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A closed Teichmüller geodesic in the moduli space ℳ  g   of Riemann surfaces of genus  g  is called  L ‐ short  if it has length at most  L / g . We show that, for any  L >0, there exist  R >ϵ>0, independent of  g , so that the  L ‐short geodesics in ℳ  g   all lie in the intersection of the ϵ‐thick part and the  R ‐thin part. We also estimate the number of  L ‐short geodesics in ℳ  g  , bounding this from above and below by polynomials in  g  whose degrees depend on  L  and tend to infinity as  L  does.

On the number and location of short geodesics in moduli space