Israel Journal of Mathematics

, Volume 127, Issue 1, pp 157–200 | Cite as

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

  • Alex Gamburd


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.


Hausdorff Dimension Cayley Graph Fundamental Domain Finite Index Congruence Subgroup 
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Copyright information

© The Hebrew University Magnes Press 2002

Authors and Affiliations

  • Alex Gamburd
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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