One example of 1-dimensional Hurwitz space (Q2702161)

Let \(f : \widehat{{\mathbb C}} \to \widehat{{\mathbb C}}\) be a 3-fold branched covering with branch points \(\{ 0,1,\infty, \lambda \}\), where \(\lambda\) is a variable point in \(\widehat{{\mathbb C}}-\{ 0,1,\infty \}\). Two such coverings \(f_1\) and \(f_2\) are said to be equivalent if there exist an (analytic) isomorphism \(g : \widehat{{\mathbb C}} \to \widehat{{\mathbb C}}\) such that \(f_1=f_2\circ g\). The set of equivalence classes, which is called an Hurwitz space, inherits a topology and a complex structure. A branched covering is said to be irreducible if no intermediate coverings exist. A branch point is said to be simple if the inverse image by \(f\) contains only one ramification point and the local degree of \(f\) at the ramification point is two. In this paper the author gives an example of 1-dimensional Hurwitz space of irreducible branched coverings of degree three with four simple branch points by using the monodromy representation \(\omega : \pi_1(\widehat{{\mathbb C}}-\{0,1, \infty, \lambda \}, i) \to \Sigma_3\), where \(\Sigma\) denotes the symmetric group on three letters, and by ''coloring the strings''. Also the author describes the Hurwitz space in relation to the Belyi curve in the case that the monodromy group is the dihedral group of order \(2p\), where \(p\) is prime.NEWLINENEWLINEFor the entire collection see [Zbl 0952.00066].