nLab HNN extension

Idea

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 subgroups which are isomorphic as abstract groups, produces a group extension in which these two become conjugate subgroups.

If the group in question is the fundamental group of a topological space $X$, and the two subgroups are the fundamental groups of homeomorphic subspaces $A,B \subset X$, then the corresponding HNN extension is the fundamental group of the result of attaching to $X$ a handlebody at $A$ and $B$.

Definition

Definition

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:

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

Notice that there is a canonical subgroup-inclusion

$$G \hookrightarrow G \ast_H$$

under which the two copies of $H$ in $G$ (given by $H$ itself and by the image $\theta(H)$) become conjugate subgroups in $G \ast_H$.

Properties

Link with graphs of groups

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 $\Pi_1(\mathcal{G})(v) \cong G \ast_H$.

As a 2-colimit

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 $(\mathbf{B}\theta,\mathbf{B}\iota) \colon \mathbf{B}H \rightrightarrows \mathbf{B}G$.

References

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]

Survey:

• Martin R. Bridson, Carl-Fredrik Nyberg-Brodda: HNN extensions and embedding theorems for groups [arXiv:2512.10800]

Textbook account:

• Roger C. Lindon, Paul E. Schupp: Free Products and HNN Extensions, Chapter IV of Combinatorial group theory, Springer (1977) [doi:10.1007/978-3-642-61896-3]

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)