The soluble subgroups and the Tits alternative in linear groups over rings of fractions of polycyclic group rings. I
In this tour-de-force the author studies subrings of the matrix ring \(D^{n\times n}\) and subgroups of \(GL(n,D)\) for a certain class of division rings \(D\) that contains the division rings of quotients of the group algebras of torsion-free polycyclic-by-finite groups \(P\) over fields \(F\) of characteristic zero or sufficiently large in comparison with the Hirsch number \(h\) of \(P\). In this first paper of the series he concentrates on the soluble subgroups of \(GL(n,D)\). The Tits alternative is promised in Part II. With \(D\) as above let \(G\) be a soluble-by-finite subgroup of \(GL(n,D)\). The author gives a fairly precise description of \(G\). For example if \(G\) is completely reducible and \(D\) is the example above, then \(G\) is abelian, by finite, by soluble of derived length at most \(h\), by finite with order bounded in terms of \(n\) and \(h\). A very wide range of techniques is used, including some recent and yet to be published number-theoretic work of R. Murty stemming from his work on Artin's Problem. At one point (Theorem Iiii) the author needs to assume the generalised Riemann hypothesis. In my view the most novel and interesting innovation in the paper is the use of Krull dimension to bound the derived length of certain soluble groups in terms of the \(h\) above. The crucial result of the author here is the following. Let \(R\) be the ring of quotients of a crossed product of a Goldie \(F\)-algebra and a polycyclic-by-finite group with Hirsch number \(h\). Then the Krull dimension (say right) of the algebra \(R^{op}\otimes_ F R\) is at least \(h\). The author's proof uses R. Resco's theory of \(r\)-sequences, but direct proofs are now available.
group algebras of torsion-free polycyclic-by-finite groups
scientific article; zbMATH DE number 179079
scientific article