HNN extension

The HNN extension group $G$ for a group $A$, is constructed from a pair of isomorphic subgroups $B\stackrel{\varphi }{\cong }C$ in $A$, according to formula

$$G=\frac{A*⟨t|-⟩}{N}$$

where$⟨t|-⟩$ is a cyclic free group, $*$ is the free product and $N$ is the normal closure of $\{tb{t}^{-1}\varphi {(b)}^{-1}:b\in B\}$.

As an example take a surface bundle $F\subset E\to S^1$, hence the homotopy long exact sequence of this bundle implies that the fundamental group ${\pi }_1(E)$ is given by

$${\pi }_1(E)=⟨x_1,\mathrm{\dots },x_k,t|\mathrm{\Pi }=1,t{x}_i{t}^{-1}=\varphi (x_i)⟩$$

where $k$ is the genus of the surface and the relation $\mathrm{\Pi }$ is $[x_1,x_2][x_3,x_4]\mathrm{\cdots }[x_{k-1},x_k]$ for an orientable surface or $x_1^2{x}_2^2\mathrm{\cdots }{x}_k^2$ is for a non-orientable one. $\varphi$ is an isomorphism induced by a self homeomorphism of $F$.