# On free subgroups in division rings

## TL;DR

This paper proves that certain division rings contain free non-cyclic subgroups unless they are commutative, addressing a conjecture and showing such subgroups exist in division algebras related to torsion-free solvable groups.

## Contribution

It demonstrates the existence of free non-cyclic subgroups in a broad class of division rings, confirming a special case of Lichtman's conjecture and extending to group algebra quotients.

## Key findings

- Division rings $K(x;\sigma,\delta)$ contain free non-cyclic subgroups unless commutative.
- Division algebras from group algebra quotients of torsion-free solvable groups have free non-cyclic subgroups.
- Answer to a special case of Lichtman's conjecture.

## Abstract

Let $K$ be a field and let $\sigma$ be an automorphism and let $\delta$ be a $\sigma$-derivation of $K$. Then we show that the multiplicative group of nonzero elements of the division ring $D=K(x;\sigma,\delta)$ contains a free non-cyclic subgroup unless $D$ is commutative, answering a special case of a conjecture of Lichtman. As an application, we show that division algebras formed by taking the Goldie ring of quotients of group algebras of torsion-free non-abelian solvable-by-finite groups always contain free non-cyclic subgroups.