Timeline for Torsion-free group that is not of type F but is virtually of type F
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Mar 22, 2019 at 18:39 | comment | added | Ben Wieland | I asserted some conjectures, but I didn't quite isolate them. The obstruction for a finitely presented finite dimensional $FP_\infty$ group to have a finite classifying space is for the $G$-module $\mathbb Z$ to be trivial in the reduced $K$-group $\bar{K}_0(\mathbb ZG)$. The $K$-theory generalization of one of the Kaplansky conjectures says that this $K$-group is zero (even if the group is just torsion-free). I don't know if this conjecture has a name, other than the $K$-theory (variant) Kaplansky conjecture. This is subsumed by the monster Farrell-Jones conjecture. | |
| Aug 24, 2014 at 19:36 | comment | added | HJRW | @Misha - I don't think the Bestvina--Brady examples come from 2-dimensional RAAGs. Indeed, they come from RAAGs defined by the 1-skeleton of a flag triangulation of a homology sphere. | |
| Aug 24, 2014 at 3:06 | vote | accept | Sarah | ||
| Aug 23, 2014 at 16:14 | comment | added | Ben Wieland | I edited to include finite presentation. As YCor says, the BB examples cannot be used for this question and the current conjecture FP+fp=F would apply. | |
| Aug 23, 2014 at 16:12 | history | edited | Ben Wieland | CC BY-SA 3.0 |
added 359 characters in body
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| Aug 23, 2014 at 12:56 | comment | added | YCor | Still, in Sarah's question finite presentability is not an issue, and thus from FP plus finite presentability we can deduce F, answering negatively the question (torsion-free + virtually F implies F). | |
| Aug 23, 2014 at 4:28 | comment | added | Misha | Bestvina and Brady disproved the conjecture $F=FP$ (Inventiones Math. 1997). Their examples are certain normal subgroups of 2-dimensional RAAGs: The subgroups they construct are $FP_2$ (and, hence, $FP$ since the groups are 2-dimensional) but are not finitely-presented. | |
| Aug 22, 2014 at 22:56 | history | answered | Ben Wieland | CC BY-SA 3.0 |