Kleinian Schottky groups, Patterson-Sullivan measures and Fourier decay

Abstract

Let $\Gamma$ be a Zariski dense Kleinian Schottky subgroup of PSL2(C). Let $\Lambda (\Gamma )$ be its limit set, endowed with a Patterson-Sullivan measure $\mu$ supported on $\Lambda (\Gamma )$. We show that the Fourier transform $\hat{\mu }(\xi )$ enjoys polynomial decay as $|\xi |$ 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.

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