HNN extensions and embedding theorems for groups

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.

💡 Research Summary

This article provides a detailed exposition and historical analysis of the seminal 1949 paper “Embedding Theorems for Groups” by Graham Higman, B. H. Neumann, and Hanna Neumann, which introduced the construction now known as the HNN extension. The paper begins by contextualizing the HNN extension within the framework of amalgamated free products, the principal pre-existing tool used in the original proofs.

The core of the 1949 work is Theorem I, which states that for any group G with an isomorphism φ between two subgroups A and B, there exists a larger group H containing G and an element t such that t⁻¹at = φ(a) for all a in A. The original proof is remarkably concise, constructing H as an amalgamated free product of two copies of G∗ℤ. The authors immediately identify the subgroup generated by G and t as the canonical, minimal such group, which is the HNN extension, denoted G∗φ.

The paper then generalizes this to Theorem II, covering families of isomorphisms between subgroups, enabling two major applications. The first application (Theorem III) shows that any torsion-free group G can be embedded into a group G* where every pair of non-trivial elements is conjugate. By iteratively applying multiple HNN extensions to conjugate all elements of the same order and taking a direct limit, they construct the first examples of infinite, countable, torsion-free simple groups. The second landmark application (Theorem IV) proves that every countable group can be embedded into a 2-generator group, effectively coding the generators of the original group into words in two new letters while preserving the defining relations.

The latter part of the article traces the profound legacy of the HNN construction. In the 1950s and 60s, it became a central tool in pure combinatorial group theory, crucial for Higman’s celebrated Embedding Theorem for finitely presented groups and for studying decision problems like the word problem. Subsequently, the focus shifted towards decompositional uses with a more topological flavor. HNN extensions arise naturally when computing fundamental groups of spaces via the Seifert-van Kampen theorem upon cutting along a subspace. This topological interpretation culminates in Bass-Serre theory, where HNN extensions are one of the two fundamental building blocks (alongside amalgamated free products) of graphs of groups. This theory describes how groups act on trees, making HNN extensions a cornerstone of modern geometric group theory and low-dimensional topology. Thus, the article highlights the HNN extension’s journey from a clever algebraic embedding trick to a fundamental structural concept in modern mathematics.