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

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