A bound for the ratio of consecutive eigenvalues of the hyperbolic Laplacian for the modular groups (Q2753160)

Let \(0<\lambda_1\leq \lambda_2\leq \dots\) be the non-zero eigenvalues of the Laplacian on the modular curve \(\mathbb{H}/\text{SL}_2(\mathbb{Z})\), where \(\mathbb{H}\) is the upper half plane. The main result of the present paper is the estimate for the consecutive quotients, NEWLINE\[NEWLINE{\lambda_{j+1}\over\lambda_j}\leq 1.628 \dotsNEWLINE\]NEWLINE for every \(j\in\mathbb{N}\). This immediately implies that the same estimate holds for all finite index subgroups of SL\(_2(\mathbb{Z})\).NEWLINENEWLINENEWLINEThe proof uses the Selberg trace formula and a growth estimate on the lengths of hyperbolic classes that is a consequence of Siegel's class number asymptotic and thus is not available for, say, cocompact groups. It is therefore not clear whether a similar result holds for other groups.