Crystallographic actions on contractible algebraic manifolds

Abstract

We study properly discontinuous and cocompact actions of a discrete subgroup Γ \Gamma of an algebraic group G G on a contractible algebraic manifold X X . We suppose that this action comes from an algebraic action of G G on X X such that a maximal reductive subgroup of G G fixes a point. When the real rank of any simple subgroup of G G is at most one or the dimension of X X is at most three, we show that Γ \Gamma is virtually polycyclic. When Γ \Gamma is virtually polycyclic, we show that the action reduces to an NIL-affine crystallographic action. Specializing to NIL-affine actions, we prove that the generalized Auslander conjecture holds up to dimension six and give a new proof of the fact that every virtually polycyclic group admits an NIL-affine crystallographic action.