HNN extension

HNN extension
Name	HNN extension
Field	Group theory
Introduced	1949
Inventor	Graham Higman; B. H. Neumann; Hanna Neumann
Notable constructions	Baumslag–Solitar groups; Higman group

HNN extension

The HNN extension is a construction in group theory introduced by Graham Higman, Bernard Neumann, and Hanna Neumann in 1949 that embeds a group into a larger group by adjoining a stable letter that conjugates one subgroup to another. It plays a central role in combinatorial and geometric group theory, connecting decomposition theorems such as the Seifert–van Kampen theorem in algebraic topology with algebraic properties studied by researchers like Max Dehn, Otto Schreier, and Reidemeister. The construction underpins key examples investigated by Boris Baumslag, Roger Lyndon, John Stallings, William Thurston, and Mikhail Gromov.

Definition and construction

Let G be a group with isomorphic subgroups A and B and an isomorphism φ: A → B. The HNN extension of G relative to φ is formed by adjoining a new element t, often called the stable letter, subject to relations t^{-1}at = φ(a) for all a in A. This yields a group described by a presentation ⟨G, t | t^{-1}at = φ(a) (a ∈ A)⟩, a method used by Max Dehn in the study of word problems and developed further by Higman, Neumann & Neumann. The construction is closely related to amalgamated free products as in the work of J. H. C. Whitehead and was employed in structural results by Hyman Bass and Jean-Pierre Serre in the theory of groups acting on trees.

Examples and basic properties

Basic examples include the Baumslag–Solitar groups BS(m,n) = ⟨a,t | t^{-1}a^m t = a^n⟩ introduced by Boris Baumslag and Donald Solitar, which exhibit phenomena studied by Gersten and Brady. HNN extensions can produce groups with specified subgroup distortion investigated by Cornelia Drutu and Pierre de la Harpe, and they can create one-relator groups analyzed by Wilhelm Magnus and L. C. Kappe. Properties of HNN extensions inform questions about torsion, residual finiteness, and the Hopf property explored by Higman and later by Graham Higman and Derek Robinson. Surface groups and fundamental groups of graphs of groups as studied by Jean-Pierre Serre often arise via iterated HNN extensions, a theme appearing in work by John Milnor and Serge Lang.

Britton's lemma and normal forms

Britton's lemma, due to John L. Britton, provides a normal form for elements in an HNN extension and is pivotal in solving the word problem for many such groups. The lemma asserts that any word representing the identity must contain a reducible pattern involving the stable letter and elements of the associated subgroups; this technique was extended by Lyndon and Schupp in their treatment of combinatorial group theory and used by Magnus, Karrass, and Solitar. Normal form theory for HNN extensions parallels the normal forms for amalgamated free products studied by Serre and underlies combination theorems used by Maskit and Bestvina–Feighn in geometric contexts. Applications of Britton-type arguments appear in proofs by Olʹshanskiĭ and Ivanov on periodic groups and small cancellation theory.

Applications in group theory

HNN extensions are used to construct groups with prescribed properties: non-Hopfian groups exemplified by Baumslag–Solitar examples studied by Baumslag and Solitar, finitely presented infinite simple groups such as constructions influenced by Higman and Thompson groups, and pathological examples in residual properties examined by Gruenberg and Longobardi. They play a role in the embedding theorems of Higman and Magnus, in the development of JSJ decompositions by Kropholler, Sela, and Rips, and in solutions to decision problems explored by Novikov, Boone, and Adian. HNN extensions connect to actions on trees in Bass–Serre theory, which Morgan and Shalen applied to 3-manifold groups studied by William Thurston and C. McMullen.

Variations and generalizations

Generalizations include multiple stable letters and graphs of groups as formalized by Jean-Pierre Serre and extended to complexes of groups considered by Haefliger and Martin Bridson. Relative HNN extensions and ascending HNN extensions appear in the work of Dunwoody and Gromov and relate to mapping tori of automorphisms studied by Nielsen, Thurston, and Bestvina–Handel. Permutation HNN extensions and HNN constructions in pro-finite and pro-p categories were developed by Ribes, Zalesskii, and Wilkes, while analogues in algebraic geometry and low-dimensional topology connect to ideas from Grothendieck and Donaldson.

Historical context and development

The HNN construction emerged from mid-20th-century efforts to understand embedding and decision problems: Max Dehn set early questions, Higman, Neumann & Neumann formalized the extension, and subsequent researchers like Magnus, Karrass, and Solitar explored consequences. The framework influenced Bass–Serre theory and was integrated into geometric group theory by Gromov, Thurston, and Stallings, feeding into modern studies by Sela on group splittings and rigidity, and by Agol and Wise on cubulation and residual properties. Contemporary work on HNN extensions continues across collaborations involving Bridson, Haglund, Wise, and Bergeron, reflecting ongoing interactions with topology, algebraic geometry, and logic.

Category:Group theory