Jump to content

HNN extension

From Wikipedia, the free encyclopedia

In mathematics, the HNN extension is an important construction of combinatorial group theory.

Introduced in a 1949 paper Embedding Theorems for Groups[1] by Graham Higman, Bernhard Neumann, and Hanna Neumann, it embeds a given group G into another group G' , in such a way that two given isomorphic subgroups of G are conjugate (through a given isomorphism) in G' .

Construction

[edit]

Let G be a group with presentation , and let 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. The original group G is called the base group for the construction, while the subgroups H and K are the associated subgroups. The new generator t is called the stable letter.

Key properties

[edit]

Since the presentation for contains all the generators and relations from the presentation for G, there is a natural homomorphism, induced by the identification of generators, which takes G to . Higman, Neumann, and Neumann proved that this morphism is injective, that is, an embedding of G into . 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.

Britton's Lemma

[edit]

A key property of HNN-extensions is a normal form theorem known as Britton's Lemma.[2] Let be as above and let w be the following product in :

Then Britton's Lemma can be stated as follows:

Britton's Lemma. If w = 1 in Gα then

  • either and g0 = 1 in G
  • or and for some i ∈ {1, ..., n−1} one of the following holds:
  1. εi = 1, εi+1 = −1, giH,
  2. εi = −1, εi+1 = 1, giK.

In contrapositive terms, Britton's Lemma takes the following form:

Britton's Lemma (alternate form). If w is such that

  • either and g0 ≠ 1 ∈ G,
  • or and the product w does not contain substrings of the form tht−1, where hH and of the form t−1kt where kK,

then in .

Consequences of Britton's Lemma

[edit]

Most basic properties of HNN-extensions follow from Britton's Lemma. These consequences include the following facts:

  • The natural homomorphism from G to is injective, so that we can think of as containing G as a subgroup.
  • Every element of finite order in is conjugate to an element of G.
  • Every finite subgroup of is conjugate to a finite subgroup of G.
  • If contains an element such that is contained in neither nor for any integer , then contains a subgroup isomorphic to a free group of rank two.

Applications and generalizations

[edit]

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. Surface bundle over the circle). 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. These two constructions allow the description of the fundamental group of any reasonable geometric gluing. This is generalized into the Bass–Serre theory of groups acting on trees, constructing fundamental groups of graphs of groups.[3][4]

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.

The idea of HNN extension has been extended to other parts of abstract algebra, including Lie algebra theory.

See also

[edit]

References

[edit]
  1. ^ Higman, Graham; Neumann, Bernhard H.; Neumann, Hanna (1949). "Embedding Theorems for Groups". Journal of the London Mathematical Society. s1-24 (4): 247–254. doi:10.1112/jlms/s1-24.4.247.
  2. ^ Roger C. Lyndon and Paul E. Schupp. Combinatorial Group Theory. Springer-Verlag, New York, 2001. "Classics in Mathematics" series, reprint of the 1977 edition. ISBN 978-3-540-41158-1; Ch. IV. Free Products and HNN Extensions.
  3. ^ Serre, Jean-Pierre (1980), Trees. Translated from the French by John Stillwell, Berlin-New York: Springer-Verlag, ISBN 3-540-10103-9
  4. ^ Warren Dicks; M. J. Dunwoody. Groups acting on graphs. p. 14. The fundamental group of graphs of groups can be obtained by successively performing one free product with amalgamation for each edge in the maximal subtree and then one HNN extension for each edge not in the maximal subtree.