# What is being braided in SL(2,Z)?

The braid group on 3 strands is a central extension of the modular group. By definition, $B_3 = \langle \sigma_1, \sigma_2: \sigma_1\sigma_2\sigma_1=\sigma_2\sigma_1\sigma_2 \rangle$ This group has a central element (commuting with both $\sigma1$ and $\sigma_2$): $\sigma_1\sigma_2\sigma_1\sigma_2\sigma_1\sigma_2$ The coset get mapped to elements of PSL(2,Z) (which can act on the hyperbolic plane). $[\sigma_1] = \left[ \begin{array}{cc} 1 & 1 \\\\ 0 & 1\end{array}\right] \text{ and } [\sigma_2] = \left[ \begin{array}{cc} 1 & 0 \\\\ -1 & 1\end{array}\right]$ I wonder, in terms of the hyperbolic plane, what is being braided here (modulo the garside elements).

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If something were being braided, I'd think the relevant group would have a map to $B_3$, rather than from $B_3$. – S. Carnahan Nov 1 '11 at 18:28
– Qiaochu Yuan Nov 1 '11 at 21:03

Don't you need some other identification between the homology of two elliptic curves in a family to get an element of $\text{SL}_2(\mathbb{Z})$? Otherwise you just get a homomorphism between two groups which are abstractly isomorphic to $\mathbb{Z}^2$. – Qiaochu Yuan Nov 1 '11 at 21:06