On the Bass Note of a Schottky Group [PDF]
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|Peter G. Doyle|
|Using a classical method from physics called Rayleigh's cutting method, Doyle proves 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 H3. 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 the Schottky group, or equivalently, for the critical exponent of the Poincaré series associated with the Schottky group. The latter implication answers a question that can be traced back to Schottky and Burnside. A paper that appeared in Acta Mathematica Universitatis Comenianae. A PostScript file is available from Doyle's site.|
|Math Topics:||Eigenvectors/Eigenvalues, Analysis, Hyperbolic Geometry, Physics|
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