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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
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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

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It has been considered by Sergey V. Ivanov in "On HNN-extensions in the class of groups of large odd exponent". Groups, rings, Lie and Hopf algebras (St. John's, NF, 2001), 39–53, Math. Appl., 555, Kluwer Acad. Publ., Dordrecht, 2003. Particular cases of HNN extensions in the class of torsion groups were also used in our paper with Olshanskii, Non-amenable finitely presented torsion-by-cyclic groups. Publ. Math. Inst. Hautes Études Sci. No. 96 (2002), 43–169 (2003).

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