Degenerations and fibrations of Riemann surfaces associated with regular polyhedra and soccer ball (Q2407559)

A \textit{degenerating family}, or a \textit{degeneration}, of Riemann surfaces, is a surjective proper holomorphic map \(\pi : M \rightarrow \Delta\) from a smooth complex surface \(M\) to the unit disc \(\Delta\) such that \(\pi ^{-1} (0)\) is singular, and every \(\pi ^{-1} (s)\) is a Riemann surface for \(s \neq 0\). The present paper deals with the five regular polyhedra and one semi-regular polyhedron, the soccer ball or truncated icosahedron. For each such polyhedron \(\mathcal {P}\), the goal is to associate to each orientation-preserving automorphism \(f\) of \(\mathcal {P}\) a degenerating family of Riemann surfaces whose topological monodromy is the automorphism \(f\). These degenerations are obtained in Section 3. A final feature of the article is to solve a question by M. Oka, namely whether there is a natural way to compactify those degenerations, obtained by the authors. This question is answered in affirmative sense in Section 4.