Robin Tucker-Drob: "Measure Equivalence, Schlichting Completions, and Baumslag-Solitar groups"

jeudi 24 nov. 2022, 14:00 → 15:00 Europe/Paris

Abstract: A subgroup H of a group Γ is commensurated by Γ if all H-orbits in Γ/H are finite. In this situation, the closure of Γ in the group of all permutations of Γ/H is a totally disconnected locally compact group called the Schlichting completion of the pair (Γ, H) and we denote it G(Γ, H). We show that if H and K are amenable commensurated subgroups of Γ and Λ respectively such that the associated Schlichting completions G(Γ, H) and G(Λ, K) are both hyperbolic with trivial amenable radical, then every measure equivalence coupling of Γ with Λ descends canonically to a measure equivalence coupling of the (possibly nonunimodular) groups G(Γ, H) and G(Λ, K). This unifies and generalizes theorems of Houdayer--Raum, Kida, and Monod--Shalom. I will discuss the consequences of this for the open problem of classifying Baumslag-Solitar groups up to measure equivalence.