# Torsion version of HNN extensions.

I am thinking of a version of HNN extensions as follows:

Assume $H,K$ are subgroups of a group $G$ and $\phi:H\to K$ is an isomorphism. We define $G_{\phi,n}$ to be the group generated by $G$ and $x\notin G$ satisfying the conditions $x h x^{-1}=\phi(h)$ and $x^n=1$.

I was wondering if such a construction has appeared in the literature before? What are the main properties (and the name) of these groups?

I appreciate any references.

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This is the quotient of the usual HNN extension $\langle x,H\mid\dots\rangle$ by the normal subgroup generated by $t^n$. This is probably the best way to view it. Unlike in the HNN extension, it is most likely not true that the natural homomorphism $G\to G_\{\phi,n}$ is always injective. –  YCor Apr 27 '13 at 9:27
Yves: Yes, it indeed need not be injective. For instance, take semidirect product $Z^2\ltimes Z$ given a hyperbolic automorphism of $Z^2$ and regard this as an HNN extension of $Z^2$. –  Misha Apr 27 '13 at 10:25