Small eigenvalues of hyperbolic surfaces with many cusps

We study topological lower bounds on the number of small Laplacian eigenvalues on hyperbolic surfaces. We show there exist constants $a,b>0$ such that when $(g+1)<a\frac{n}{\log n}$, any hyperbolic surface of genus-$g$ with $n$ cusps has at least $b\frac{2g+n-2}{\log\left(2g+n-2\right)}$ Laplacian eigenvalues below $\frac{1}{4}$. We also show that, under certain additional constraints on the lengths of short geodesics, the lower bound can be improved to $b\left(2g+n-2\right)$ with the weaker condition $(g+1)<an$.