The shape of the filling-systole subspace in surface moduli space and critical points of the systole function

This paper studies the space Xg ⊂ Mg consisting of surfaces with filling systoles and its subset, critical points of the systole function.In the first part, we obtain a surface with Teichmüller distance 1 5 log log g to Xg and in the second and third part, prove that most points in Mg have Teichmüller distance 1 5 log log g to Xg and Weil-Petersson distance 0.6521( √ log g -√ 7 log log g) respectively.Therefore we prove that the radius-r neighborhood of Xg is not able to cover the thick part of Mg for any fixed r > 0. In last two parts, we get critical points with small and large (comparable to diameter of thick part of Mg) distance respectively.