HNN extensions and embedding theorems for groups

TL;DR

This paper reviews the historical significance and foundational role of HNN extensions in group theory, highlighting their use in constructing groups and their central role in Bass-Serre theory.

Contribution

It provides a detailed overview of the original HNN paper and discusses subsequent developments and applications in group theory.

Findings

HNN extensions are fundamental tools in combinatorial group theory.

They enable the construction of new groups and decompositions.

HNN extensions are central to Bass-Serre theory.

Abstract

The Higman-Neumann-Neumann (HNN) paper of 1949 is a landmark of group theory in the twentieth century. The proof of its main theorem covers less than a page and uses only pre-existing technology, but the construction that it introduced -- the HNN extension -- quickly became one of the principal tools of combinatorial group theory, widely used to build new groups and to describe enlightening decompositions of existing groups. In this article, we shall describe the contents of the HNN paper, and then discuss some of the important developments that followed in its wake, leading up to the central role that HNN extensions play in the Bass--Serre theory of groups acting on trees.