The essential and Cremona dimensions of a group

The Cremona dimension of a group G is the minimal n such that G is isomorphic to a subgroup of the Cremona group of birational transformations of an n -dimensional rational variety. In this survey article, we give many examples that give evidence to the conjecture that the Cremona dimension of a finite group over the field of complex numbers is less than or equal to the essential dimension of the group.