Kesten's theorem for uniformly recurrent subgroups

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We prove an inequality on the difference between the spectral radius of the Cayley graph of a group G and the spectral radius of the Schreier graph H G for any subgroup H. As an application we extend Kesten's theorem on spectral radii to uniformly recurrent subgroups and give a short proof that the result of Lyons and Peres on cycle density in Ramanujan graphs holds on average. More precisely, we show that if G is an infinite deterministic Ramanujan graph, then the time spent in short cycles by a random walk of length n is o(n).