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

We prove the Virtual Surjection Conjecture for discrete groups, both for property $F_n$ and for property $F{P}_n$: given a product of groups of type $F_k$ (respectively $F{P}_k$), a subgroup that virtually surjects onto $k$-tuples must be $F_k$ (respectively $F{P}_k$) as well. We prove the homological $n$-$(n+1)$-$(n+2)$ Conjecture for discrete groups under the assumption the common quotient is finitely presented, and prove that this assumption cannot be dropped. We deduce the $n$-$(n+1)$-$(n+2)$ Conjecture for property $F_n$.