Title: Signature morphisms of Cremona groups

Abstract: A Cremona group Cr(n) is the groupe of birational self-maps of a projective space of dimension n. It is an algebraic group if n=1 and it is not of finite dimension for n>1, in fact, it contains a polynomial ring in n-1 variables. We are interested in homomorphism from Cr(n) to a finite group. For n=2 and the base-field over complex numbers, no such quotient can exist, basically because birational maps only contract rational curves. Over non-closed fields and in higher dimension, there are many birational maps contracting non-rational subvarieties, and it turns out that there are many homomorphisms from Cr(n) to a finite group. In this talk I explain and motivate this phenomenon in dimension 2 and in dimension 3.