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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. 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