On second eigenvalues of closed hyperbolic surfaces for large genus

Abstract: In this article, we study the second eigenvalues of closed hyperbolic surfaces for large genus. We show that for every closed hyperbolic surface $X_g$ of genus $g$ $(g\geq 3)$, up to uniform positive constants multiplications, the second eigenvalue $\lambda_2(X_g)$ of $X_g$ is greater than $\frac{\mathcal{L}_2(X_g)}{g^2}$ and less than $\mathcal{L}_2(X_g)$; moreover these two bounds are optimal as $g\to \infty$. Here $\mathcal{L}_2(X_g)$ is the shortest length of simple closed multi-geodesics separating $X_g$ into three components. Furthermore, we also investigate the quantity $\frac{\lambda_2(X_g)}{\mathcal{L}_2(X_g)}$ for random hyperbolic surfaces of large genus. We show that as $g\to \infty$, a generic hyperbolic surface $X_g$ has $\frac{\lambda_2(X_g)}{\mathcal{L}_2(X_g)}$ uniformly comparable to $\frac{1}{\ln(g)}$.