Polynomial structures on Polycyclic-by-finite groups: Existence and Uniqueness
by
Karel Dekimpe
K.U.Leuven Campus Kortrijk, Universitaire Campus, B-8500 Kortrijk, Belgium
Coauthors: Paul Igodt (K.U.Leuven Campus Kortrijk), Yves Benoist (ENS, Paris, France)

Given a space X and a group S of homeomorphisms of X (e.g. S=Isom(X), the group of isometries of a Riemannian space X, S=Diff(X), the group of diffeomorphisms of a smooth manifold X, ...). Problem: What is the class C of groups acting on X via maps in S, where eventually there are extra conditions on the action.

There has been special interest in the case where X=E, the Euclidean space in dimension n, S=Aff(E), the group of invertible affine transformations of E and where it was required that a group G in the class C acts cocompactly and properly discontinuously. It was conjectured for a long time that the class C of groups G arising from this setting was exactly the class of polycyclic-by-finite groups. However, by the work of Yves Benoist (1992-1995), we know that there are polycyclic-by-finite groups (even nilpotent ones) not belonging to the class C. Moreover, also the other other direction, namely the question wether or not C is a subclass of the class of polycyclic-by-finite groups, is uncertain and is worldwide known as the Auslander conjecture. Although a lot of work has been done around this conjecture, it is only known to be true up to dimension 6.

In our research we try to replace the group S=Aff(E), by another group of homeomorphisms, which does also have an important geometrical/toplogical meaning. It turned out that the group S=P(E), the group of polynomial automorphisms of E (these are polynomial maps of E onto itself, which are bijective and for which also the inverse is polynomial), is especially interested in this respect. (Note that the group P(E) consists exactly of the automorphisms of E seen as a real affine algebraic set).

I would like to describe how we were able to prove that the class of polycyclic-by-finite groups is a subclass of the class C determined by this setting. Moreover, very recently we were able to show that each polycyclic-by-finite group acts in essentially one way on E in the required way.

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Date received: December 2, 1999