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, $G$, and a subgroup, $H \subset G$, equipped with a(nother) monomorphism $\theta \colon H \hookrightarrow G$, then the HNN-extension $G \ast_H$ is obtained by adjoining an element $t$ to $G$ subject to the condition:
Notice that there is a canonical subgroup-inclusion
under which the two copies of $H$ in $G$ (given by $H$ itself and by the image $\theta(H)$) become conjugate subgroups in $G \ast_H$.
Examine the fundamental group? of the graph of groups, $\mathcal{G}$, with underlying graph the graph with one vertex, $v$ and one edge, $e$ and nothing else.
Take
the vertex group, $G_v$, to be $G$,
the edge group, $G_e$, to be $H$,
the two morphisms from $G_e$ to $G_v$ to be the canonical inclusion of $H$ into $G$ and the given monomorphism, $\theta$,
then $\Pi_1(\mathcal{G})(v) \cong G \ast_H$.
Consider groups $G$ seen in terms of their delooping groupoid $\mathbf{B}G$ as objects of the (2,1)-category Grpd of groupoids.
Beware that this construction $\mathbf{B}(-) \;\colon\; Grps \to Grpds$ 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 $(\varphi,\theta) \colon H \rightrightarrows G$, a 2-morphism $\mathbf{B}\varphi \Rightarrow \mathbf{B}\theta$ between their deloopings is equivalently (a natural isomorphism whose unique component is) an element $g \in G$ such that $g\varphi g^{-1}=\theta$.
Now within this (2,1)-category, the (delooping of the) HNN-extension $G\ast_H$ (Def. ) is a 2-colimit:
Namely, if $\iota \colon H \hookrightarrow G$ denotes the given subgroup inclusion and $\theta \colon H \hookrightarrow G$ the (other) monomorphism, then $\mathbf{B}(G\ast_H)$ is the coinserter of $(\mathbf{B}\theta,\mathbf{B}\iota) \colon \mathbf{B}H \rightrightarrows \mathbf{B}G$.
The original article:
See also:
Wikipedia, HNN extension
Jean-Pierre Serre, Arbres, amalgames, $SL_2$, 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.