... there is a connected component of the moduli space, which has only one other connected component, given by the conjugate of the previous one.Proof. By Theorem 3.3 it suffices to show the first statement, ... corresponding curves C, Cwith an action of G, and the productaction of G on C × Cis free if and only if, defining c, cby the propertiesabc = abc=1 ,and letting Σ be the union of the ... simply connected).His result on the one side uses methods and results developed in a longsequel of papers ([Cat1,2,3], [Man1,2,3]), on the other, it uses a rather elaborateconstruction.About the...