Timeline for Non-finitely presented FP groups with cohomological dimension $2$
Current License: CC BY-SA 4.0
7 events
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| Feb 15, 2022 at 15:17 | comment | added | IJL | This uses the Morse function on the universal covering space of the Salvetti complex, which gets called $X_L$. The RAAG $A_L$ acts freely cocompactly on $X_L$, and $BB_L$ acts freely cocompactly on the level set (= inverse image of a point under the Morse function $f:X_L\rightarrow \mathbb{R}$). The whole of $X_L$ is built by attaching cones on $L$ to the level set. In the case when $L$ is acyclic, it follows that the level set is itself acyclic, since it must have the same homology as $X_L$ which is contractible. Thus the dimension of the level set is an upper bound for $cd(BB_L)$. | |
| Feb 15, 2022 at 10:14 | comment | added | HJRW | Ian, could you explain why "For an acyclic flag complex [$L$] the cohomological dimension of the Bestvina-Brady group [$BB_L$] is equal to the dimension of the flag complex"? Clearly the dimension of the RAAG $A_L$ is at most $\mathrm{dim}L+1$, so this also bounds $\mathrm{dim}BB_L$. But how do we lose the extra 1? | |
| Feb 14, 2022 at 14:04 | vote | accept | Maxime Ramzi | ||
| Feb 14, 2022 at 14:04 | comment | added | Maxime Ramzi | Thanks, this is exactly the kind of information I wanted ! | |
| Feb 14, 2022 at 10:43 | history | edited | IJL | CC BY-SA 4.0 |
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| Feb 14, 2022 at 10:35 | history | edited | IJL | CC BY-SA 4.0 |
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| Feb 14, 2022 at 10:14 | history | answered | IJL | CC BY-SA 4.0 |