affine actions on R3 are classified; this leads to a classification of affine crystallographic groups acting on R3. 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 R3 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.

This article was published in Advances in Mathematics and referenced in Journal of Generalized Lie Theory and Applications

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