Explicit spectral gap for Schottky subgroups of SL(2,Z).

Let F be a family of finite coverings of a hyperbolic surface S. A spectral gap of F is an interval I = [0, epsilon] such that the eigenvalues in I  (counted with multiplicity) of the Laplacian \Delta_S of S and \Delta_X, any X ∈ F, are the same. I will present a joint work with M. Magee where we give a spectral gap for congruence coverings when S is the surface associated to a Schottky subgroup of SL(2, Z) with thick enough limit set. The proof exploits the link between eigenvalues of the Laplacian and zeros of dynamical zeta functions attached to S via the thermodynamic formalism.