On free subgroups in maximal subgroups of skew linear groups

TL;DR

This paper investigates the existence of free non-cyclic subgroups within maximal subgroups of almost subnormal subgroups of skew linear groups over locally finite division rings, extending classical results like Tits' Alternative.

Contribution

It extends the understanding of free subgroup existence to maximal subgroups of skew linear groups over locally finite division rings, a less explored area.

Findings

Identifies conditions for free subgroup existence in these maximal subgroups

Extends Tits' Alternative to skew linear groups over division rings

Provides new insights into subgroup structure in non-commutative settings

Abstract

The study of the existence of free groups in skew linear groups have been begun since the last decades of the 20-th century. The starting point is the theorem of Tits (1972), now often is referred as Tits' Alternative, stating that every finitely generated subgroup of the general linear group $\GL_n(F)$ over a field $F$ 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.