18 research outputs found
Kleinian Schottky groups, Patterson-Sullivan measures and Fourier decay
Let be a Zariski dense Kleinian Schottky subgroup of PSL2(C). Let
be its limit set, endowed with a Patterson-Sullivan measure
supported on . We show that the Fourier transform
enjoys polynomial decay as goes to
infinity. This is a PSL2(C) version of the result of Bourgain-Dyatlov [8], and
uses the decay of exponential sums based on Bourgain-Gamburd sum-product
estimate on C. These bounds on exponential sums require a delicate
non-concentration hypothesis which is proved using some representation theory
and regularity estimates for stationary measures of certain random walks on
linear groups.Comment: 2 figure
Sharp resonances on hyperbolic manifolds
Non UBCUnreviewedAuthor affiliation: Université d'Avignon (France)Facult