On the bass note of a Schottky group
Peter G. Doyle · Acta Mathematica · 1988
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 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 Hausdor dimension of the limit set of , or equivalently, for the critical exponent of the Poincare series associated with . The latter implication answers a question that can be traced back to Schottky and Burnside. 1