One-relator groups with torsion are coherent

TL;DR

This paper proves that all one-relator groups with torsion are coherent, meaning every finitely generated subgroup is finitely presented, resolving a question posed by Baumslag in 1974.

Contribution

It establishes the coherence of one-relator groups with torsion, a significant open problem in group theory, providing a complete answer for this class.

Findings

All one-relator groups with torsion are coherent.

Every finitely generated subgroup of such groups is finitely presented.

Answers Baumslag's 1974 question for this class.

Abstract

We show that any one-relator group $G=F/\langle\langle w\rangle\rangle$ with torsion is coherent -- i.e., that every finitely generated subgroup of $G$ is finitely presented -- answering a 1974 question of Baumslag in this case.