On the cokernel of the Baumslag rationalization

TL;DR

This paper proves that the cokernel of the homomorphism from a free group of rank two to its Baumslag rationalization is non-abelian and contains a free subgroup of countable rank, answering a question by Farjoun.

Contribution

It establishes the non-abelian nature of the cokernel and shows it contains a large free subgroup, providing new insights into the structure of Baumslag rationalizations.

Findings

Cokernel of the homomorphism is not abelian

Cokernel contains a free subgroup of countable rank

Answers a question posed by Emmanuel Farjoun

Abstract

We prove that for the free group of rank two $F$ the cokernel of the homomorphism to its Baumslag rationalization $F\to {\sf Bau}(F)$ is not abelian. Moreover, this cokernel contains a free subgroup of countable rank. This answers a question of Emmanuel Farjoun.