On the bass note of a Schottky group

Peter G. Doyle

Version 1.01A1
29 August 1994

Abstract:

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 for the lowest eigenvalue of the quotient manifold of a classical Schottky group acting on hyperbolic 3-space . By work of Patterson and Sullivan, this implies that there is a universal upper bound for the Hausdorff dimension of the limit set of , or equivalently, for the critical exponent of the Poincaré series associated with . The latter implication answers a question that can be traced back to Schottky and Burnside.

• Introduction
  • Classical Schottky groups
  • A lower bound for the bass note
  • Implications
  • Some history
  • Rayleigh's cutting method
  • Infinitely skinny tubes that grow more or less exponentially
  • Plan
• Cutting a manifold into tubes
  • The lowest eigenvalue
  • The cutting method
  • Cutting into tubes
  • Inhomogeneous strings
  • Using piecewise differentiable test functions
  • What the tubes tell us about the original space
  • Twiddling and
  • Estimating for a string
  • Tubes that grow more or less exponentially
  • Growth criteria
• Cutting up
  • The geometrical problem
  • Cutting up the fundamental domain
  • The picture in the upper half-space model of
  • The case of no circles
  • The case of one circle
  • The case of two tangent circles
  • The case of two non-tangent circles
  • The case of three or more circles
  • The obvious strategy doesn't work
  • Keeping the tubes more or less vertical
  • Working our way down
  • Associating pieces of the plane to relevant hemispheres
  • Taking care of the babies
  • Room to grow
  • Annexing annuli
  • Avoiding conflicts between neighboring hemispheres
  • Overlap of renounced territory
  • Avoiding overlap
  • An elementary geometric lemma
  • How little area to ask for
  • Laying out the tubes
  • Taking stock
  • What is the constant?
• References