Sofic profile and computability of Cremona groups

In this paper, we show that Cremona groups are sofic. We actually introduce a quantitative notion of soficity, called sofic profile, and show that the group of birational transformations of a d-dimensional variety has sofic profile at most polynomial of degree d. We also observe that finitely generated subgroups of the Cremona group have a solvable word problem. This provides examples of finitely generated groups with no embeddings into any Cremona group, answering a question of S. Cantat.