Free structures in division rings

TL;DR

This paper investigates the existence of free subalgebras in division rings, specifically proving Makar-Limanov's conjecture for certain skew polynomial rings over function fields of algebraic varieties.

Contribution

It proves Makar-Limanov's conjecture for division rings of fractions of skew polynomial rings over specific algebraic function fields.

Findings

Confirmed the conjecture for function fields of abelian varieties

Confirmed the conjecture for function fields of projective spaces

Established conditions under which free subalgebras exist in these division rings

Abstract

Makar-Limanov's conjecture states that if a division ring D is finitely generated and infinite dimensional over its center k then D contains a free k-subalgebra of rank 2. In this work, we will investigate the existence of such structures in D, the division ring of fractions of the skew polynomial ring L[t;\sigma], where t is a variable and $\sigma$ is a k-automorphism of L. For instance, we prove Makar-Limanov's conjecture when either L is the function field of an abelian variety or the function field of the n-dimensional projective space.