If $A$ and $B$ are groups we have the following short exact sequence: $$ 0 \to [A,B] \to A * B \to A \times B \to 0, $$ where the group $[A,B]$ is free (see e.g. Serre's Trees).
I am wondering if there is a "similar" sequence if we replace $A * B$ by $A *_C B$ (or by $A *_C$ an HNN-Extension). By "similar" I mean a short exact sequence where we have some particular knowledge about the left group (the "kernel") and the right group (the "quotient").
In this spirit (since I was thinking about the abelianization) I am searching for some references about the commutator subgroup of a free product with amalgation (or HNN-Extension), since I am sure someone has already studied this subgroup.