- 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 be a group with
presentation , and let be an isomorphism between two subgroups and of .Let be a new symbol not in , and define:The group is called the "HNN extension of" "relative to" . The original group G is called the "base group" for the construction, while the subgroups and are the "associated subgroups". The new generator is called the "stable letter".Key properties
Since the presentation for contains all the generators and relations from the presentation for, there is a natural homomorphism, induced by the identification of generators, which takes to . Higman, Neumann and Neumann proved that this morphism is injective, that is, an embedding of 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
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 as above, whenever in a product of the form
:,
where for ,
satisfies in
then either "n=0" and in "G" or else for some one of the following holds:
* ,
* .Restated in contrapositive terms, Britton's Lemma says that if "w" is a product of the above form such that
*either "n=0" and in "G",
*or "n>0" and the product "w" does not contain substrings of the form , where and of the form where ,then in .
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 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 and then 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
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