Nonamenable subforests of multi-ended quasi-pmp graphs

TL;DR

This paper demonstrates the almost everywhere nonamenability of certain complex graphs by constructing special Borel subforests and introduces a novel weighted cycle-cutting algorithm and a random spanning forest to analyze nonunimodularity in percolation theory.

Contribution

It introduces a weighted cycle-cutting algorithm for constructing nonamenable subforests and a random spanning forest generalizing the Free Minimal Spanning Forest for nonunimodular graphs.

Findings

Proves a.e. nonamenability of specific quasi-pmp graphs.

Constructs Borel subforests with multiple nonvanishing ends.

Develops a new random spanning forest model.

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.