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.
Minimal length of nonsimple closed geodesics on hyperbolic surfaces
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In the present paper, we show that the minimal length of closed geodesics on finite-type hyperbolic surfaces with self-intersection number $k$ has order $2\log k$ as $k$ gets large.
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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.