Free subgroups in maximal subgroups of skew linear groups

Abstract

The study of the existence of free groups in skew linear groups have begun since the last decades of the 20th century. The starting point is the theorem of Tits (1972), now often referred to as Tits’ Alternative, stating that every finitely generated subgroup of the general linear group [Formula: see text] over a field [Formula: see text] either contains a non-cyclic free subgroup or it is solvable-by-finite. In this paper, we study the existence of non-cyclic free subgroups in maximal subgroups of an almost subnormal subgroup of the general skew linear group over a locally finite division ring.

On locally solvable subgroups in division rings

“…The former case implies F (T ) = D, hence D is a locally finite division ring by Lemma 2.4. Since M is locally solvable, it contains no non-cyclic free subgroups, and thus [D : F ] < ∞ by [3,Theorem 3.1]. If the latter case occurs, then T ⊆ F (T ) * ∩ G is subnormal in F (T ) * .…”