FREE SYMMETRIC AND UNITARY PAIRS IN DIVISION RINGS WITH INVOLUTION

Abstract

Let D be a division ring with an involution and characteristic different from 2. Then, up to a few exceptions, D contains a pair of symmetric elements freely generating a free subgroup of its multiplicative group provided that (a) it is finite-dimensional and the center has a finite sufficiently large transcendence degree over the prime field, or (b) the center is uncountable, but not algebraically closed in D. Under conditions (a), if the involution is of the first kind, it is also shown that the unitary subgroup of the multiplicative group of D contains a free subgroup, with one exception. The methods developed are also used to exhibit free subgroups in the multiplicative group of a finite-dimensional division ring provided the center has a sufficiently large transcendence degree over its prime field.