Limit points of the branch locus of 𝓜 g

Abstract

Abstract Let 𝓜 g be the moduli space of compact connected hyperbolic surfaces of genus g ≥ 2, and 𝓑 g ⊂ 𝓜 g its branch locus. Let M g ^ $\begin{array}{} \widehat{{\mathcal{M}}_{g}} \end{array} $ be the Deligne–Mumford compactification of the moduli space of smooth, complete, connected surfaces of genus g ≥ 2 over ℂ. The branch locus 𝓑 g is stratified by smooth locally closed equisymmetric strata, where a stratum consists of hyperbolic surfaces with equivalent action of their orientation-preserving isometry group. Any stratum can be determined by a certain epimorphism Φ. In this paper, for any of these strata, we describe the topological type of its limits points in 𝓜͡ g in terms of Φ. We apply our method to the 2-complex dimensional stratum corresponding to the pyramidal hyperbolic surfaces.