Nonamenable subforests of multi-ended quasi-pmp graphs
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912790041591808 |
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| author | Chen, Ruiyuan Terlov, Grigory Tserunyan, Anush |
| author_facet | Chen, Ruiyuan Terlov, Grigory Tserunyan, Anush |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_07908 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Nonamenable subforests of multi-ended quasi-pmp graphs Chen, Ruiyuan Terlov, Grigory Tserunyan, Anush Dynamical Systems Logic Probability Primary 37A20, 03E15, 60K35, Secondary 37A40, 05C22, 60B99 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. |
| title | Nonamenable subforests of multi-ended quasi-pmp graphs |
| topic | Dynamical Systems Logic Probability Primary 37A20, 03E15, 60K35, Secondary 37A40, 05C22, 60B99 |
| url | https://arxiv.org/abs/2211.07908 |