6 - On unipotent groups

Summary

We saw in the last chapter that a finitely generated torsion-free nilpotent group has a natural representation as a unipotent group of matrices over ℚ. A good way to study a unipotent group is to apply the ‘logarithm’ map, which embeds the group into a certain Lie algebra of matrices. The grouptheoretic operations are then reflected in the Lie algebra operations; moreover, if the group is finitely generated, its logarithm will be almost (but not quite) a lattice in the Lie algebra. In this way, certain questions about unipotent groups get translated into questions about lattices in a Lie algebra (which is, in particular, a finite-dimensional vector space over ℚ), and these things are usually easier to deal with.

In section A we develop the necessary formal properties of the logarithm operation, use them to construct the Lie algebra of a unipotent matrix group over ℚ, and as an application construct the radicable hull (or ‘Mal'cev completion’) of such a group. Section B explores the connection between finitely generated unipotent groups and lattices: we shall see that such a group is only ‘a finite distance away’ from a lattice group, that is a group whose logarithm is actually a lattice. These results are applied in section C to show that the automorphism group of a finitely generated nilpotent group is in a natural way isomorphic to an arithmetic group: what this means, and some of its implications, will be discussed when we get there.