Free pairs of symmetric and unitary units in normal subgroups of a division ring

Abstract

Let [Formula: see text] be the field of fractions of the group algebra [Formula: see text] of the Heisenberg group [Formula: see text], over the field [Formula: see text] of characteristic [Formula: see text]. We show that for some involutions of [Formula: see text] that are not induced from involutions of [Formula: see text], [Formula: see text] contains free symmetric and unitary pairs. We also give a general condition for a normal unitary subgroup of a division ring to contain a free group, and prove a generalization of Lewin’s Conjecture.