Free subgroups in linear groups over some skew fields (Q1083548)

Let D be the division ring of quotients of the universal enveloping algebra of a finite dimensional Lie algebra over a field of characteristic zero. In Theorem A the author analyses the structure of a subgroup of GL(n,D) with no non-cyclic free subgroups. There are several corollaries. For example Corollary 2 gives a bound to the derived length of a soluble subgroup of GL(n,D) in terms of n and the dimension of L. The proof uses a translation of the problem to one in positive characteristic and the application of an interesting theorem (Theorem B) on the structure of subgroups of GL(n,E), where E is a division ring of positive characteristic p and finite dimension \(q^ m\) over its centre, q a prime with \(n<q-1\). Finally the author considers similar questions over the division ring of quotients of the group algebra (over a field) of a free metabelian group and over the universal field of fractions of the group algebra of a free group.