Borel subgroups of the Cremona group

Following Demazure and Serre, the Cremona group is endowed with a natural topology. Borel subgroups are then defined as the maximal closed connected solvable subgroups. In this talk, we describe all Borel subgroups of the complex plane Cremona group. It turns out that their rank may be equal to 0,1 or 2 (the rank of a closed subgroup of the Cremona group being defined as the maximal dimension of a torus contained in it). In principle, this fact answers a question of Popov. We also show that all Borel subgroups of rank 1 or 2 are conjugate, but that this result no longer holds for Borel subgroups of rank 0. This is a joint work with I. Hedén.