On largeness and multiplicity of the first eigenvalue of hyperbolic surfaces

Abstract: We apply topological methods to study the smallest non-zero number $\lambda_1$ in the spectrum of the Laplacian on finite area hyperbolic surfaces. For closed hyperbolic surfaces of genus two we show that the set ${S \in {\mathcal{M}_2}: {\lambda_1}(S) > 1/4 }$ is unbounded and disconnects the moduli space ${\mathcal{M}_2}$.