Let two finite locally isometric covers carry uniquely length-identified primitive closed geodesics . If their geometric intersection numbers differ, the covering surfaces are not isometric, because an isometry preserves both lengths and intersections. For base simple curves intersecting once, lift components correspond to cycles of their monodromy permutations. Their intersection number is the number of sheets common to the two cycles. This distinguishes covers even when every individual permutation has the same cycle type in both actions, as in Gassmann equivalence.
In a finite hyperbolic covering, primitive lifts of a primitive closed geodesic correspond to cycles of its monodromy permutation; an -cycle gives length . 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.
If two simple base geodesics meet once, their lift components correspond to the cycles of their monodromy permutations. A lifted crossing belongs to the sheet labels common to both cycles, giving the displayed geometric intersection count. When their relevant lifted lengths identify these components independently of the covering map, different multisets of intersection counts obstruct an isometry between the covering surfaces.
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