Action of Cremona groups of CAT(0) complex

When we study a group it is useful to consider objects on which the group acts. An interesting object with an action of the group of birational automorphisms of a surface is the cube complex. Its vertices are birational models of the surface and its faces are subsequent blow-ups of several points. Such a complex has a property CAT(0); more precisely, any finite number of its vertices lie in a common cube. This geometric construction allows to give new proofs of facts describing the structure of groups of birational automorphisms of surfaces, in particular, to reprove some regularizability criterions. Also using this method one can show that a birational action of a finitely generated group on a surface defined over a finite field is regularizable if each generator is a regularizable automorphism. In my talk I am going to speak about the construction of the CAT(0) complex associated with a surface and to prove these facts following papers by Anthony Genevois, Anne Lonjou and Christian Urech.