Three articles on one-relator groups by Wilhelm Magnus

TL;DR

This collection of three foundational articles by Wilhelm Magnus introduces key results in one-relator group theory, including the Freiheitssatz and decidability of the word problem, laying groundwork for combinatorial group theory.

Contribution

First articles on one-relator groups, proving the Freiheitssatz and decidability of the word problem, pioneering combinatorial group theory research.

Findings

Proof of the Freiheitssatz

Decidability of the word problem in certain one-relator groups

Analysis of automorphisms and subgroups in related groups

Abstract

This is an English translation of three articles, originally written in German, by Wilhelm Magnus (1907--1990). The articles are from 1930, 1931, and 1932, respectively, and were the first articles published on one-relator group theory. The first article gives a proof of the famous Freiheitssatz, and many applications of the same. The second investigates various topics in combinatorial group theory, including determining the automorphism group of the figure-eight knot group, listing the subgroups of the modular group, and a proof of the decidability of the word problem in a restricted class of one-relator groups. The third gives a proof of the decidability of the word problem in all one-relator groups. A short preface is provided, summarizing the articles.