Non-standard limits of graphs and some orbit equivalence invariants

Abstract

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 2 , d'invariant β et des variantes de dimension supérieure. Nous proposons aussi, sous de faibles hypothèses de finitude, divers résultats de convergence de nombres de Betti 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.

Asymptotic invariants of lattices in locally compact groups

“…Farber proved in [17] that the sequence b i (Γ n )/[Γ : Γ n ] also converges to β i (Γ) whenever {Γ n } n is a Farber and nested chain of finite index subgroups of Γ. The approximation problem for b i was later studied in several different occasion, for example in the case of non Farber chains in [10], for sofic approximations in [16] and in a more general setting in [13].…”

“…actions of a fixed countable group have the same cost, [20]. Abért and Nikolov's theorem was later generalized to non-nested sequences in [8] and [13], where in general only one inequality is proved.…”

“…Our understanding of the lattice approximation problem will be derived from the study of the regular ultraproduct and its cross section. Indeed using that the cross section of the regular ultraproduct is the ultraproduct of the cross sections we will reduce the problem to the case of ultraproducts of finite graphed equivalence relations studied in [13]. The idea behind the proof is not hard but since we will have to work with non standard probability spaces we have to be very careful about the measurability problems.…”

“…When we take the nerve of the associated covering by balls of radius 2η we are just considering a finite simplicial complex and we are choosing its root at random. Now as above we let n go to infinity and we take the ultraproduct of the sequence of pointed finite simplicial complexes as in [13] (which is closely related to the Benjamini-Schramm convergence of finite simplicial complexes). This will give us a random (not anymore finite) simplicial complex.…”

$$\ell ^2$$-Betti numbers of random rooted simplicial complexes

“…Petersen, Sauer and Thom generalized the notion of Farber sequences to lattices in a totally disconnected, second countable unimodular group G, which is also a instance of local weak convergence and proved that the normalized Betti numbers of a uniformly discrete Farber sequence converge to the ℓ 2 -Betti number of the group G [28]. Recently Carderi, Gaboriau and de la Salle proved a theorem for graphed groupoids and ultra products of them, which for instance implies the convergence for Farber sequences [8].…”