Crystallographic actions on contractible algebraic manifolds

Abstract

Abstract. We study properly discontinuous and cocompact actions of a discrete subgroup Γ of an algebraic group G on a contractible algebraic manifold X. We suppose that this action comes from an algebraic action of G on X such that a maximal reductive subgroup of G fixes a point. When the real rank of any simple subgroup of G is at most one or the dimension of X is at most three, we show that Γ is virtually polycyclic. When Γ is virtually polycyclic, we show that the action reduces to a 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 a NIL-affine crystallographic action.

Nil-Affine Crystallographic Actions of Virtually Polycyclic Groups

“…Every reductive subgroup of Aff(N ) has a fixed point n 0 ∈ N . This lemma is exactly [13,Lemma 3.4], but for the convenience of the reader we recall the proof here.…”

“…The first version of this theorem without the extra condition of acting translationlike is due to K. Dekimpe in [11] and a new proof was given in [13] more recently. The techniques we develop in this paper lead to a different proof of this result, giving us more information for NIL-affine actions which are only properly discontinuous and not necessarily cocompact.…”

NIL-affine crystallographic actions of virtually polycyclic groups

“…The first version of this theorem without the extra condition of acting translation-like is due to K. Dekimpe in [10] and a new proof was given in [12] more recently. The techniques we develop in this paper give a new and simplified proof of this result and are moreover important for studying NIL-affine actions of virtually polycyclic groups which are not necessarily cocompact.…”