There are only finitely many infra-nilmanifolds under each nilmanifold: a new proof

AbstractInfra-nilmanifolds are a natural generalization of the flat Riemannian manifolds. Together they share the property of being completely determined (up to a well understood diffeomorphism) by their fundamental group. In the case of the flat Riemannian manifolds this goes back to the well known three theorems of Bieberbach. The first two theorems of Bieberbach were generalized for the case of infra-nilmanifolds in [1] and [7] respectively. In [6] Kyung Bai Lee announced a generalization of the third Bieberbach theorem. We will show here, by means of an example, that the proof of this theorem, as presented there, contains a counting principle which is incorrect. However, this will not reject the theorem: in fact, we propose a completely different proof of the statement using a deep result on conjugacy classes of subgroups of arithmetic groups. Moreover, a key observation in [6], most likely allows a more useful approach to an explicit classification of these manifolds as already has been shown in [4]