Hyperbolic Y-piece
= Hyperbolic Y-piece
{c}
{title2=$Y(\ell_1,\ell_2,\ell_3)$}
= Y-piece
{c}
{synonym}
A <Y-piece> is a compact curvature-$-1$ <hyperbolic surface> homeomorphic to a <pair of pants (mathematics)>, with three geodesic boundary curves. Its labelled boundary lengths $\ell_1,\ell_2,\ell_3>0$ determine it up to <isometry>. Cutting along the three mutually perpendicular seams produces two congruent <right-angled hyperbolic hexagons> with alternating sides $\ell_i/2$. Conversely, doubling the hexagon along its other three sides constructs the <Y-piece> for arbitrary positive lengths.