Cluster Algebras and Scattering Diagrams, Part III. Cluster Scattering Diagrams

This is a self-contained exposition of several fundamental properties of cluster scattering diagrams introduced and studied by Gross, Hacking, Keel, and Kontsevich. In particular, detailed proofs are presented for the construction, the mutation invariance, and the positivity of theta functions of cluster scattering diagrams. Throughout the text we highlight the fundamental roles of the dilogarithm elements and the pentagon relation in cluster scattering diagrams.