Kleinian Schottky groups, Patterson Sullivan measures and Fourier Decay

Géométrie Dynamique

Salle Duhem M3
Pan Wenyu
Pennsylvania State University
Vendredi, 21 Juin, 2019 - 10:15 - 11:15

We show that the Fourier transform of the Patterson-Sullivan measure of a Kleinian Schottky subgroup of PSL2(C) enjoys polynomial decay. This generalises a result of Bourgain-Dyatlov for convex co-compact Fuchsian groups, 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.