= Generic isolation of lifted simple geodesics
{title2=$\ell(\widetilde\gamma)=r\ell(\gamma)$}
In a finite hyperbolic covering, primitive lifts of a primitive closed <geodesic> correspond to cycles of its <monodromy permutation>; an $r$-cycle gives length $r\ell(\gamma)$. In a real-analytic family of compact <hyperbolic surfaces> or hyperbolic cone orbifolds, a prescribed multiple of a simple curve's length can be isolated generically from unrelated lifted lengths. To exclude an identity of length functions, pinch the simple curve: only its peripheral powers can have length tending to zero, while intersecting curves are long by the <collar lemma>. The countably many remaining length equalities have empty interior, so the <Baire category theorem> supplies a parameter avoiding them. Only finitely many closed geodesics lie below a fixed local length bound; thus the isolation persists in an open parameter neighbourhood. This makes cycle-intersection information an intrinsic obstruction to an <isometry>, rather than an invariant of a chosen covering map alone.
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