Timeline for Extension of $FP_{n}$ group
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 24, 2019 at 17:24 | vote | accept | J.L. | ||
| Oct 18, 2019 at 19:51 | history | edited | LSpice | CC BY-SA 4.0 |
Authors and arXiv link
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| Oct 18, 2019 at 15:53 | answer | added | J.L. | timeline score: 7 | |
| Oct 18, 2019 at 8:29 | comment | added | YCor | 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. | |
| Oct 18, 2019 at 7:46 | history | edited | J.L. | CC BY-SA 4.0 |
added 144 characters in body
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| Oct 18, 2019 at 7:45 | comment | added | J.L. | @LSpice The paper is 'Finitely presented residually free groups', citeseerx.ist.psu.edu/viewdoc/… , Thm 5.2 | |
| Oct 18, 2019 at 7:43 | comment | added | J.L. | @tj_ Sorry, I do not understand your counterexample. $C\times C$ is finitely presentable, that is, of type $F_{2}$, so in particular it is of type $FP_{2}(\mathbb{Q})$, and so it is of type $FP_{1}(\mathbb{Q})$. I am not saying that my group $T$ can't be of type $FP_{n+1}(\mathbb{Q})$, only that we can ensure that it is $FP_{n}(\mathbb{Q})$. | |
| Oct 18, 2019 at 2:47 | comment | added | LSpice | @YCor, ah, fair enough. Still it’s appropriate to mention the paper! | |
| Oct 17, 2019 at 23:01 | comment | added | YCor | @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 | |
| Oct 17, 2019 at 20:59 | comment | added | LSpice | What is the paper? Or, at least, what is a group of type $\operatorname{FP}_n(\mathbb Q)$? | |
| Oct 17, 2019 at 19:33 | history | asked | J.L. | CC BY-SA 4.0 |