Properly discontinuous actions of subgroups in amenable algebraic groups and its application to affine motions
Let \(\pi\) be a group acting properly discontinuously on a contractible manifold X, such that the action extends to an action of a real algebraic group G containing \(\pi\). If such a \(\pi\) has a nontrivial radical (unique maximal normal solvable subgroup), then it is shown in the present paper that the quotient X/\(\pi\) has the structure of an injective Seifert fiber space, as introduced by Conner-Raymond, with typical resp. exceptional fiber diffeomorphic to a solv- respectively infrasolv- manifold. Moreover if G is an amenable algebraic group, then such an action is essentially unique (up to conjugation by diffeomorphisms). Several applications are given, among them: 1) A complete affinely flat manifold admits a nontrivial (maximal) torus action if its fundamental group has nontrivial center; 2) if the fundamental group of such a manifold is virtually polycyclic, then it has the structure of an infrasolv-manifold; 3) two virtually polycyclic affine crystallographic groups are conjugate in \(Diff({\mathbb{R}}^ n)\) iff they are isomorphic (the last 2 results are due to \textit{D. Fried} and \textit{W. M. Goldman} [Adv. Math. 47, 1-49 (1983; see the preceding review)]).
polycyclic affine crystallographic groups
scientific article; zbMATH DE number 3911324
scientific article