On the coherence of one-relator groups and their group algebras

TL;DR

This work resolves Baumslag's conjecture by proving that every one-relator group is coherent and that its group algebra over a field of characteristic zero is coherent under reducible presentations. The authors develop a two-tier strategy: first establish homological coherence for broad two-dimensional classes (notably fundamental groups of two–complexes with non-positive immersions and locally indicable CD$^2$ groups with $b_2^{(2)}=0$), then promote FP$_2$ control to full finite presentability using Bass–Serre theory and the Magnus hierarchy. Central tools include NTPI and bireducible complexes, flat module constructions feeding Tor calculations, and the use of Hughes-free/Linnell/Dubrovin division rings to control homological dimensions. The results yield broad applications: coherence for mapping tori of free groups, right-angled Artin groups and certain Coxeter groups, groups with staggered presentations, and an alternative route to the rank-1 Hanna Neumann conjecture. Together, these contributions deepen the connection between geometric group theory, homological algebra, and the algebraic structure of group rings, with potential implications for understanding coherence in broader group classes and their algebras.

Abstract

We prove that one-relator groups are coherent, solving a well-known problem of Gilbert Baumslag. Our proof strategy is readily applicable to many classes of groups of cohomological dimension two. We show that fundamental groups of two-complexes with non-positive immersions are homologically coherent, we show that groups with staggered presentations and many Coxeter groups are coherent and we show that group algebras over fields of characteristic zero of groups with reducible presentations without proper powers are coherent.

Theorems & Definitions (117)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1
• Proposition 2.2