# HNN-extension

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In 1949, G. Higman, B.H. Neumann and H. Neumann [a4] proved several famous embedding theorems for groups using a construction later called the HNN-extension. The theory of HNN-groups is central to geometric and combinatorial group theory and should be viewed in parallel with amalgamated products (cf. also Amalgam of groups).

The easiest way to define an HNN-group is in terms of presentations of groups.

## Presentation of groups.

A presentation of a group is a pair where is a subset of , the free group on the set , and is isomorphic (cf. also Isomorphism) to the quotient group , where is the intersection of all normal subgroups of containing (cf. also Normal subgroup). The subgroup is called the normal closure of in . See also Presentation.

Given an arbitrary group , there is an obvious homomorphism such that for all . Clearly, is a presentation for .

## HNN-extensions.

Suppose is an isomorphism of subgroups of a group and is not in . The HNN-extension of with respect to has presentation The generator is called the stable letter, the base group and and the associated subgroups of this HNN-extension. When , the HNN-extension is called ascending.

Shorthand notation for the above group is or .

In [a4] it was shown that the mapping taking for all is a monomorphism. The rest of the normal form theorem for HNN-extensions was proved by J.L. Britton in 1963 [a1] (Britton's lemma): Let be a sequence of elements of and let the letter , with or without subscripts, denote . A sequence will be called reduced if there is no consecutive subsequence with or with . For a reduced sequence and , the element of is different from the unit element.

In the original reference [a4], the following theorem is proved: Every group can be embedded in a group in which all elements of the same order are conjugate (cf. also Conjugate elements). In particular, every torsion-free group can be embedded in a group with only two conjugacy classes. If is countable, so is . Also, every countable group can be embedded in a group generated by two elements of infinite order. The group has an element of finite order if and only if does. If is finitely presentable, then so is .

For an excellent account of the history of HNN-extensions, see [a2]. See [a5], Chap. IV, for basic results and landmark uses of HNN-extensions, such as: the torsion theorem for HNN-extensions; the Collins conjugacy theorem for HNN-extensions; the construction of finitely-presented non-Hopfian groups (in particular, the Baumslag–Solitar group is non-Hopfian; cf. also Non-Hopf group); decompositions of -relator groups; Stallings' classification of finitely-generated groups with more than one end in terms of amalgamated products and HNN-extensions; and Stallings' characterization of bipolar structures on groups.

HNN-extensions are of central importance in, e.g., the modern version of the Van Kampen theorem (based on topological results in [a8], [a9]); the Bass–Serre theory of groups acting on trees and the theory of graphs of groups (see [a7]); Dunwoody's accessibility theorem [a3]; and JSJ decompositions of groups [a6].

How to Cite This Entry:
HNN-extension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=HNN-extension&oldid=17603
This article was adapted from an original article by Mike Mihalik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article