Generalization of Selberg’s $ \frac{3}{{16}} $ theorem and affine sieve

Abstract

An analogue of the well-known $ \frac{3}{{16}} $ lower bound for the first eigenvalue of the Laplacian for a congruence hyperbolic surface is proven for a congruence tower associated with any non-elementary subgroup L of SL(2, Z). The proof in the case that the Hausdorff of the limit set of L is bigger than $ \frac{1}{2} $ is based on a general result which allows one to transfer such bounds from a combinatorial version to this archimedian setting. In the case that delta is less than $ \frac{1}{2} $ we formulate and prove a somewhat weaker version of this phenomenon in terms of poles of the corresponding dynamical zeta function. These “spectral gaps” are then applied to sieving problems on orbits of such groups.