Constructing free groups in a normal subgroup of the multiplicative group of division rings

Abstract

Let D be a division ring with center k and multiplicative group D , and let N be a normal subgroup of D . Assuming that N contains a subgroup G which is nonabelian torsion-free polycyclic-by-finite (not abelian-by-finite), we construct a free noncyclic subgroup of N in terms of elements of G. We also show that if char k ¤ 2, D is generated over k by the torsion-free polycyclic-by finite group G (not abelian-by-finite), and if it is possible to extend the involution x D x 1 of G to a k involution of D, then D contains a free symmetric pair.