Crystallographic actions on contractible algebraic manifolds

TL;DR

This paper investigates the actions of discrete subgroups on algebraic manifolds, establishing conditions under which these groups are virtually polycyclic and connecting to NIL-affine crystallographic actions, with implications for the generalized Auslander conjecture.

Contribution

It proves that under certain conditions, such group actions are virtually polycyclic and reduces the problem to NIL-affine actions, advancing understanding of algebraic group actions on manifolds.

Findings

Groups are virtually polycyclic under specified conditions.

Action reduces to NIL-affine crystallographic action when virtually polycyclic.

Generalized Auslander conjecture holds up to dimension six.

Abstract

We study properly discontinuous and cocompact actions of a discrete subgroup $\Gamma$ 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 $\Gamma$ is virtually polycyclic. When $\Gamma$ is virtually polycyclic, we show that $\Gamma$ is virtually polycyclic. When $\Gamma$ is virtually polycyclic, we show that the action reduces to a NIL-affine crystallographic action. As applications, we prove that the generalized Auslander conjecture for NIL-affine actions holds up to dimension six and give a new proof of the fact that every virtually polycyclic group admits a NIL-affine crystallographic action.