= Rigid ramification type
{title2=$\mathrm{Ni}(G;C_1,\ldots,C_r)/G$}
Fix ordered branch positions and conjugacy classes $C_i$ in a finite group $G$. The generating product-one tuples $(g_1,\ldots,g_r)\in\prod C_i$ form the inner Nielsen set. The type is rigid when this set is nonempty and a single simultaneous-conjugation orbit. Over $\mathbb C(T)$ this fixes the connected Galois cover up to isomorphism: tuples prescribe the <monodromy> epimorphism of the punctured sphere, and conjugate epimorphisms give isomorphic covers; the local completions at punctures are uniquely determined by their cyclic <monodromy>.
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