HNN extension

In mathematics , the HNN extension is a construction from group theory . The theory of HNN extensions is fundamental in the combinatorial and geometrical study of groups. HNN extensions and amalgamated products form the basis of the Bass-Serre theory . They were introduced by Graham Higman , Bernhard Neumann and Hanna Neumann in 1949 in the article "Embedding Theorems for Groups", where some basic properties were also proven.

An HNN extension is an inclusion of a given group in another group , so that a given isomorphism of two subgroups and of in is realized by conjugation with an element . ${\displaystyle G}$ ${\displaystyle G_{\alpha }}$ ${\displaystyle J}$ ${\displaystyle K}$ ${\displaystyle G}$ ${\displaystyle G_{\alpha }}$ ${\displaystyle t\in G_{\alpha }}$

In this case one speaks of an HNN extension above the group , and one speaks of a nontrivial HNN extension if is. ${\displaystyle J}$ ${\displaystyle J\not =G}$

definition

Let a group , two subgroups and an isomorphism be given . ${\displaystyle G}$ ${\displaystyle J,K\subset G}$ ${\displaystyle \alpha \colon J\to K}$

If the presentation has then the HNN extension of is defined by the following presentation: ${\displaystyle G}$ ${\displaystyle \langle S|R\rangle }$ ${\displaystyle G_{\alpha }}$ ${\displaystyle G}$ ${\displaystyle \alpha }$

${\displaystyle \langle S,t\ |\ R,\ tjt^{-1}=\alpha (j)\ \forall j\in J\rangle }$

Because the group contains the generators and relations of , it is clear that there is a canonical homomorphism from to . Higman, Neumann and Neumann proved that this morphism is injective. ${\displaystyle G_{\alpha }}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle G_{\alpha }}$

Britton's normal forms and lemma

It is often useful for calculations to be able to bring elements of into a normal form. This normal form is not unique, Britton's lemma describes exactly when two normal forms correspond to the same element. ${\displaystyle G_{\alpha }}$

Normal form :

Each element can be written as: ${\displaystyle w\in G_{\alpha }}$

${\displaystyle w=g_{0}t^{\varepsilon _{1}}g_{1}t^{\varepsilon _{2}}\cdots g_{n-1}t^{\varepsilon _{n}}g_{N},\qquad g_{i}\in G,\varepsilon _{i}=\pm 1}$

Britton's lemma, proven in 1963 in "The word problem", offers a way of describing the nontrivial elements of an HNN extension:

Britton's lemma : Let be in the above normal form such that ${\displaystyle w\in G_{\alpha }}$

• either and , ${\displaystyle N=0}$ ${\displaystyle g_{0}\not =1\in G}$
• or and in w there are no subwords of the form with or with , ${\displaystyle N>0}$ ${\displaystyle tjt^{-1}}$ ${\displaystyle j\in J}$ ${\displaystyle tkt^{-1}}$ ${\displaystyle k\in K}$

then is in . ${\displaystyle w\not =1}$ ${\displaystyle G_{\alpha }}$

properties

Let be a group and its HNN extension using an isomorphism of two subgroups. ${\displaystyle G}$ ${\displaystyle G_{\alpha }}$ ${\displaystyle \alpha \colon J\to K}$

• If countable , then too . ${\displaystyle G}$ ${\displaystyle G_{\alpha }}$
• If it is finally generated , then too . ${\displaystyle G}$ ${\displaystyle G_{\alpha }}$
• If there is no torsion , then too . ${\displaystyle G}$ ${\displaystyle G_{\alpha }}$

Topological interpretation

Let it be a connected space with two connected subsets for which there is a homeomorphism . We define through an equivalence relation ${\displaystyle X}$ ${\displaystyle A,B\subset X}$ ${\displaystyle \phi \colon A\to B}$ ${\displaystyle X}$

${\displaystyle x\sim y\Longleftrightarrow x\in A,y\in B,y=\phi (x)}$ or ${\displaystyle x\in B,y\in A,y=\phi ^{-1}(x)}$

and denote with the quotient space of this equivalence relation. Then the fundamental group of is an HNN extension of the fundamental group of . ${\displaystyle Y=X/\sim }$ ${\displaystyle Y}$ ${\displaystyle X}$

More precisely: be a base point, and for base points choose ways from or to and corresponding identifications of with subgroups . Homeomorphism induces an isomorphism and thus an isomorphism . Then ${\displaystyle x_{0}\in X}$ ${\displaystyle y_{0}=\left[x_{0}\right]\in X}$ ${\displaystyle a_{0}\in A,b_{0}=\phi (a_{0})\in B}$ ${\displaystyle a_{0}}$ ${\displaystyle b_{0}}$ ${\displaystyle x_{0}}$ ${\displaystyle \pi _{1}(A,a_{0}),\pi _{1}(B,b_{0})}$ ${\displaystyle J,K\in \pi _{1}(X,x_{0})}$ ${\displaystyle \phi }$ ${\displaystyle \phi _{*}\colon \pi _{1}(A,a_{0})\to \pi _{1}(B,b_{0})}$ ${\displaystyle \alpha \colon J\to K}$

${\displaystyle \pi _{1}(Y,y_{0})=\pi _{1}(X,x_{0})*_{\alpha }}$.

The proof uses Seifert and van Kampen's theorem .

Bass Serre Theory

The HNN extension can be interpreted as a fundamental group of the group graph with a corner v and an edge e, edge group , corner group and monomorphisms ${\displaystyle G_{\alpha }}$ ${\displaystyle G_{e}:=G}$ ${\displaystyle G_{v}:=J}$

${\displaystyle \alpha _{0},\alpha _{1}\colon G_{e}\to G_{v}}$

given by and . ${\displaystyle \alpha _{0}=id}$ ${\displaystyle \alpha _{1}=\alpha }$

literature

• Zieschang, H .; Vogt, E .; Coldewey, H.-D .: Areas and plane discontinuous groups. Lecture Notes in Mathematics, Vol. 122 Springer-Verlag, Berlin-New York 1970
• Jean-Pierre Serre: Arbres, amalgames, SL ₂ . Avec un sommaire anglais. Rédigé avec la collaboration de Hyman Bass. Astérisque, No. 46. ​​Société Mathématique de France, Paris, 1977. (Chapter 1.4)
• Stocker, Ralph; Zieschang, Heiner: Algebraic Topology. An introduction. Second edition. Math guides. BG Teubner, Stuttgart, 1994. ISBN 3-519-12226-X
• Scott, Peter; Wall, Terry: Topological methods in group theory. Homological group theory (Proc. Sympos., Durham, 1977), pp. 137-203, London Math. Soc. Lecture Note Ser., 36, Cambridge Univ. Press, Cambridge-New York, 1979. ( online )
• Stillwell, John: Geometry of surfaces. Corrected reprint of the 1992 original. University text. Springer-Verlag, New York, 1992. ISBN 0-387-97743-0 (Chapter 9.2.2)

Web links

• HNN extension (Encyclopedia of Mathematics)
• Groups and Graphs

Individual evidence

1. ↑ Higman, Graham; Neumann, BH; Neumann, Hanna: Embedding theorems for groups. J. London Math. Soc. 24, (1949). 247-254.
2. ^ Britton, John L .: The word problem. Ann. of Math. (2) 77 (1963) 16-32.