The Ribes-Zalesski\u{i} product theorem via foldings and extensions

We present an exposition of the Auinger-Steinberg proof of the Ribes-Zalesski\u{i} product theorem for pro-V topologies, where V is a pseudovariety of groups closed under extensions with abelian kernel. This proof is self-contained and is accessible to those acquainted with coverings of graphs, and as such, it provides an easy entry point to various other deep theorems for which the product theorem is formally equivalent to.