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I am familiar with the definition of the HNN extension of a group relative to an isomorphism between two of its subgroups. For comparison's sake let me make that explicit. For groups $G_1, G_2\leq G=\langle S\mid R\rangle$ with isomorphism $\alpha:G_1\rightarrow G_2$, and new element $t\notin S$, we denote $G*_{\alpha}:=\langle S,t\mid R, \forall g\in G_1 \;tgt^{-1}=\alpha(g)\rangle$.

What I want to know is, what is meant by the notation $G*_H$, where $H$ is just some other group? In case the following is needed to make sense of this, in what I'm reading all my groups are finitely generated, and Kleinian, and the group $H$ is non-elementary (though I expect the notation has a definition in a broader setting).

Does the notation imply that there are isomorphisms from $H$ to two possibly different subgroups of $G$, implying existence of a $G_1, G_2$, and isomorphism $\alpha$ between them as in the more familiar notation described in the 1st paragraph? If so, doesn't the structure of $G*_H$ depend a lot on the choices for these objects? How can we get a well defined extension when we only specify $H$?

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    $\begingroup$ I think that the answer to both of your questions is "yes". Specifying only H is not enough, unless the additional information is supplied by the context, somehow. $\endgroup$ Dec 10, 2013 at 1:25
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    $\begingroup$ I'm thinking now that the author merely meant to refer to an arbitrary HNN extension of the form I describe in the 3rd paragraph, and did not care about the details, because he wants to discuss some general properties of objects of that nature. $\endgroup$
    – j0equ1nn
    Dec 10, 2013 at 1:52

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