We demonstrate measurable one-ended spanning trees in measurable graphs, indicating a connection to measure-hyperfinite properties.
We show that a one-ended, locally finite, measurable graph on a standard probability space admits a measurable one-ended spanning subtree if and only if it is measure-hyperfinite. This answers a question posed by Bowen, Poulin, and Zomback and extends recent results of Timár and Conley, Gaboriau, Marks, and Tucker-Drob.
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Bowen et al. (2025) studied this question.
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