A Y-piece is a compact curvature- hyperbolic surface homeomorphic to a pair of pants, with three geodesic boundary curves. Its labelled boundary lengths determine it up to isometry. Cutting along the three mutually perpendicular seams produces two congruent right-angled hyperbolic hexagons with alternating sides . Conversely, doubling the hexagon along its other three sides constructs the Y-piece for arbitrary positive lengths.
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