Free $\mathbb Q$-groups are residually torsion-free nilpotent

TL;DR

The paper solves Baumslag's conjecture by proving that free $m{ ext{$ olinebreak$}$Q}$-groups are residually torsion-free nilpotent. The authors reduce the problem to embedding finitely generated subgroups into free pro-$p$ groups through strong, iterated centralizer extensions, and they develop a broad toolbox—R-rings, completed group algebras, Sylvester rank functions, and division rings—to control dimensions and ideals in this setting. Key contributions include the construction of universal division rings for pro-$p$ contexts, a mod-$p$ L$^2$-Betti number framework, and a robust theory of $m{ ext{D}}$-torsion-free modules that underpins the inductive embedding steps. These results illuminate the structure of $m{Q}$-completions and limit groups, yielding residually torsion-free nilpotent realizations and connecting to parafree and ICE groups, with broader implications for linearity questions in free $m{ ext{Q}}$-groups and free pro-$p$ groups.</p>

Abstract

We develop a method to show that some (abstract) groups can be embedded into a free pro-$p$ group. In particular, we show that a finitely generated subgroup of a free $\mathbb Q$-group can be embedded into a free pro-$p$ group for almost all primes $p$. This solves an old problem raised by G. Baumslag: free $\mathbb Q$-groups are residually torsion-free nilpotent.

Theorems & Definitions (84)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• Proposition 2.3
• Lemma 2.4
• Remark 2.5