On the bass note of a Schottky group
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.
scientific article; zbMATH DE number 4059801