Cutkosky Rules from Outer Space

We overview recent results on the mathematical foundations of Cutkosky rules. We emphasize that the two operations of shrinking an internal edge or putting internal lines on the mass-shell are natural operation on the cubical chain complex studied in the context of geometric group theory. This together with Cutkosky's theorem regarded as a theorem which informs us about variations connected to the monodromy of Feynman amplitudes allows for a systematic approach to normal and anomalous thresholds, dispersion relations and the optical theorem. In this report we follow [1] closely.