= Finite length coordinates for hyperbolic surfaces
{title2=$9g-9\text{ marked geodesic lengths}$}
For a fixed <pants decomposition> of a closed <genus>-$g$ <hyperbolic surface>, its $3g-3$ cuff lengths determine all its <Y-pieces>. Two suitably chosen transverse simple closed curves at each cuff, one obtained from the other by a <Dehn twist>, determine the corresponding twist: hyperbolic seam identities express their lengths through shifted hyperbolic cosines of that twist, removing the sign ambiguity of just one transversal. Thus finitely many marked lengths determine the <Fenchel–Nielsen coordinates>. The unmarked <length spectrum> lacks these labels, which is why this fact alone does not imply global spectral uniqueness.
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