Ground state and lowest eigenvalue of the Laplacian for non-compact hyperbolic surfaces

Let M be a complete Riemannian surface with constant curvature -1, infinite volume, and a finitely generated fundamental group. Denote by λ(M) the lowest eigenvalue of the Laplacian on M, and let Φ_M be the associated eigenfunction. We estimate the size of λ(M) and the shape of Φ_M by a finite procedure which has an electrical circuit analogue. Using the Margulis lemma, we decompose M into its thick and thin parts. On the compact thick components, we show that Φ_M varies from a constant value by no more than O( {Mathematical expression}). The estimate for λ(M) is calculable in terms of the topology of M and the lengths of short geodesics of M. An analogous theorem of the compact case was treated in [SWY].