Shape of filling-systole subspace in surface moduli space and critical points of systole function

Abstract

This paper studies the space $X_g\subset {\mathcal{M}}_g$ 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 $\frac{1}{5}\log \log g$ to $X_g$ and in the second and third part, prove that most points in ${\mathcal{M}}_g$ have Teichmüller distance $\frac{1}{5}\log \log g$ to $X_g$ and Weil-Petersson distance $0.6521(\sqrt{\log g}-\sqrt{7\log \log g})$ respectively.Therefore we prove that the radius-$r$ neighborhood of $X_g$ is not able to cover the thick part of ${\mathcal{M}}_g$ for any fixed $r>0$. In last two parts, we get critical points with small and large (comparable to diameter of thick part of ${\mathcal{M}}_g$) distance respectively.

33 pages, 16 figures

References in corpus (3)

• Large genus asymptotics for lengths of separating closed geodesics on random surfaces
• A new uniform lower bound on Weil-Petersson distance
• Maximal systole of hyperbolic surface with largest $S^3$ extendable abelian symmetry