Free products and amalgamated products are coproducts in the context of discrete groups. As exhibited by Bass-Serre theory, they play a central role in the structure theory of discrete groups and the geometric approach to group theory.
In this talk I will explain how to generalise these constructions to the setting of étale groupoids by constructing the amalgamated product of two étale groupoids over a common subgroupoid. I will then discuss approximation properties and the reduced groupoid C*-algebra of an amalgamated product of groupoids. This is based on joint work in progress with Max Durrant.
