By Marcin Sabok
Appears in collection : XVI International Luminy Workshop in Set Theory / XVI Atelier international de théorie des ensembles
We characterize hyperfinite bipartite graphings that admit measurable perfect matchings. In particular, we prove that every regular hyperfinite one-ended bipartite graphing admits a measurable perfect matching. We give several applications of this result. We extend the Lyons-Nazarov theorem by showing that a bipartite Cayley graph admits a factor of iid perfect matching if and only if the group is not iso-morphic to the semidirect product of Z and a finite group of odd order, answering a question of Kechris and Marks in the bipartite case. We also answer an open question of Bencs, Hruskova and Toth arising in the study of balanced orientations in graphings. Finally, we show how our results generalize and lead to a simple approach to recent results on measurable circle squaring.
By Juliette Schabanel
By Christoph Koutschan
