Non-standard limits of graphs and some orbit equivalence invariants

Carderi, Alessandro; Gaboriau, Damien; de la Salle, Mikael

Annales Henri Lebesgue, Tome 4 (2021), pp. 1235-1293.

Nous nous intéressons aux groupoïdes discrets préservant une mesure de probabilité, aux actions de groupes et aux relations d’équivalence dans le contexte d’espaces de probabilité généraux. Pour ces objets, nous considérons les notions de coût, de nombres de Betti ${\ell }^2$, d’invariant $\beta$ et des variantes de dimension supérieure. Nous proposons aussi, sous de faibles hypothèses de finitude, divers résultats de convergence de nombres de Betti ${\ell }^2$ et de gradient de rang pour des suites d’actions, de groupoïdes et de relations d’équivalence. En particulier, nous établissons le lien entre le coût combinatoire et le coût de la relation d’équivalence ultralimite. Enfin, nous étudions une version relative de la propriété de Stuck–Zimmer.

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.

Classification : 37A20, 05C75, 20E18, 20F69, 28E05, 46M07
Mots clés : Orbit equivalence, Asymptotic Properties of Graphs and Groups, Ultraproducts, Cost, L2 Betti Numbers, Soficity, rank gradient

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