Property (T) for Unimodular random graphs and manifolds

This is a report on a joint work with Hector Jardon Sanchez and Sam Mellick. I will start by explaining how the questions about homology growth in closed manifolds motivate extending results about $L^2$-homology and the theory of cost from the context of universal coverings of manifolds to the context of unimodular random graphs and manifolds. Then I will defeine property (T) for URG's, explain the analogs of Connes-Weiss and Glasner-Weiss theorems in this context, and discuss the cost of such URG's.