A filling multi-geodesic on a genus g hyperbolic surface has length at least half the perimeter of a regular right-angled (8g-4)-gon, and this bound is sharp.
The geometry and combinatorics of closed geodesics on hyperbolic surfaces
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Shortest filling geodesics on hyperbolic surfaces
A filling multi-geodesic on a genus g hyperbolic surface has length at least half the perimeter of a regular right-angled (8g-4)-gon, and this bound is sharp.