Bourgain and Gamburd, using combinatorial growth estimates of Helfgott, Breuillard-Green-Tao and Pyber-Szabo showed that the Cayley graph of SL(d,Z/p) with respect to a generating set that is the projection mod p of an appropriate set of elements of SL(d,Z) form a family of expanders. This was extended by Salehi-Golsefidy and Varju to perfect groups such as the skew product of SL(d,Z) with Z^d.

One of the main open problems that remain is to which extent the expansion properties of Cayley graphs of groups such as SL(d,Z) depends on the generating set.

We study this problem in the skew product setting, showing that there is a uniform spectral gap for all choice of generators for which the projection to the SL(d,Z) is "good" (has a spectral gap of given size).

This study was motivated by an analogous question regarding random walks on the isometry group of R^n and the study of self similar sets.

Joint work with Péter Varju.