Random walks in compact groups

Let X_1,X_2,… be independent identically distributed random elements of a compact group G . We discuss the speed of convergence of the law of the product X_l⋯ X_1 to the Haar measure. We give poly-log estimates for certain finite groups and for compact semi-simple Lie groups. We improve earlier results of Solovay, Kitaev, Gamburd, Shahshahani and Dinai.