Book contents
- Frontmatter
- Contents
- Introduction
- Finite Simple Groups and Fusion Systems
- Finite and Infinite Quotients of Discrete and Indiscrete Groups
- Local-Global Conjectures and Blocks of Simple Groups
- A Survey on Some Methods of Generating Finite Simple Groups
- One-Relator Groups: An Overview
- New Progress in Products of Conjugacy Classes in Finite Groups
- Aspherical Relative Presentations all Over Again
- Simple Groups, Generation and Probabilistic Methods
- Irreducible Subgroups of Simple Algebraic Groups – A Survey
- Practical Computation with Linear Groups Over Infinite Domains
- Beauville p-Groups: A Survey
- Structural Criteria in Factorised Groups Via Conjugacy Class Sizes
- Growth in Linear Algebraic Groups and Permutation Groups: Towards a Unified Perspective
- L2-Betti Numbers and their Analogues in Positive Characteristic
- On the Pronormality of Subgroups of Odd Index in Finite Simple Groups
- Vertex Stabilizers of Graphs with Primitive Automorphism Groups and a Strong Version of the Sims Conjecture
- On the Character Degrees of a Sylow p-Subgroup of a Finite Chevalley Group G(pf) Over a Bad Prime
- Patterns on Symmetric Riemann Surfaces
- Subgroups of Twisted Wreath Products
- Some Remarks on Self-Dual Codes Invariant Under Almost Simple Permutation Groups
- Test Elements: From Pro-p to Discrete Groups
- References
One-Relator Groups: An Overview
Published online by Cambridge University Press: 15 April 2019
Book contents
- Frontmatter
- Contents
- Introduction
- Finite Simple Groups and Fusion Systems
- Finite and Infinite Quotients of Discrete and Indiscrete Groups
- Local-Global Conjectures and Blocks of Simple Groups
- A Survey on Some Methods of Generating Finite Simple Groups
- One-Relator Groups: An Overview
- New Progress in Products of Conjugacy Classes in Finite Groups
- Aspherical Relative Presentations all Over Again
- Simple Groups, Generation and Probabilistic Methods
- Irreducible Subgroups of Simple Algebraic Groups – A Survey
- Practical Computation with Linear Groups Over Infinite Domains
- Beauville p-Groups: A Survey
- Structural Criteria in Factorised Groups Via Conjugacy Class Sizes
- Growth in Linear Algebraic Groups and Permutation Groups: Towards a Unified Perspective
- L2-Betti Numbers and their Analogues in Positive Characteristic
- On the Pronormality of Subgroups of Odd Index in Finite Simple Groups
- Vertex Stabilizers of Graphs with Primitive Automorphism Groups and a Strong Version of the Sims Conjecture
- On the Character Degrees of a Sylow p-Subgroup of a Finite Chevalley Group G(pf) Over a Bad Prime
- Patterns on Symmetric Riemann Surfaces
- Subgroups of Twisted Wreath Products
- Some Remarks on Self-Dual Codes Invariant Under Almost Simple Permutation Groups
- Test Elements: From Pro-p to Discrete Groups
- References
Summary
In 1985, at Groups St Andrews, Gilbert Baumslag gave a short course on one-relator groups which provided a look at the subject up to that point. In this paper we partially update the massive amount of work done over the past three decades. For the most part we concentrate on areas and results to which the authors have made contributions. We look at the important connections with surface groups and elementary theory, and describe the surface group conjecture and the Gromov conjecture on surface subgroups. We look at the solution by Wise of Baumslag’s residual finiteness conjecture and discuss a new Baumslag conjecture on virtually free-by-cyclic groups. We examine various amalgam decompositions of one-relator groups and the Baumslag-Shalen conjectures. We then look at a series of open problems in one-relator group theory and their status. Finally we introduce a concept called plainarity based on the Magnus breakdown of a one-relator group which might provide a systematic approach to the solution of problems in one-relator groups.
- Type
- Chapter
- Information
- Groups St Andrews 2017 in Birmingham , pp. 119 - 157Publisher: Cambridge University PressPrint publication year: 2019
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