# Boundary of the pyramidal equisymmetric locus of M_g

## TL;DR

This paper characterizes the boundary stratification of the pyramidal locus within the moduli space of hyperbolic surfaces, providing a complete classification of the boundary components for surfaces with pyramidal symmetry.

## Contribution

It provides a complete list of boundary strata for the pyramidal locus in the moduli space of hyperbolic surfaces, clarifying the topological types of degenerations with pyramidal symmetry.

## Key findings

- Complete classification of boundary strata for P_n
- Identification of topological types of degenerations
- Clarification of the stratification structure

## Abstract

The augmented moduli space is a compactification of moduli space M_n obtained by adding stable hyperbolic surfaces. The different topological types of the added stable surfaces produces a stratification of the boundary of M_n . Let P_n be the pyramidal locus in the moduli space, i.e., the set of hyperbolic surfaces of genus n such that the topological action of its preserving-orientation isometry group is the pyramidal action of the dihedral group Dn. The purpose of this paper is to state the complete list of strata in the boundary of Pn