Three embeddings of the Klein simple group into the Cremona group of rank three

We study the action of the Klein simple group PSL2() consisting of 168 elements on two rational threefolds: the three-dimensional projective space and a smooth Fano threefold of anticanonical degree 22 and index 1. We show that the Cremona group of rank three has at least three non-conjugate subgroups isomorphic to PSL2(). As a by-product, we prove that admits a Kahler-Einstein metric, and we construct a smooth polarized 3 surface of degree 22 with an action of the group PSL2(). Unless explicitly stated otherwise, varieties are assumed to be projective, normal and complex.