A celebrated theorem of Selberg states that for congruence subgroups of the modular group 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 the modular group. For such subgroups with a high enough Hausdorff dimension of the limit set we establish a spectral gap property and consequently obtain uniform expansion results pertaining to Lubotzky's 1-2-3 problem. For related families of Cayley graphs (in recent joint work with M. Shahshahani) we obtain uniform diameter bounds by extending Solovay-Kitaev theorem in Quantum Computation.
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