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Let $\mathcal{M}_{g}$ be the moduli space of compact connected hyperbolicsurfaces of genus $g\geq2$, and ${\mathcal B}_g \subset {\mathcal M}_{g} $ itsbranch locus. Let $\widehat{{\mathcal{M}}_{g}}$ be the Deligne-Mumfordcompactification of the moduli space of smooth, complete, connected surfaces ofgenus $g\geq 2$ over $\mathbb{C}$. The branch locus ${\mathcal B}_g$ isstratified by smooth locally closed equisymmetric strata, where a stratumconsists of hyperbolic surfaces with equivalent action of their preservingorientation isometry group. Any stratum can be determined by a certainepimorphism $\Phi$. In this paper, for any of these strata, we describe thetopological type of its limits points in $\widehat{\mathcal{M}}_g$ in terms of$\Phi$. We apply our method to the $2$-complex dimensional stratumcorresponding to the pyramidal hyperbolic surfaces.

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