Non-standard limits of graphs and some orbit equivalence invariants
Annales Henri Lebesgue, Volume 4 (2021), pp. 1235-1293.

KeywordsOrbit equivalence, Asymptotic Properties of Graphs and Groups, Ultraproducts, Cost, L2 Betti Numbers, Soficity, rank gradient

### Abstract

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

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