Cremona Groups and the Icosahedron focuses on the Cremona groups of ranks 2 and 3 and describes the beautiful appearances of the icosahedral group A₅ in them. The book surveys known facts about surfaces with an action of A₅, explores A₅-equivariant geometry of the quintic del Pezzo threefold V₅, and gives a proof of its A₅-birational rigidity.

The authors explicitly describe many interesting A₅-invariant subvarieties of V₅, including A₅-orbits, low-degree curves, invariant anticanonical K3 surfaces, and a mildly singular surface of general type that is a degree five cover of the diagonal Clebsch cubic surface. They also present two birational selfmaps of V₅ that commute with A₅-action and use them to determine the whole group of A₅-birational automorphisms. As a result of this study, they produce three non-conjugate icosahedral subgroups in the Cremona group of rank 3, one of them arising from the threefold V₅.

This book presents up-to-date tools for studying birational geometry of higher-dimensional varieties. In particular, it provides readers with a deep understanding of the biregular and birational geometry of V₅.