Perfect matchings in bipartite graphings
We proved with Matt Bowen and Marcin Sabok that hyperfinite, bipartite, one-ended graphings with a non-integral fractional perfect matching admit a measurable perfect matching a.e. We apply this to prove the amenable version of the Lyons-Nazarov theorem, to give a new proof of the measurable circle squaring and to find balanced orientations in hyperfinite, one-ended graphing. Timar used our rounding theorem to give factor matchings between independent Poisson point processes in the euclidean space.
If time allows I will give an example of a d-regular measurably bipartite treeing for d >2 without measurable perfect matching answering a question of Kechris and Marks. Moreover, this example admits no absolutely integrable circulation. In particular, it admits no free Z-action on any subset of vertices with positive measure.
The talk will focus on the proofs, especially on the structure of extreme points of the set of measurable fractional perfect matchings and the use of connected toasts (derived from one-ended spanning trees).