Any virtually polycyclic group admits a NIL-affine crystallographic action
by
Karel Dekimpe
K.U.Leuven Campus Kortrijk (Belgium)

In 1977, John Milnor asked wether it was true that any (torsion free) virtually polycyclic group admits an affine crystallographic action. Ten years ago, Yves Benoist answered this question negatively by producing a nilpotent counter example. As an alternative we could prove that any virtually polycyclic group does admit a polynomial crystallographic action. Although this is a good alternative for the failing affine crystallographic actions, there are two main problems with this approach.

1. There is no knowledge on the degree of the polynomials needed.

2. The polynomial crystallographic actions do not have a rich geometrical meaning.

In this talk, we show that another alternative is possible, namely that of NIL-affine crystallographic actions. In fact, we explain how we can prove that any polycyclic-by-finite group G admits a representation into the affine group Aff(N)=NxAut(N) of a simply connected and connected nilpotent Lie group, in such a way that G acts properly discontinuously and cocompactly on N. Such an action is called a NIL-affine crystallographic action and has a nice geometrical meaning in terms of the natural affine connection on N. This shows that with respect to the second problem above NIL-affine crystallographic actions are perhaps preferable above polynomial crystallographic actions. Moreover, we can interpret this NIL-affine crystallographic action as a polynomial crystallographic action which is of degree bounded above by the Hirsch length of the group. So this gives a very sharp answer to the first problem above.

Date received: September 20, 2002