Quantitative spectral gap for thin groups of hyperbolic isometries

The Laplacian on a geometrically finite hyperbolic manifold M has some discrete spectra above 0, provided the Hausdorff dimension of the limit set associated to M is large enough. In particular there is a spectral gap between the two lowest eigenvalues of the Laplacian. For sieving purposes related to arithmetic M it is important that the spectral gap can be made explicit and uniform as one passes through a tower of congruence covers. I’ll talk about what goes into proving such a quantitative spectral gap in the case that the manifold has infinite volume. This generalizes a result of Gamburd (2002), following a method of Sarnak and Xue (1991).