HNN extension

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In mathematics, the HNN extension is a basic construction of combinatorial group theory.

Introduced in a 1949 paper Embedding Theorems for Groups[1] by Graham Higman, B. H. 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 G = <S|R>, and let α : HK be an isomorphism between two subgroups of G. Let t be a new symbol not in S, and define

G*_{\alpha} = \left \langle S,t \Big| R, tht^{-1}=\alpha(h), \forall h\in H \right \rangle.

The group Gα 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 Gα 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 Gα. Higman, Neumann and Neumann proved that this morphism is injective, that is, an embedding of G into Gα. 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 Gα be as above and let w be the following product in Gα:

w=g_0 t^{\varepsilon_1} g_1 t^{\varepsilon_2} \cdots g_{n-1} t^{\varepsilon_n}g_n, \qquad g_i \in G, \varepsilon_i = \pm 1.

Then Britton's Lemma can be stated as follows:

Britton's Lemma. If w = 1 in Gα then

  • either n = 0 and g0 = 1 in G
  • or n > 0 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 n = 0 and g0 ≠ 1 ∈ G,
  • or n > 0 and the product w does not contain substrings of the form tht−1, where hH and of the form t−1kt where kK,
then w ≠ 1 in Gα.

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 Gα is injective, so that we can think of Gα as containing G as a subgroup.
  • Every element of finite order in Gα is conjugate to an element of G.
  • Every finite subgroup of Gα is conjugate to a finite subgroup of G.
  • If HG and KG then Gα contains a subgroup isomorphic to a free group of rank two.

Applications[edit]

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.[3]

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

Generalizations[edit]

HNN extensions are elementary examples of fundamental groups of graphs of groups, and as such are of central importance in Bass–Serre theory.

References[edit]

  1. ^ Higman, Graham; B. H. Neumann; Hanna Neumann (1949). "Embedding Theorems for Groups" (PDF). Journal of the London Mathematical Society. s1-24 (4): 247–254. doi:10.1112/jlms/s1-24.4.247. Retrieved 2008-03-15. 
  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. ^ Jean-Pierre Serre. Trees. Translated from the French by John Stillwell. Springer-Verlag, Berlin-New York, 1980. ISBN 3-540-10103-9