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arXiv · 1812.00704

Non-standard limits of graphs and some orbit equivalence invariants

Alessandro Carderi·Damien Gaboriau·Mikael de la Salle
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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, $\beta$-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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Alessandro Carderi, Damien Gaboriau, Mikael de la Salle. 2018-12-03. Non-standard limits of graphs and some orbit equivalence invariants. https://doi.org/10.5802/ahl.102

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