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 $1\to N_i\to {\Gamma }_i\stackrel{{\pi }_i}{\to }Q\to 1$ be short exact sequences of groups and let $P=\{({\gamma }_1,{\gamma }_2)\in {\Gamma }_1\times {\Gamma }_2:{\pi }_1({\gamma }_1)={\pi }_2({\gamma }_2)\}$ 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 ${\Gamma }_1$ and ${\Gamma }_2$ 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.

arXiv comment26 pages