The analysis of the spectrum of an element of the group ring of SU(2),
SL(2,Z),... under finite dimensional irreducible unitary
representations, arises in many contexts (in analysis-the Ruziewizc
problem, in combinatorics-expanders, in quantization-semi classical
theory). One crucial feature is the apparently difficult problem of a
spectral gap. For SU(2) the construction of an element with a spectral
gap was first carried out by Drinfeld. We give a general and elementary
method for achieving such a gap and discuss some other features of the
spectra of such elements. This is joint work with Gamburd and
Jakobson.