In 1949, G. Higman, B.H. Neumann and H. Neumann [a4] proved several famous embedding theorems for groups using a construction later called the HNN-extension. The theory of HNN-groups is central to geometric and combinatorial group theory and should be viewed in parallel with amalgamated products (cf. also Amalgam of groups).
The easiest way to define an HNN-group is in terms of presentations of groups.
Presentation of groups.
A presentation of a group $G$ is a pair $\langle X | R \rangle$ where $R$ is a subset of $F(X)$, the free group on the set $X$, and $G$ is isomorphic (cf. also Isomorphism) to the quotient group $F(X)/N(R)$, where $N(R)$ is the intersection of all normal subgroups of $F(X)$ containing $R$. The subgroup $N(R)$ is called the normal closure of $R$ in $F(X)$.
Given an arbitrary group $G$, there is an obvious homomorphism $\tau_G : F(G) \rightarrow G$ such that $\tau_G(g) = g$ for all $g \in G$. Clearly, $\langle G | \ker \tau_G \rangle$ is a presentation for $G$.
Suppose is an isomorphism of subgroups of a group and is not in . The HNN-extension of with respect to has presentation
The generator is called the stable letter, the base group and and the associated subgroups of this HNN-extension. When , the HNN-extension is called ascending.
Shorthand notation for the above group is or .
In [a4] it was shown that the mapping taking for all is a monomorphism. The rest of the normal form theorem for HNN-extensions was proved by J.L. Britton in 1963 [a1] (Britton's lemma): Let be a sequence of elements of and let the letter , with or without subscripts, denote . A sequence will be called reduced if there is no consecutive subsequence with or with . For a reduced sequence and , the element
of is different from the unit element.
In the original reference [a4], the following theorem is proved: Every group can be embedded in a group in which all elements of the same order are conjugate (cf. also Conjugate elements). In particular, every torsion-free group can be embedded in a group with only two conjugacy classes. If is countable, so is . Also, every countable group can be embedded in a group generated by two elements of infinite order. The group has an element of finite order if and only if does. If is finitely presentable, then so is .
For an excellent account of the history of HNN-extensions, see [a2]. See [a5], Chap. IV, for basic results and landmark uses of HNN-extensions, such as: the torsion theorem for HNN-extensions; the Collins conjugacy theorem for HNN-extensions; the construction of finitely-presented non-Hopfian groups (in particular, the Baumslag–Solitar group is non-Hopfian; cf. also Non-Hopf group); decompositions of -relator groups; Stallings' classification of finitely-generated groups with more than one end in terms of amalgamated products and HNN-extensions; and Stallings' characterization of bipolar structures on groups.
HNN-extensions are of central importance in, e.g., the modern version of the Van Kampen theorem (based on topological results in [a8], [a9]); the Bass–Serre theory of groups acting on trees and the theory of graphs of groups (see [a7]); Dunwoody's accessibility theorem [a3]; and JSJ decompositions of groups [a6].
|[a1]||J.L. Britton, "The word problem" Ann. of Math. , 77 (1963) pp. 16–32|
|[a2]||B. Chandler, W. Magnus, "The history of combinatorial group theory: A case study in the history of ideas" , Studies History Math. and Phys. Sci. , 9 , Springer (1982)|
|[a3]||M.J. Dunwoody, "The accessibility of finitely presented groups" Invent. Math. , 81 (1985) pp. 449–457|
|[a4]||G. Higman, B.H. Neumann, H. Neumann, "Embedding theorems for groups" J. London Math. Soc. , 24 (1949) pp. 247–254; II.4, 13|
|[a5]||R. Lyndon, P. Schupp, "Combinatorial group theory" , Springer (1977)|
|[a6]||E. Rips, Z. Sela, "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition" Ann. of Math. (2) , 146 : 1 (1997) pp. 53–109|
|[a7]||J.P. Serre, "Arbres, amalgams, " Astéerisque , 46 (1977)|
|[a8]||E.R. Van Kampen, "On the connection between the fundamental groups of some related spaces" Amer. J. Math. , 55 (1933) pp. 261–267|
|[a9]||E.R. Van Kampen, "On some lemmas in the theory of groups" Amer. J. Math. , 55 (1933) pp. 268–273|
HNN-extension. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=HNN-extension&oldid=41859