Timeline for Topological interpretation for groups of type $FP_2$
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15 events
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| Jun 13, 2018 at 11:52 | answer | added | IJL | timeline score: 5 | |
| Jan 21, 2014 at 22:40 | comment | added | Gaelan Hanlon | Thanks for the answer Yves. @HJRW I'm not aware of a reference. I had sketched a proof of the claim, but I assumed that since $G$ is finitely generated the acyclic $2$-complex coming from Eilenberg-Ganea would have a cocompact $1$-skeleton. However this doesn't seem to necessarily be the case (at least an argument is needed). | |
| Jan 21, 2014 at 7:10 | comment | added | YCor | I think the question has a positive answer: indeed it's shown in Brown's book that a FP2 group $F/R$ can be written as quotient of a fp group $F/R_1$ ($R_1$ finite) by a perfect normal subgroup $N/R_1$. Thus adding 2-gons corresponding to $F/R_1$ to the Cayley graph of $F/R$ works. | |
| Jan 21, 2014 at 4:48 | comment | added | HJRW | What's a reference for the assertion that the converse holds when the cohomological dimension is 2? | |
| Jan 21, 2014 at 1:38 | comment | added | Benjamin Steinberg | Anyway I now understand what you are asking. | |
| Jan 21, 2014 at 1:36 | comment | added | Benjamin Steinberg | My point is an acyclic 2-dim complex already implies cd 2. | |
| Jan 21, 2014 at 0:31 | comment | added | Gaelan Hanlon | In the case of Bestvina and Brady's group of $cd(G)=2$, type $FP_2$, not $F_2$, $G$ acts freely and cocompactly on a $2$-dimensional acyclic space $X$ which is not simply connected. So $X/G$ is not a classifying space for $G$, so I am not claiming that $gd(G) = 2$, nor that $FP_2$ implies geometric dimension $2$ in general. | |
| Jan 21, 2014 at 0:29 | comment | added | Gaelan Hanlon | Prehaps I wasn't clear, provided $G$ is $FP_2$, does $G$ act freely on a space $X$ (not necessarily $2$-dimensional) such that $H_1(X) = 0$ and the $2$-skeleton of $X$ is finite mod $G$? | |
| Jan 21, 2014 at 0:08 | comment | added | Benjamin Steinberg | Eilenberg-Ganea shows cd=geometric dimension in dimension >2. I think you confuse FP with geometric dimension. Thompson's group F has FP_2 but no finite dimensional classifying space. | |
| Jan 21, 2014 at 0:06 | comment | added | Benjamin Steinberg | Eilenberg-Ganea conjecture is open for n=2. It is known either this is false or Whitehead conjecture is false. Any group acting freely on a 2-dimensional acyclic complex has cd 2. The augmented cellular chain complex would give a free resolution of the trivial module. | |
| Jan 20, 2014 at 23:59 | comment | added | Gaelan Hanlon | Thanks for the reply, however I don't see how $FP_n$ not being equivalent to $F_n$ implies this is false. Perhaps you could explain more? Yes, Bestvina and Brady construct a group $G$ which has $cd(G) = 2$, is of type $FP_2$, but not finitely presented. However, by the Eilenberg-Ganea Theorem, $G$ acts on freely on a $2$-dimensional acyclic (not necessarily simply connected) space $X$. Using Schanuel's Lemma and the fact that $G$ is of type $FP_2$ one gets that $X$ is $G$-cocompact. | |
| Jan 20, 2014 at 23:44 | comment | added | Benjamin Steinberg | This is not true. Bestvina-Brady group show FP_n is not F_n. perhaps you are thinking of geometric dimension vs cohomological dimension. There either geometric dimension 2 is not cd 2 or whitehead conjecture is false. | |
| Jan 20, 2014 at 23:22 | review | First posts | |||
| Jan 20, 2014 at 23:22 | |||||
| Jan 20, 2014 at 23:10 | history | edited | Gaelan Hanlon |
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| Jan 20, 2014 at 23:03 | history | asked | Gaelan Hanlon | CC BY-SA 3.0 |