Finiteness Properties of Fibre Products over Virtually Nilpotent Quotients

Abstract

The $n$-$(n+1)$-$(n+2)$ theorem, recently established by Cohen and Shusterman, says that if two groups of type $\mathrm{F}_{n+1}$ map onto a common quotient $Q$ of type $\mathrm{F}_{n+2}$ and one of the two kernels is of type $\mathrm{F}_n$, then the associated fibre product is of type $\mathrm{F}_{n+1}$. We prove a stronger, symmetric variant of this theorem when $Q$ is virtually nilpotent. More precisely, for $i\in\{1,2\}$, let $1\to N_i\to Γ_i\overset{\smallπ_i}{\to} Q\to 1$ be short exact sequences of groups and let $P=\{(γ_1,γ_2)\inΓ_1\timesΓ_2\colon π_1(γ_1)=π_2(γ_2)\}$ be their fibre product. Let $k,l,m,n\in\mathbb N_0$, assume that $N_1$ and $N_2$ are of type $\mathrm{FP}_k$ and $\mathrm{FP}_l$, respectively, and that $Γ_1$ and $Γ_2$ are of type $\mathrm{F}_m$ and $\mathrm{F}_n$, respectively. We prove that $P$ is of type $\mathrm{F}_{\min\{k+l+1,m,n\}}$. The same formula holds with the homological finiteness properties $\mathrm{FP}_r$ in place of $\mathrm{F}_r$. This provides an alternative proof of the homotopical and homological Virtual Surjections Theorem, which gives a finiteness criterion for subgroups of direct products in terms of their embedding in the ambient product.