May 2017, UK

**Title:** Rationality of del Pezzo fibrations and the Cremona group

**Abstract:** I will talk about embeddings of PSL_2(7) into the Cremona group of rank 3. Study of embeddings of a finite group G into the Cremona group is equivalent to study of G-equivariant birational geometry of rational GQ-Mori fiber spaces. Thus to classify embeddings of PSL_2(7) one has to classify rational PSL_2(7)Q-Mori fiber spaces up to PSL_2(7)-equivariant birational equivalence. I will discuss classification of PSL_2(7)Q-del Pezzo fibrations and their birational rigidity, in particular, rationality.