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.
Nonsimple closed geodesics with given intersection number on hyperbolic surfaces
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We prove that the minimal length of a closed geodesic with self-intersection number $k$ on any finite-type hyperbolic surface is $2\cosh^{-1}(1+2k)$ for $k>1750$. This improves the previously known threshold $k > 10^{13350}$. Our proof is independent of it.
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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.