Small eigenvalues of hyperbolic surfaces with many cusps

Abstract

We study topological lower bounds on the number of small Laplacian eigenvalues on hyperbolic surfaces. We show that for any $a>0$, there exists $b>0$ such that if $g<an$ and $X$ has genus $g$ and $n$ cusps, then $X$ has at least $b(2g+n-2)$ small eigenvalues. This saturates the sharp upper bound $2g+n-2$ of Otal and Rosas up to a constant multiplicative factor.