Virtual Surjection and the $n$-$(n+1)$-$(n+2)$ Theorem for Profinite Groups

We prove the Virtual Surjection Conjecture for profinite groups. Namely, given a product of $n$ profinite ${\mathrm{F}\mathrm{P}}_k$ groups, a subgroup that virtually surjects onto $k$-tuples must be ${\mathrm{F}\mathrm{P}}_k$ as well. We also prove the $n$-$(n+1)$-$(n+2)$ Conjecture for profinite groups, as well as a few other ${\mathrm{F}\mathrm{P}}_n$ permanence results for fibre products. Our main tool is a numerical criterion for property ${\mathrm{F}\mathrm{P}}_n$ of modules of profinite groups. Our work suggests a new finiteness property to investigate.