nLab HNN-extension




In combinatorial group theory, by the HNN construction — named after Higman, Neumann and Neumann (1949) — one means a universal construction which from a group and two abstractly isomorphic subgroups produces a group extension in which these two become conjugate subgroups.



Given a group, GG, and a subgroup, HGH \subset G, equipped with a(nother) monomorphism θ:HG\theta \colon H \hookrightarrow G, then the HNN-extension G* HG \ast_H is obtained by adjoining an element tt to GG subject to the condition:

hHt 1ht=θ(h). \underset{h \in H}{\forall} \;\;\;\; t^{-1} \cdot h \cdot t \;=\; \theta(h) \,.

Notice that there is a canonical subgroup-inclusion

GG* H G \hookrightarrow G \ast_H

under which the two copies of HH in GG (given by HH itself and by the image θ(H)\theta(H)) become conjugate subgroups in G* HG \ast_H.


Examine the fundamental group? of the graph of groups, 𝒢\mathcal{G}, with underlying graph the graph with one vertex, vv and one edge, ee and nothing else.


  • the vertex group, G vG_v, to be GG,

  • the edge group, G eG_e, to be HH,

  • the two morphisms from G eG_e to G vG_v to be the canonical inclusion of HH into GG and the given monomorphism, θ\theta,

then Π 1(𝒢)(v)G* H\Pi_1(\mathcal{G})(v) \cong G \ast_H.

As a 2-colimit

Consider groups GG seen in terms of their delooping groupoid BG\mathbf{B}G as objects of the (2,1)-category Grpd of groupoids.

Beware that this construction B():GrpsGrpds\mathbf{B}(-) \;\colon\; Grps \to Grpds is not a fully faithful (2,1)-functor: The fully-faithful embedding is obtained only by regarding the delooping groupoids as pointed objects in Grpd, see at looping and delooping. The 2-morphisms between (1-morphisms between) delooping groupoids which do not respect the base point are precisely those natural transformations referred to in the following.

Explicitly, this means that for a parallel pair of group homomorphisms (φ,θ):HG(\varphi,\theta) \colon H \rightrightarrows G, a 2-morphism BφBθ\mathbf{B}\varphi \Rightarrow \mathbf{B}\theta between their deloopings is equivalently (a natural isomorphism whose unique component is) an element gGg \in G such that gφg 1=θg\varphi g^{-1}=\theta.

Now within this (2,1)-category, the (delooping of the) HNN-extension G* HG\ast_H (Def. ) is a 2-colimit:

Namely, if ι:HG\iota \colon H \hookrightarrow G denotes the given subgroup inclusion and θ:HG\theta \colon H \hookrightarrow G the (other) monomorphism, then B(G* H)\mathbf{B}(G\ast_H) is the coinserter of (Bθ,Bι):BHBG(\mathbf{B}\theta,\mathbf{B}\iota) \colon \mathbf{B}H \rightrightarrows \mathbf{B}G.


The original article:

  • Graham Higman, B. H. Neumann, Hanna Neuman, Embedding Theorems for Groups, J. of the London Mathematical Society s1-24 4 (1949) 247-254 [doi:10.1112/jlms/s1-24.4.247]

See also:

  • Wikipedia, HNN extension

  • Jean-Pierre Serre, Arbres, amalgames, SL 2SL_2, volume 46 of Astérisque , Société mathématique

    de France (1977)

  • Jean-Pierre Serre, , Trees , Springer Monographs in Mathematics, Springer-Verlag Berlin.

    150, 157, 161 (2003)

Last revised on December 11, 2022 at 11:02:15. See the history of this page for a list of all contributions to it.