Abstract: In this talk, I will first survey some results on the problem of spectral gaps for averaging operators on simple Lie groups, a natural question related to the speed of convergence for random walks on these groups. I will first describe the case of compact simple Lie groups, explaining briefly the remarkable method due to Bourgain-Gamburd and further esults. Then I will describe what result of Boutonnet-Ioana-Salehi Golsefidy in the non-compact case. Finally, I will explain how one cap apply these, following Bourgain, to a question related to fractal geometry, and more precisely to the study of the geometric properties of a natural measure associated to a random matrix products called the Furstenberg measure, in analogy with similar questions on iterated function systems, like Bernoulli convolutions.
