Trace-formula methods in spectral geometry (Q1378953)

Let \(\Gamma\) be the modular group acting on the upper half-plane \(\mathbb{H}\) and denote by \(\Gamma_n\) the principal congruence subgroup of \(\Gamma\). If \(\lambda_1\) denotes the least positive eigenvalue of \(-\Delta\) on \(\Gamma_n \setminus \mathbb{H}\), then a famous theorem of Selberg asserts that \(\lambda_1 \geq {3\over 16}\) and a famous conjecture of Selberg claims that \(\lambda_1 \geq {1\over 4}\). The paper under review arose from the question of understanding Selberg's theorem in a geometric way. First, the author proves a trace formula for graphs. An application of this trace formula yields an analogue of the Selberg \({3 \over 16}\)-theorem for graphs. This theorem is then adapted to Riemann surfaces and yields an asymptotic bound on \(\lambda_1\) for certain families of Riemann surfaces which have many properties in common with the family of congruence surfaces.