Boundary of the pyramidal equisymmetric locus of $${\mathcal M}_n$$

Abstract

The augmented moduli space $$\widehat{\mathcal M}_n$$ is a compactification of moduli space $$\mathcal M_n$$ obtained by adding stable hyperbolic surfaces. The different topological types of the added stable surfaces produce a stratification of $$\partial \widehat{\mathcal M}_n$$. Let $$\mathcal {P}_n\subset {\mathcal M}_n$$ be the pyramidal locus in 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 $$D_n$$. The purpose of this paper is to state the complete list of strata in the boundary of $$\mathcal {P}_n$$.