I will present recent and on-going joint work with Oren Becker on spectral gaps for actions of linear groups and consequences regarding expander graphs. We prove a uniform spectral gap (uniform Kazhdan constant for quasi-regular representations) for all strongly irreducible actions of linear groups on projective and affine spaces. This strong uniformity (w.r.t. the generating set and the field of coefficients) has its source in diophantine considerations and the theory of heights. It allows to obtain uniform expansion for finite simple groups of Lie type of bounded rank over almost all primes as well as uniform non-concentration estimates à la Littlewood-Offord for random walks on matrix groups.
