Degenerating families with finite monodromy groups
A degenerating family of Riemann surfaces is a proper surjective holomorphic map \(\pi :M\rightarrow S\) from a smooth complex surface \(M\) to a smooth complex curve \(S\) with connected fibers such that it has at most finitely many singular fibers. In the semi-local case, when \(S\) is the unit disk in \(\mathbb{C}\) and there is a unique singular fiber over the origin, \(\pi\) is a degeneration of Riemann surfaces. If both \(M\) and \(S\) are compact, then \(\pi\) is a global degenerating family. A degenerating family of Riemann surfaces over a Riemann surface defines a monodromy representation, which is a homomorphism from the fundamental group of a punctured surface to the mapping class group. The authors show that, given such a homomorphism, if its image is finite, then there exists an (isotrivial) degenerating family of Riemann surfaces whose monodromy representation coincides with it. They discuss the special sections of such a degenerating family.
scientific article; zbMATH DE number 7370867