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

## TL;DR

This paper investigates the limit points of the branch locus in the moduli space of hyperbolic surfaces, describing their topological types in the Deligne-Mumford compactification, with a focus on pyramidal hyperbolic surfaces.

## Contribution

It provides a detailed description of the topological types of limit points of strata in the branch locus of the moduli space, using epimorphisms to classify strata.

## Key findings

- Topological types of limit points are characterized in terms of epimorphisms.
- The method is applied to the 2-dimensional stratum of pyramidal hyperbolic surfaces.
- Results enhance understanding of the boundary behavior of the branch locus.

## Abstract

Let $\mathcal{M}_{g}$ be the moduli space of compact connected hyperbolic surfaces of genus $g\geq2$, 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 $g\geq 2$ 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 $\widehat{\mathcal{M}}_g$ in terms of $\Phi$. We apply our method to the $2$-complex dimensional stratum corresponding to the pyramidal hyperbolic surfaces.