THE AUSLANDER CONJECTURE FOR NIL–AFFINE CRYSTALLOGRAPHIC GROUPS

DIETRICH BURDE

A classical crystallographic group is a discrete subgroup of Isom(ℝ^{n}). Such groups act properly discontinuously and cocompactly on ℝ^{n}. The structure of such groups is well known by the three Bieberbach theorems. In fact, all these groups are finitely generated virtually abelian. As a generalization one also studies affine crystallographic groups and NIL-affine crystallographic groups. The latter are subgroups of Aff(N) = N ⋊ Aut(N) acting properly discontinuously and cocompactly on N. Here N is a simply connected, connected real nilpotent Lie group of finite dimension n. We prove that for all dimensions 1 ≤ n ≤ 5 the generalized Auslander conjecture holds, i.e., that such subgroups are virtually polycyclic.