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Peer-reviewedCC BYProceedings of the London Mathematical Society

Nonamenable subforests of multi‐ended quasi‐pmp graphs

  • Ruiyuan Chen1
  • Grigory Terlov2
  • Anush Tserunyan3
  1. 1Faculty of Mathematics Informatics, and Mechanics University of Warsaw Warsaw Poland
  2. 2Department of Statistics and Operations Research University of North Carolina Chapel Hill North Carolina USA
  3. 3Mathematics and Statistics Department McGill University Montréal QC Canada
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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.

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Cite

BibTeXRIS
BibTeX
@article{chen2026nonamenable,
  title = {{Nonamenable subforests of multi‐ended quasi‐pmp graphs}},
  author = {Ruiyuan Chen and Grigory Terlov and Anush Tserunyan},
  journal = {Proceedings of the London Mathematical Society},
  year = {2026},
  month = sep,
  doi = {10.1112/plms.70217},
  url = {https://doi.org/10.1112/plms.70217},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Nonamenable subforests of multi‐ended quasi‐pmp graphs
AU  - Ruiyuan Chen
AU  - Grigory Terlov
AU  - Anush Tserunyan
PY  - 2026
DA  - 2026/09/01
JO  - Proceedings of the London Mathematical Society
DO  - 10.1112/plms.70217
UR  - https://doi.org/10.1112/plms.70217
AB  - 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.
SN  - 0024-6115
SN  - 1460-244X
ER  -