Free subgroups in almost subnormal subgroups of general skew linear groups

TL;DR

This paper studies the existence of non-cyclic free subgroups within certain almost subnormal subgroups of the general linear group over a weakly locally finite division ring, providing new insights and classifications.

Contribution

It establishes conditions for free subgroup existence in these subgroups and characterizes all infinite finitely generated almost subnormal subgroups of ${ m GL}_n(D).

Findings

Existence of non-cyclic free subgroups in certain almost subnormal subgroups.

Classification of all infinite finitely generated almost subnormal subgroups.

Applications to understanding subgroup structure in linear groups.

Abstract

Let $D$ be a weakly locally finite division ring and $n$ a positive integer. In this paper, we investigate the problem on the existence of non-cyclic free subgroups in non-central almost subnormal subgroups of the general linear group ${\rm GL}_n(D)$. Further, some applications of this fact are also investigated. In particular, all infinite finitely generated almost subnormal subgroups of ${\rm GL}_n(D)$ are described.