Timeline for $P^1$ minus k points

4 events

when toggle format	what		by	license	comment
May 25, 2013 at 12:57	comment	added	Misha		Lee: What Mohammad Tehrani wants is an "explicit" map from the tuple $(z_1,...,z_n)$ to the matrices generating the uniformizing Fuchsian group. This is essentially the "accessory parameters" problem which perplexed Klein and Poincare, and to which no satisfactory solution is known to this day, even if one allows infinite series as "explicit" solutions. The best results are due to A.Venkov (real case, he even wrote an explicit infinite series!), Zograf and Takhtajian. Needless to say, none of this is needed for constructing coordinates on the moduli space.
May 23, 2013 at 16:45	comment	added	Lee Mosher		The derivation of the matrices should be covered in any textbook on hyperbolic geometry. In outline, if you fix the polygon $P$ in the upper half plane model, each side pairing $a_i \mapsto \bar a_i$ takes the endpoints of $a_i$ to the endpoints of $\bar a_i$. Once the image of a third point at infinity is determined, the matrix is determined. There is also a completeness condition for each cusp: the monodromy around that cusp must be parabolic. With that, you get a very explicit set of formulas parameterizing the matrices.
May 23, 2013 at 15:08	comment	added	Mohammad Farajzadeh-Tehrani		Sure, but how do the matrices corresponding to these isometries look like? Is it discussed in those references?
May 22, 2013 at 22:24	history	answered	Lee Mosher	CC BY-SA 3.0