8 - Soluble ℤ-linear groups

Summary

In this chapter we are going to take a closer look at the way in which a polyclic group can sit inside GL_n(ℤ). As we saw in Chapter 7, a good understanding of isomorphisms between polycyclic groups would seem to depend on knowing when soluble subgroups of GL_n(ℤ) are conjugate, and an algorithm for deciding this question is sketched in section D, below. Sections B and C investigate the normalizer in GL_n(ℤ) of an arbitrary soluble-by-finite subgroup. Using the fact (proved in Chapter 5) that the holomorph of a polycyclic-by-finite group G can be embedded in some GL_n(ℤ), we shall derive some rather striking consequences: (a) Aut G has a normal subgroup K, isomorphic to an arithmetic group, with (Aut G)/K finitely generated and abelian-by-finite; (b) Aut G is a finitely presented group; (c) the finite subgroups of Aut G lie in finitely many conjugacy classes; and (d) for each natural number m, there exist only finitely many non-isomorphic extensions of G by a group of order m.

An important tool in these investigations is the Zariski topology in a linear group. This has been lurking in the background to some of the previous chapters; but the time has come for a fuller discussion of the topic, and this is given in section A.

The Zariski topology

Let k be a field and m a natural number.