The Margulis-Soifer property for the planar Cremona group

In 1981, Margulis and Soifer established a remarkable property of finitely generated linear groups: such a group contains uncountably many maximal subgroups if and only if it is not virtually solvable. In particular, this yields the existence of maximal subgroups of infinite index in SLn(Z). We will show how their methods can be adapted to establish an analogous result for groups of birational transformations of the plane. This is based on joint work with Serge Cantat, Julie Déserti and Seung Uk Jang.