Atlas: Plane Polynomial Crystallographic Groups by Karel Dekimpe

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G^3, Special Session in Geometric Group Theory
January 10-13, 2001
part of the AMS/MAA joint meeting
New Orleans, LA, USA

Organizers
Phil Bowers, Martin Bridson, Stephen Brick, Jon Corson, Igor Mineyev

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Plane Polynomial Crystallographic Groups
by
Karel Dekimpe
Katholieke Universiteit Leuven, Campus Kortrijk

We study questions of the following form:

Let X be a space and let S be a set of homeomorphisms of X. Problem: What groups act properly discontinuously and cocompactly on X via maps in S?

When X=E, the Euclidean space in dimension n, and S=Isom(E), the groups obtained are the crystallographic groups. If S=Aff(E), we get the so-called affine crystallographic groups. A intriguing open question in this area is due to Auslander (1964):

Is it true that all affine crystallographic groups are polycyclic-by-finite?

If we enlarge S to the group P(E) of polynomial diffeomorphisms of E, we obtain the so-called polynomial crystallographic groups. We were able to make some progress for the analogue of Auslander's problem. Our main results are:

Any polycyclic-by-finite and planar polynomial crystallographic group is of bounded degree.

Any planar polynomial crystallographic group of bounded degree is a polycyclic-by-finite group.

This last result gives a positive answer to the problem of Auslander in dimension 2 and for polynomial actions.

Date received: September 19, 2000