In mathematics, the HNN extension is an important construction of combinatorial group theory.
be an isomorphism between two subgroups of G. Let t be a new symbol not in S, and define The group
is called the HNN extension of G relative to α.
A consequence is that two isomorphic subgroups of a given group are always conjugate in some overgroup; the desire to show this was the original motivation for the construction.
A key property of HNN-extensions is a normal form theorem known as Britton's Lemma.
.Most basic properties of HNN-extensions follow from Britton's Lemma.
These consequences include the following facts: Applied to algebraic topology, the HNN extension constructs the fundamental group of a topological space X that has been 'glued back' on itself by a mapping f : X → X (see e.g.
Thus, HNN extensions describe the fundamental group of a self-glued space in the same way that free products with amalgamation do for two spaces X and Y glued along a connected common subspace, as in the Seifert-van Kampen theorem.
The HNN extension is a natural analogue of the amalgamated free product, and comes up in determining the fundamental group of a union when the intersection is not connected.
[3] These two constructions allow the description of the fundamental group of any reasonable geometric gluing.
[4][5] HNN-extensions play a key role in Higman's proof of the Higman embedding theorem which states that every finitely generated recursively presented group can be homomorphically embedded in a finitely presented group.
Most modern proofs of the Novikov–Boone theorem about the existence of a finitely presented group with algorithmically undecidable word problem also substantially use HNN-extensions.