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

TL;DR

This paper provides a comprehensive survey of the history and recent advances in the theory of one-relator groups, highlighting key developments over nearly a century.

Contribution

It offers a detailed overview of the historical evolution and recent progress in the study of one-relator groups, structured into two independent chapters.

Findings

Historical development of one-relator groups until geometric group theory

Recent progress in the theory of one-relator groups

Minimal overlap between historical and recent developments

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.