On the bass note of a Schottky group

Using a classical method from physics called Rayleigh’s cutting method, we prove the conjecture of Phillips and Sarnak that there is a universal lower bound L2 > 0 for the lowest eigenvalue of the quotient manifold of a classical Schottky group Γ acting on hyperbolic 3-space H 3. By work of Patterson and Sullivan, this implies that there is a universal upper bound U2 < 2 for the Hausdorff dimension of the limit set of Γ, or equivalently, for the critical exponent of the Poincaré series associated with Γ. The latter implication answers a question that can be traced back to Schottky and Burnside.