Emmanuel Breuillard Université Paris-Sud (Orsay) Mathematics This mini-course will consist of a survey of results and techniques about diameter and spectral gaps bounds for finite groups, and some of their applications to the study of thin subgroups of arithmetic groups, culminating in the super-strong approximation theorem. In particular we will describe the role played by multiplicative combinatorics via approximate subgroups and by ergodic theory via random matrix products.