Smoothness criterion from cone-monodromy cycles
= Smoothness criterion from cone-monodromy cycles
{title2=$\text{residual cone order}=m/\ell$}
Over a cone point of order $m$, a cycle of length $\ell$ in the <monodromy> on sheets has ramification degree $\ell$ and residual cone angle $2\pi\ell/m$. The lifted inherited <hyperbolic metric> is smooth precisely when every such cycle has length $m$. Equivalently the subgroup defining the cover meets no conjugate of a nonidentity power of a cone generator.