What kind of amalgamated product is supposed to describe $A*_F$? I'm currently following a course in geometric group theory and I've stumbled upon this notation $$G = A*_F$$ where $A$ and $F$ are supposed to be groups, $F$ assumed to be finite.
Unfortunately, I can't find (neither in my course notes nor in my books) any definition of this.
I suppose that this could be in fact $G = A*_F \mathbb{Z}$ since this appeared in the context of HNN extensions and we used to talk about one-loop quotients that we described via amalgamations with $\mathbb{Z}$.
Could anyone confirm whether $G = A*_F$ is in fact $G = A*_F \mathbb{Z}$ or if not, what exactly the former is?
Thanks in advance for any help!
 A: This is simply the notation for an HNN-extension of $A$ across $F$. It does not denote a free product with amalgamation, and is not in general isomorphic to $A\ast_F\mathbb{Z}$. (Indeed, if $F$ is finite and non-trivial then it does not embed into $\mathbb{Z}$, so we cannot form the product $A\ast_F\mathbb{Z}$.)
HNN-extensions and free products with amalgamation are really very similar, and it is helpful to think of them as being two variations on the same construction. This notation emphasises this similarity. Indeed, a free product with amalgamation $A\ast_CB$ pins together the two groups $A$ and $B$ across (isomorphic copies of) a subgroup $C$, and it is often helpful to think of an HNN-extension $A\ast_C$ a pinning together a single copy of $A$ across (isomorphic copies of) a subgroup (and the stable letter lets us do this). The above notation emphasises this "pinning together a single copy of $A$".
The notation also reflects Bass-Serre theory, where HNN-extensions and free products with amalgamation are the building blocks of the theory of groups acting on trees. Here, $A\ast_CB$ acts on its "Bass-Serre tree", with fundamental domain of the form $A-_C-B$, where $A$ and $B$ are vertex stabilisers and $C$ is an edge stabiliser. Similarly, an HNN-extension $A\ast_C$ acts on its Bass-Serre tree, with fundamental domain a single vertex and a single loop edge, with the vertex stabiliser $A$ and the edge stabiliser $C$ (sorry, can't really draw that one here!).
