EXPANDING GRAPHS AND INVARIANT-MEANS

All the known explicit constructions of expander families are essentia lly obtained by considering a sequence of finite index normal subgroup s N-i < the broken barey graphs of Gamma/N-i w.r. to the projection of a global finite set of generators of Gamma. For many of these example s (e.g. Gamma = SL2(Z), Gamma/N-i congruent to SL2(F-p)), we present f irst constructions of new, different, sets of generators for the finit e quotients, which make the Cayley graphs an expander family. An intri nsic connection between the expanding property and uniqueness of the H aar measure on an appropriate compact group, as an invariant mean, is established and used in the construction of such generators.