Birational automorphisms of Severi-Brauer surfaces and the Cremona group

A Severi-Brauer surface over a field $k$ is an algebraic $k$-surface which is isomorphic to the projective plane over the algebraic closure of $k$. I will describe the group of birational transformations of a non-trivial Severi-Brauer surface, proving in particular that “in most cases” it is not generated by elements of finite order. This is already a very curious feature, since the group of birational self-maps of a trivial Severi-Brauer surface, i.e. of a projective plane, is always generated by involutions (at least over a perfect field). Then I will demonstrate how to use this result to get some insights into the structure of the groups of birational transformations of some higher-dimensional varieties, including the projective space of dimension $> 3$.