What is being braided in SL(2,Z)? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T12:36:48Z http://mathoverflow.net/feeds/question/79734 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79734/what-is-being-braided-in-sl2-z What is being braided in SL(2,Z)? John Mangual 2011-11-01T17:51:07Z 2011-11-01T19:24:17Z <p><a href="http://en.wikipedia.org/wiki/Braid_group#Relation_between_B3_and_the_modular_group" rel="nofollow">The braid group on 3 strands is a central extension of the modular group</a>. 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). $[\sigma_1] = \left[ \begin{array}{cc} 1 &amp; 1 \\ 0 &amp; 1\end{array}\right] \text{ and } [\sigma_2] = \left[ \begin{array}{cc} 1 &amp; 0 \\ -1 &amp; 1\end{array}\right]$ I wonder, in terms of the hyperbolic plane, what is being braided here (modulo the garside elements).</p> http://mathoverflow.net/questions/79734/what-is-being-braided-in-sl2-z/79752#79752 Answer by JSE for What is being braided in SL(2,Z)? JSE 2011-11-01T19:24:17Z 2011-11-01T19:24:17Z <p>Without thinking about this too carefully: I think what's getting braided are three of the Weierstrass points of an elliptic curve. More precisely: consider the space of distinct 3-tuples of points p,q,r on A^1. On the one hand, you can braid these points around. On the other hand, every path in this space (i.e. every braid) gives a family of elliptic curves</p> <p>y^2 = (x-p)(x-q)(x-r)</p> <p>and you can ask what the braid does to the homology of the elliptic curve; that's an element of SL_2(Z).</p>