Relative expanders or weakly relatively Ramanujan graphs

Let G be a fixed graph with largest (adjacency matrix) eigenvalue λ 0 and with its universal cover having spectral radius $rho$. We show that a random cover of large degree over G has its "new" eigenvalues bounded in absolute value by roughly √λ 0ρ. This gives a positive result about finite quotients of certain trees having "small" eigenvalues, provided we ignore the "old" eigenvalues. This positive result contrasts with the negative result of A. Lubotzky and T. Nagnibeda which showed that there is a tree all of whose finite quotients are not "Ramanujan" in the sense of Lubotzky, R. Phillips, and P. Sarnakand of Y. Greenberg. Our main result is a "relative version" of the Broder-Shamir bound on eigenvalues of random regular graphs. Some of their combinatorial techniques are replaced by spectral techniques on the universal cover of G. For the choice of G that specializes our main theorem to the Broder-Shamir setting, our result slightly improves theirs.