Graphs of groups with homomorphisms not necessarily injective I'm wondering if there is any literature on graphs of groups where the maps $G\to H$ from an edge group $G$ to its endpoint group $H$ are not necessarily $\pi_1$-injective. Or is this just too general to actually say anything meaningful? Are there any results on this subject?
 A: Let us call an "HNN extension" with non-injective homomorphism between associated subgroups ni-HNN extension as opposite to the ordinary HNN extensions. I know three examples where ni-HNN extensions were used. 


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*Ilya Kapovich result that an ascending ni-HNN extension of a free group $F$ with non-injective homomorphism $\phi : F\to F$ is actually isomorphic to an ascending HNN extension of some free group. 

*A result of Igor Lysenok and Rostislav Grigorchuk that a finitely presented amenable group containing the Grigorchuk group is a ni-HNN extension of some finitely presented group ("A set of defining relations for the Grigorchuk group", Mat. Zametki 38 (1985), no. 4, 503–516, "An example of a finitely presented amenable group that does not belong to the class EG". Mat. Sb. 189 (1998), no. 1, 79--100). 

*Our with A. Yu. Olshanskii finitely presented non-amenable torsion-by-cyclic group is a ni-HNN extension of some finitely presented group containing the free Burnside group ("Non-amenable finitely presented torsion-by-cyclic groups." Publ. Math. Inst. Hautes Études Sci. No. 96 (2002), 43–169). 
A: As Yves has already indicated in comments, any "non-injective" graph of groups $\mathcal{G}$ canonically describes a graph of groups $\overline{\mathcal{G}}$ with the usual injectivity hypothesis.  I'll briefly explain the construction.
For $\mathcal{G}$ a "non-injective" graph of groups, the fundamental group $\pi_1\mathcal{G}$ still makes sense -- you can either go through Serre's definition in Trees and check that injectivity wasn't used, or a quick and dirty justification is to notice that you can build a corresponding graph of spaces $\mathcal{X}$ in the usual way, and define $\pi_1\mathcal{G}=\pi_1\mathcal{X}$.
At this point, for every edge or vertex $x$ of $\mathcal{G}$, there is a homomorphism $G_x\to\pi_1\mathcal{G}$; let $\overline{G}_x$ denote its image.  The data $\overline{G}_x$ attached to the underlying graph of $\mathcal{G}$ now define an "injective" graph of groups of the usual kind.
For this reason, there isn't much literature on "non-injective" graphs of groups, although I can think of a few places where the above construction is used: in Bestvina and Feighn's Inventiones paper on the Rips machine, for instance.
Of course, passing from $\mathcal{G}$ to $\overline{\mathcal{G}}$ is a very destructive process, which often yields something trivial (in the sense that the inclusion of some vertex map is surjective). So for this construction to be useful, you have to have some reason why $\overline{\mathcal{G}}$ is non-trivial in your particular case. One useful remark is that HNN extensions, or more generally graphs of groups with underlying graphs that aren't trees, are always non-trivial. 
A: You can interpret $\pi_1$ of the graph of groups as $\pi_1$ of the homotopy colimit of the corresponding diagram of classifying spaces.  When the maps in the graph are not injective, this homotopy colimit is not necessarily a $K(\pi_1 \mathcal G, 1)$.  In other words,  even if $\pi_1 \mathcal G$ is boring, a graph of groups with non injective maps can have a nontrivial $\pi_2$!
