We proved with Matt Bowen and Gábor Kun 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 graphings. Timar used our rounding theorem to give factor matchings between independent Poisson point processes in the euclidean space.

This talk will focus on the applications, especially the characterization of groups whose bipartite Cayley graphs admit a factor of iid perfect matching, solving the bipartite case of the Lyons-Nazarov problem. We will also see how our result implies the measurable circle squaring (with two different proofs, if time permits).