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I am reading a paper (Finitely presented residually free groups by Bridson, Howie, Miller III, and Short, Theorem 5.2) where they write the following:

Since $S_{0}$ is a group of type $FP_{n}(\mathbb{Q})$ and there is a series $$S_{0}\triangleleft S_{1}\triangleleft \cdots \triangleleft S_{l}=T$$ where each $S_{i+1}/S_{i}$ is finite or infinite cyclic, then by the obvious induction, $T$ is of type $FP_{n}(\mathbb{Q})$.

I understand that if $S_{i+1}/S_{i}$ is finite and $S_{i}$ of type $FP_{n}(\mathbb{Q})$, then so is $S_{i+1}$, because that property is inherited in finite extensions. Nevertheless, I really can't prove the same when $S_{i+1}/S_{i}$ is infinite cyclic.

Why is this true? Maybe it does not always hold, but in my case yes: I am working in direct products of limit groups, so $S_{0}$ is a full subdirect product of limit groups, $S_{0}\leq \Gamma_{1}\times \cdots \times \Gamma_{n}$.

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  • $\begingroup$ What is the paper? Or, at least, what is a group of type $\operatorname{FP}_n(\mathbb Q)$? $\endgroup$
    – LSpice
    Commented Oct 17, 2019 at 20:59
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    $\begingroup$ @LSpice this is a standard finiteness property of group cohomology. One says that $G$ has Property $FP_n(R)$ if the trivial $RG$-module $R$ has a resolution by projective $RG$-modules $P_n\to P_{n-1}\to\dots P_0\to R\to 0$ with each $P_i$ finitely generated. See Wikipedia: finiteness properties of groups $\endgroup$
    – YCor
    Commented Oct 17, 2019 at 23:01
  • $\begingroup$ @YCor, ah, fair enough. Still it’s appropriate to mention the paper! $\endgroup$
    – LSpice
    Commented Oct 18, 2019 at 2:47
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    $\begingroup$ @LSpice The paper is 'Finitely presented residually free groups', citeseerx.ist.psu.edu/viewdoc/… , Thm 5.2 $\endgroup$
    – J.L.
    Commented Oct 18, 2019 at 7:45
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    $\begingroup$ Note: the result is trivial for $n=1$, since $FP_1(Q)$ means finitely generated. It can also be checked to be true for $n=2$, using that a group is $FP_2(Q)$ iff is isomorphic to $G/N$ for some finitely presented group group $G$ and some normal subgroup $N$ such that $N/[N,N]$ is torsion. $\endgroup$
    – YCor
    Commented Oct 18, 2019 at 8:29

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If $0\to A \to B \to C\to 0$ is a short exact sequence of groups and $A$, $C$ are of type $FP_{n}$, then so is $B$. Reference: Proposition 2.2 in Homological finiteness properties of fibre products by Kochloukova and Ferreira Lima (arXiv link).

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