Quotients of birational automorphism groups of rationally connected threefolds

Based on approach of Blanc–Lamy–Zimmermann, I will explain how to find a non-trivial surjective homomorphism from the birational automorphism group of a rationally connected variety to an abelian group. After explaining main ideas, I will focus on the case of del Pezzo fibrations, extending results of the mentioned authors. This is a work in progress with Jeremy Blanc.