We study the behavior of the co-spectral radius of a subgroup H of a discrete group $\Gamma $ under taking intersections. Our main result is that the co-spectral radius of an invariant random subgroup does not drop upon intersecting with a deterministic co-amenable subgroup. As an application, we find that the intersection of independent co-amenable invariant random subgroups is co-amenable.

We prove a lower bound on the difference between the spectral radius of the Cayley graph of a group $G$ and the spectral radius of the Schreier graph $H\backslash G$ 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 [Lyons and Peres. Cycle density in infinite Ramanujan graphs. Ann. Probab.43(6) (2015), 3337–3358, Theorem 1.2] holds on average. More precisely, we show that if ${\mathcal{G}}$ is an infinite deterministic Ramanujan graph then the time spent in short cycles by a random trajectory of length $n$ is $o(n)$.