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

Cutkosky Rules from Outer Space