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The theory of one-relator groups: history and recent progress

Marco Linton, Carl-Fredrik Nyberg-Brodda

TL;DR

The paper traces the theory of one-relator groups from its Dehn–Magnus heritage through a modern geometric and homological toolkit. It integrates classical results (Freiheitssatz, Lyndon’s Identity Theorem) with contemporary methods (Brown’s criterion, Friedl–Tillmann polytope, Wise’s negative immersions) to map the landscape of splittings, subgroup structure, and group-ring properties. Key advances include a complete coherence proof for all one-relator groups, a detailed understanding of torsion and torsion-free cases via the Newman Spelling Theorem and Brown–Moldavanskii frameworks, and a robust program describing JSJ-type decompositions and L^2-invariants for these groups. The work highlights how geometry, topology, and algebra combine to resolve longstanding problems (e.g., decidability of word conjugacy and isomorphism in broad classes) while posing new questions about two-relator groups and finer invariants with practical implications for hyperbolic and CAT(0) techniques.

Abstract

The theory of one-relator groups is now almost a century old. The authors therefore feel that a comprehensive survey of this fascinating subject is in order, and this document is an attempt at precisely such a survey. This article is divided into two chapters, reflecting the two different phases in the story of one-relator groups. The first chapter, written by the second author, covers the historical development of the theory roughly until the advent of geometric group theory. The second chapter, written by the first author, covers the recent progress in the theory up until the present day. The two chapters can be read independently of one another and have minimal overlap.

The theory of one-relator groups: history and recent progress

TL;DR

The paper traces the theory of one-relator groups from its Dehn–Magnus heritage through a modern geometric and homological toolkit. It integrates classical results (Freiheitssatz, Lyndon’s Identity Theorem) with contemporary methods (Brown’s criterion, Friedl–Tillmann polytope, Wise’s negative immersions) to map the landscape of splittings, subgroup structure, and group-ring properties. Key advances include a complete coherence proof for all one-relator groups, a detailed understanding of torsion and torsion-free cases via the Newman Spelling Theorem and Brown–Moldavanskii frameworks, and a robust program describing JSJ-type decompositions and L^2-invariants for these groups. The work highlights how geometry, topology, and algebra combine to resolve longstanding problems (e.g., decidability of word conjugacy and isomorphism in broad classes) while posing new questions about two-relator groups and finer invariants with practical implications for hyperbolic and CAT(0) techniques.

Abstract

The theory of one-relator groups is now almost a century old. The authors therefore feel that a comprehensive survey of this fascinating subject is in order, and this document is an attempt at precisely such a survey. This article is divided into two chapters, reflecting the two different phases in the story of one-relator groups. The first chapter, written by the second author, covers the historical development of the theory roughly until the advent of geometric group theory. The second chapter, written by the first author, covers the recent progress in the theory up until the present day. The two chapters can be read independently of one another and have minimal overlap.

Paper Structure

This paper contains 89 sections, 148 theorems, 171 equations, 8 figures.

Table of Contents

  1. History
  2. Introduction
  3. Dehn, free groups, and presentations (1880--1930)
  4. Presentations and Dehn's fundamental problems
  5. Surface groups and knot groups
  6. The Nielsen--Schreier Theorem, amalgamated free products
  7. Magnus, the Freiheitssatz, and the word problem (1930-1932)
  8. The Freiheitssatz
  9. Early applications of the Freiheitssatz.
  10. The word problem for one-relator groups
  11. The Lull Phase (1932--1960)
  12. Some results on free groups
  13. Lyndon's Identity Theorem
  14. Residual properties of groups
  15. Why did progress halt?
  16. ...and 74 more sections

Key Result

Theorem 1.2.2

Let $G = \pi_1(\Sigma_g)$ be as in Eq:pres-of-sigma-g. Let $w$ be a non-trivial reduced word, in the generators of $G$, which represents the identity element of $G$. Then there exists a subword $u$ of $w$ such that $u$ is a subword of some cyclic conjugate of the defining relator of $G$ or its inver

Figures (8)

  • Figure 1: A simple illustration of \ref{['one-relator_splitting']} in the case of a genus two surface.
  • Figure 2: A one-relator $\mathbb{Z}$-domain.
  • Figure 3: The trace of the word $a^2b^3ab^{-2}a^{-1}bab^{-1}a^{-2}bab^{-1}a^{-2}b^{-1}$.
  • Figure 4: The 1-skeleton of $Z_{\phi}$, obtained by projecting $\Lambda_w$ onto $L_{\phi}$.
  • Figure 5: An example of a one-relator complex $Z_{\phi}$ with $\phi$ freely descending and ascending.
  • ...and 3 more figures

Theorems & Definitions (211)

  • Remark 1.2.1
  • Theorem 1.2.2: Dehn's Spelling Theorem, De12
  • Theorem 1.2.3
  • Proposition 1.2.4: Dehn/Nielsen, 1917 Ni17
  • Theorem 1.2.5: The Nielsen--Schreier Theorem
  • Theorem 1.2.6: Schreier, 1927
  • Theorem 1.3.1: Magnus' Freiheitssatz, 1930 Ma30
  • Corollary 1.3.2: Magnus, 1930
  • Theorem 1.3.3: Magnus' Conjugacy Theorem, Ma30
  • Proposition 1.3.4: Magnus, Ma30
  • ...and 201 more