Limit points of the branch locus of ${\mathcal{M}}_g$

Abstract

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