Timeline for In what sense is the braid group $B_3$ the universal central extension of the modular group $\Gamma$?
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| when toggle format | what | by | license | comment | |
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| Nov 21, 2015 at 23:48 | comment | added | Ian Agol | Of the possible central extensions by $\mathbb{Z}$ of $\mathbb{Z}/2\mathbb{Z}\ast \mathbb{Z}/3\mathbb{Z}$, this is the only one (up to isomorphism) which is torsion-free. | |
| Nov 21, 2015 at 19:26 | comment | added | Qiaochu Yuan | @Chris: I don't buy this. For example, $\frac{1}{n} \mathbb{Z}/\mathbb{Z}$ sits as a lattice inside $S^1$. The corresponding subgroup of the universal cover $\mathbb{R}$ is $\frac{1}{n} \mathbb{Z}$, which is in no sense (that I'm familiar with) the universal central extension of $\frac{1}{n} \mathbb{Z}/\mathbb{Z}$. | |
| Nov 21, 2015 at 19:24 | comment | added | Chris Gerig | It has something to do with the fact that it sits (as a lattice) inside the universal cover of $PSL_2(\mathbb{R})$ (as topological group), and we have the commutative diagram of short exact sequences of groups (with the same factor of $\mathbb{Z}$). Googling more, there is this book Moonshine beyond the Monster which argues that $B_3$ is "universal" in some sense, and Baez tries to explain it in his blog post here: math.ucr.edu/home/baez/week233.html | |
| Nov 21, 2015 at 18:15 | history | asked | Qiaochu Yuan | CC BY-SA 3.0 |