Birational maps of Severi-Brauer surfaces, with applications to Cremona groups of higher rank

Cremona groups are groups of birational transformations of a projective space. Their structure depends on the dimension and the field. In this talk, we will first focus on birational transformations of (non-trivial) Severi-Brauer surfaces, that is, surfaces that become isomorphic to the projective plane over the algebraic closure of K. Such surfaces do not contain any K-rational point. We will prove that if such a surface contains a point of degree 6, then its group of birational transformations is not generated by elements of finite order as it admits a surjective group homomorphism to the integers.
As an application, we use this result to study Mori fiber spaces over the field of complex numbers for which the generic fiber is a non-trivial Severi-Brauer surface. We prove that any group of cardinality at most the one of the complex numbers is a quotient of the Cremona group of rank 4 (and higher).
This is joint work with Jérémy Blanc and Egor Yasinsky.