Ends of stationary metric measure spaces

We prove that stationary random metric measure spaces have 0,1,2 or a Cantor space of ends. This notion includes stationary random graphs, manifolds and discrete subgroups. In the case of surfaces, we classify all possible homeomorphism types, in analogy with the work of Biringer and Raimbault on unimodular Riemannian manifolds. Our approach relies on a general “no geometric core” principle and an analysis of finite versus infinite expected return times.