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LetM 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 onM, and let ΦM be the associated eigenfunction. We estimate the size of λ(M) and the shape of ΦM 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 ΦM varies from a constant value by no more thanO(<img src="/fulltext-image.asp?format=htmlnonpaginated&src=U34125QJ292U3256_html\220_2005_Article_BF01211062_TeX2GIFIE1.gif" border="0" alt="$$\sqrt {\lambda (M)}$$" />). The estimate for λ(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].

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