The general Ramanujan Conjectures for congruence subgroups of arithmetic groups and approximations that have been proven towards them are central to many modern diophantine applications. Recently some robust “softer” tools have been developed and results established which deal with very general subgroups of the group of n by n integer matrices with determinant equal to 1 or –1, which go by the name “thin matrix groups.” We will describe some one of these tools (“expanders”) and review some novel diophantine applications of this theory. We also discuss the ubiquity of these thin matrix groups.

Aimed at a general mathematical audience.