On the spectral gap for infinite index “congruence” subgroups of SL2(Z)

A celebrated theorem of Selberg states that for congruence subgroups of SL2(Z) there are no exceptional eigenvalues below 3/16. Extending the work of Sarnak and Xue for cocompact arithmetic lattices, we prove a generalization of Selberg’s theorem for infinite index “congruence” subgroups of SL2(Z). For such subgroups with a high enough Hausdorff dimension of the limit set we establish a spectral gap property and consequently solve a problem of Lubotzky pertaining to expander graphs.