On the bass note of a Schottky group
type: Item
instance of: scholarly article
title: On the bass note of a Schottky group
zbMATH Open document ID: 0649.30036
DOI: 10.1007/BF02392277
author: Peter G. Doyle
published in: Acta Mathematica
publication date: 1988-01-01T00:00:00Z
review text: For a Kleinian group \(\Gamma\) let \(\delta\) (\(\Gamma)\) denote the exponent of convergence. In dimensions \(\geq 4\) it is known (Phillips-Sarnak) that there exists a constant \(c_ n<n-1\) so that for any Schottky group one has \(\delta (\Gamma)\leq c_ n\). It had been unclear for a long time as to whether the analogous statement should hold in the classical case \(n=3\); in the Fuchsian case \(n=2\) it is false (Beardon). The author proves here that indeed the analogous statement is true when \(n=3\). His proof, like that of Phillips-Sarnak, uses the connection of \(\delta\) (\(\Gamma)\) with a certain Rayleigh quotient which, by an ingenious parametrization and cutting argument is shown to be bounded away from zero.
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 4059801
zbMATH Keywords: Kleinian group
zbMATH Keywords: Schottky group
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
cites work: Poincaré Theta Series and Singular Sets of Schottky Groups
cites work: (3/2)-dimensional measure of singular sets of some Kleinian groups
cites work: The spectral geometry of the apollonian packing
cites work: On the deformation theory of classical Schottky groups
cites work: Q3706275
cites work: A minimum principle for the principal eigenvalue for second-order linear elliptic equations with natural boundary conditions
cites work: The limit set of a Fuchsian group
cites work: The Laplacian for domains in hyperbolic space and limit sets of Kleinian groups
cites work: Discrete conformal groups and measurable dynamics
cites work: Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups
description: scientific article; zbMATH DE number 4059801