Non-standard limits of graphs and some orbit equivalence invariants

We consider probability measure preserving discrete groupoids, group actions and equivalence relations in the context of general spaces. study for these objects notions cost, $\beta$-invariant some higher-dimensional variants. also propose various convergence results about $\ell^2$-Betti numbers rank gradient sequences actions, groupoids or under weak finiteness assumptions. In particular we connect combinatorial cost with ultralimit relations. Finally a relative version Stuck-Zimmer property is considered.