Applications
In terms of the fundamental group in algebraic topology, the HNN extension is the construction required to understand the fundamental group of a topological space X that has been 'glued back' on itself by a mapping f (see e.g. Surface bundle over the circle). That is, HNN extensions stand in relation of that aspect of the fundamental group, as free products with amalgamation do with respect to the Seifert-van Kampen theorem for gluing spaces X and Y along a connected common subspace. Between the two constructions essentially any geometric gluing can be described, from the point of view of the fundamental group.
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.
Both HNN-extensions and amalgamated free products are basic building blocks in the Bass–Serre theory of groups acting on trees.
The idea of HNN extension has been extended to other parts of abstract algebra, including Lie algebra theory.
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