Timeline for Cohomological dimension of finitely presented group
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jul 12, 2012 at 9:32 | comment | added | HJRW | Well, in this case $G$ is a central cyclic extension of a 2-generator abelian group, so this problem should be fairly straightforward (though I haven't got time to work out the answer). The comments aren't very noticeable - if you asked this as a separate question, I bet you'd get an answer very quickly. | |
| Jul 12, 2012 at 4:44 | vote | accept | Earthliŋ | ||
| Jun 15, 2012 at 6:11 | comment | added | Earthliŋ | Thanks, that answers my question. Cohomological dimension 2 really doesn't mean much, then... Pity. How about if the group is almost Abelian, e.g. $[[a,b],a]=[[a,b],b]=e$ (for $a,b$ the two generators). My intuition told me that in this case $\mathrm{cd}\ G=2$ implies that $G$ must in fact be Abelian, but then my intuition tells me all sorts of funny things... | |
| Jun 15, 2012 at 2:52 | comment | added | Lee Mosher | I wasn't thinking so much about Gromov's random groups as "any old random thing you want to do", the point being that cd G = 2 implies just about nothing about relators, and that it's quite easy to construct oodles of examples with cd G = 2 and no particular pattern to the relators. But I quite forgot about one relator groups! Which really nails in the point. | |
| Jun 14, 2012 at 22:53 | comment | added | Ian Agol | Ok, I guess it follows from the Magnus-Moldovanskii hierarchy. Wise's recent results also imply the virtual cohomological dimension is $\leq 2$ in the presence of torsion. | |
| Jun 14, 2012 at 17:55 | comment | added | Steve D | Lyndon, "Cohomology Theory of Groups with a Single Defining Relation". | |
| Jun 14, 2012 at 17:44 | comment | added | Ian Agol | Do you know an old reference? Gromov's random group theorem is overkill, one could just use that a random relator will be small-cancellation, but was this observation made before Gromov? Anyway, an answer is more useful if it provides at least some reference. | |
| Jun 14, 2012 at 17:38 | comment | added | Steve D | In fact, quotient any relator that isn't a proper power, or primitive, works as well. | |
| Jun 14, 2012 at 14:23 | history | answered | Lee Mosher | CC BY-SA 3.0 |