Coarse non-amenability and covers with small eigenvalues

Given a closed Riemannian manifold M and a (virtual) epimorphism from the fundamental group of M onto a free group of rank 2, we construct a tower of finite sheeted regular covers {M_n}_{n=0}^{\infty} of M such that the first non-zero eigenvalues \lambda_1(M_n) of the Laplacian converge to zero as n tends to infinity. This is the first example of such a tower which is not obtainable up to uniform quasi-isometry (or even up to uniform coarse equivalence) by the previously known methods where the fundamental group of M is supposed to surject onto an amenable group.