A degenerating family of Riemann surfaces over a Riemann surface gives us a monodromy representation, which is a homomorphism from the fundamental group of a punctured surface to the mapping class group. We 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. Moreover, we discuss the special sections of such a degenerating family.
The author would like to thank Shigeru Takamura for very fruitful discussion and warm encouragement. The author is partially supported by the Fujukai Foundation.
The author also thanks the anonymous referee for giving very useful comments on an earlier version of the paper.
"Degenerating families with finite monodromy groups." Kodai Math. J. 44 (1) 1 - 19, March 2021. https://doi.org/10.2996/kmj44101