Leírás
Measurable perfect matchings have been intensively studied in the last decade. Most of the theorems use either the expansion of the corresponding measurable graph (Lyons-Nazarov theorem, Banach-Ruziewicz problem) or its hyperfiniteness (measurable and Borel solution to Tarski's circle squaring problem). However, not many examples are known when there is no measurable perfect matching though the measurable Hall condition holds. Most examples admit two-ended components (i.e., of linear growth): in the hyperfinite case we characterized with Bowen and Sabok graphings without measurable perfect matchings via such obstructions. I explain this characterization and give applications from the measurable circle squaring to the amenable Lyons-Nazarov theorem.
In the second half of the talk I construct for every d ≥ 3 a d-regular acyclic measurably bipartite graphing that admits no measurable perfect matching. This answers a question of Kechris and Marks. A denser version refutes a conjecture of Peled and Gurel-Gurevich on couplings by constructing a permuton that admits atomless sections, but its support does not contain the graph of a pmp [0,1]->[0,1] mapping. The talk is based on two papers under revision: the hyperfinite one with Bowen and Sabok at Duke and the one with the examples at Inventiones. (I also hope to talk about a follow-up work with Laci Tóth if time and
patience allows.)