Let S be a finite symmetric subset of SL_d(Z). I will discuss some recent results on how words of a given length formed from the elements of S distribute among the residue classes modulo an integer q. In particular I will show that the corresponding Cayley graphs of the groups SL_d(Z/qZ) are expanders, and this yields rapid mixing of the random walk.

Some of the results are joint with Jean Borugain, and others by Alireza Salehi Golsefidy.