Ends of unimodular random manifolds

We study the ends of a generic manifold, with respect to a unimodular measure on the space of pointed Riemannian manifolds with bounded curvatures. We apply our general result to the case of surfaces and obtain as corollaries a very precise description of generic leaves for foliations with invariant measures and of quotients of the hyperbolic plane by invariant random subgroups of the isometry group.