Co-spectral radius of intersections

We study the behavior of the co-spectral radius of a subgroup $H$ of a discrete group $\mathrm{\Gamma }$ 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.