TITLE: Amalgamated Free Product: Practical Uses.
QUESTION [5 upvotes]: Suppose $H$ is embedded in $G$ and $H'$ is isomorphic to $H$ and embedded in $G'$. Then we can simultaneously embed $H$, $H'$, $G$ and $G'$ into a single object (the amalgamated free product) such that $H$ and $H'$ become identifiable. This seems similar to the Isomorphism Extension Theorem for fields which is important for developing Galois Theory. What are the practical uses of the amalgamated free product?

REPLY [3 votes]: Another important use of amalgamated products arises from groups acting on trees. One easy way to interpret $G=\mbox{SL}_2(\mathbb{Z})$ as an amalgam is to observe how it naturally acts on a tree embedded in the upper half plane $H$ (the tree being simply the boundary of the tessellation of $H$ by fundamental domains of the action of $G$).
See Serre's "Trees" ("Arbres, Amalgames, SL_2"), or these nice lecture notes.