- HNN extension
In
mathematics , the HNN extension is a basic construction ofcombinatorial group theory .Introduced in a 1949 paper "Embedding Theorems for Groups" [ cite journal|title=Embedding Theorems for Groups|journal=Jornal of the London Mathematical Society|date=1949|first=Graham|last=Higman|coauthors=B. H. Neumann, Hanna Neumann|volume=s1-24|issue=4|pages=247–254|doi= 10.1112/jlms/s1-24.4.247|url=http://jlms.oxfordjournals.org/cgi/reprint/s1-24/4/247.pdf|format=PDF|accessdate=2008-03-15 ] by
Graham Higman ,B. H. Neumann andHanna 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
Let G be a group with
presentation G=langle Smid R angle, and letalpha be an isomorphism between two subgroups H and K of G.Let t be a new symbol not in S, and define:G*_{alpha} = langle S,t mid R, tht^{-1}=alpha(h), forall hin H angleThe group G*_{alpha} is called the "HNN extension of"G "relative to" alpha. 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
Since the presentation for G*_{alpha} contains all the generators and relations from the presentation forG, there is a natural homomorphism, induced by the identification of generators, which takes G to G*_{alpha}. Higman, Neumann and Neumann proved that this morphism is injective, that is, an embedding of Ginto G*_{alpha}. 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
A key property of HNN-extensions is a normal form theorem known as "Britton's Lemma". [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-13: 9783540411581; Ch. IV. Free Products and HNN Extensions.] It states that for a group G*_{alpha} as above, whenever in G*_{alpha} a product of the form
:w=g_0 t^{epsilon_1} g_1 t^{epsilon_2} dots t^{epsilon_n}g_n,
where g_iin G, epsilon_iin{1,-1} for i=1,dots, n,
satisfies w=1 in G*_{alpha}
then either "n=0" and g_0=1 in "G" or else for some i=1,dots, n-1 one of the following holds:
* epsilon_i}=1, {epsilon_{i+1=-1, g_iin H ,
* epsilon_i}=-1, {epsilon_{i+1=1, g_iin K .Restated in contrapositive terms, Britton's Lemma says that if "w" is a product of the above form such that
*either "n=0" and g_0 e 1 in "G",
*or "n>0" and the product "w" does not contain substrings of the form tht^{-1}, where hin H and of the form t^{-1}kt where kin K,then w e 1 in G*_{alpha}.
Consequences of Britton's Lemma
Most basic properties of HNN-extensions follow from Britton's Lemma. These consequences include the following facts:
*The natural homomorphism from "G" to G*_{alpha} is injective, so that we can think of G*_{alpha} as containing "G" as a
subgroup .
*Every element of finite order in G*_{alpha} is conjugate to an element of "G".
*Every finite subgroup of G*_{alpha} is conjugate to a finite subgroup of "G".
*If H e G and K e G then G*_{alpha} contains a subgroup isomorphic to afree group of rank two.Applications
In terms of the
fundamental group inalgebraic topology , the HNN extension is the construction required to understand the fundamental group of atopological 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, asfree products with amalgamation do with respect to theSeifert-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 afinitely presented group . Most modern proofs of the Novikov-Boone theorem about the existence of afinitely 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. [ Jean-Pierre Serre. "Trees." Translated from the French by John Stillwell. Springer-Verlag, Berlin-New York, 1980. ISBN: 3-540-10103-9]The idea of HNN extension has been extended to other parts of
abstract algebra , includingLie algebra theory.Generalizations
HNN extensions are elementary examples of fundamental groups of graphs of groups, and as such are of central importance in
Bass–Serre theory .References
Wikimedia Foundation. 2010.