Abstract:We prove the a.e. nonamenability of locally finite quasi-pmp Borel graphs whose every component admits at least three nonvanishing ends with respect to the underlying Radon--Nikodym cocycle. We witness their nonamenability by constructing Borel subforests with at least three nonvanishing ends per component, and then applying Tserunyan and Tucker-Drob's recent characterization of amenability for acyclic quasi-pmp Borel graphs. Our main technique is a weighted cycle-cutting algorithm, which yields a weight-maximal spanning forest. We also introduce a random version of this forest, which generalizes the Free Minimal Spanning Forest, to capture nonunimodularity in the context of percolation theory.
From: Anush Tserunyan [view email]
[v1]
Tue, 15 Nov 2022 05:27:29 UTC (80 KB)
[v2]
Fri, 17 Nov 2023 19:04:39 UTC (83 KB)
[v3]
Thu, 11 Sep 2025 17:36:40 UTC (87 KB)
[v4]
Sun, 28 Dec 2025 04:24:59 UTC (89 KB)
[v5]
Thu, 27 Aug 2026 20:29:25 UTC (91 KB)
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