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.
CITATION STYLE
Gamburd, A. (2002). On the spectral gap for infinite index “congruence” subgroups of SL2(Z). Israel Journal of Mathematics, 127, 157–200. https://doi.org/10.1007/BF02784530
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