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Perfect matchings in hyperfinite graphings

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.

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Citation data

  • DOI 10.24350/CIRM.V.19809703
  • Cite this video Sabok Marcin (9/16/21). Perfect matchings in hyperfinite graphings. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19809703
  • URL https://dx.doi.org/10.24350/CIRM.V.19809703

Bibliography

  • Bowen, Matthew, Gabor Kun, and Marcin Sabok. "Perfect matchings in hyperfinite graphings." arXiv preprint arXiv:2106.01988 (2021). - https://arxiv.org/abs/2106.01988

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