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

TL;DR

This paper demonstrates that free $ ext{Q}$-groups are residually torsion-free nilpotent by embedding finitely generated subgroups into free pro-$p$ groups for almost all primes, solving a longstanding problem.

Contribution

It introduces a new method to embed groups into free pro-$p$ groups, establishing residual torsion-free nilpotency of free $ ext{Q}$-groups.

Findings

Finitely generated subgroups of free $ ext{Q}$-groups embed into free pro-$p$ groups for almost all primes

Free $ ext{Q}$-groups are residually torsion-free nilpotent

Provides a solution to G. Baumslag's old problem

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.