My question is the following:
Suppose that we are given the amalgamated product $ G = G_1 * _{G_3} G_2 $, and subgroups $ H_i \le G_i $ for $i=1,2,3$, such that in addition $H_3$ is as large as possible - namely, $H_3 = G_3 \cap H_1 \cap H_2 $.
Is is true that $ H = H_1 * _{H_3} H_2 $ embeds as a subgroup of $G$ in the obvious way, or it there some counter-example?
The standard results in this direction require, for example, that $H_1 \cap G_3 = H_2 \cap G_3$ , which of course might not hold in general.