Coarse non-amenability and covers with small eigenvalues

Given a closed Riemannian manifold M and a (virtual) epimorphism $${\pi_1(M)\twoheadrightarrow \mathbb{F}_2}$$ of the fundamental group onto a free group of rank 2, we construct a tower of finite sheeted regular covers $${\left\{M_n\right\}_{n=0}^{\infty}}$$ of M such that λ1(Mn) → 0 as n → ∞. 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 π1(M) is supposed to surject onto an amenable group.