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Nonamenable subforests of multi-ended quasi-pmp graphs

Ruiyuan Chen, Grigory Terlov, Anush Tserunyan

TL;DR

This work extends amenability theory from the probability-measure-preserving (pmp) setting to locally finite quasi-pmp Borel graphs by tying end-geometry to the Radon–Nikodym cocycle. The authors introduce a weighted cycle-cutting procedure that yields a $\w$-maximal spanning forest and its random analogue $\RandomMF_{\w}$, generalizing the Free Minimal Spanning Forest to nonunimodular contexts and percolation. The main result proves that if a.e. components have at least $3$ $\w$-nonvanishing ends, there exists a Borel, ergodic subforest whose ends are nonempty and perfect, implying nowhere amenability; conversely, components with $2$ ends are hyperfinite. Applications to coinduced actions and cluster graphings demonstrate how these forests reveal nonamenability, end-indistinguishability, and percolation phenomena in nonunimodular graphs, connecting measured group theory with probabilistic graph theory.

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.

Nonamenable subforests of multi-ended quasi-pmp graphs

TL;DR

This work extends amenability theory from the probability-measure-preserving (pmp) setting to locally finite quasi-pmp Borel graphs by tying end-geometry to the Radon–Nikodym cocycle. The authors introduce a weighted cycle-cutting procedure that yields a -maximal spanning forest and its random analogue , generalizing the Free Minimal Spanning Forest to nonunimodular contexts and percolation. The main result proves that if a.e. components have at least -nonvanishing ends, there exists a Borel, ergodic subforest whose ends are nonempty and perfect, implying nowhere amenability; conversely, components with ends are hyperfinite. Applications to coinduced actions and cluster graphings demonstrate how these forests reveal nonamenability, end-indistinguishability, and percolation phenomena in nonunimodular graphs, connecting measured group theory with probabilistic graph theory.

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.
Paper Structure (24 sections, 37 theorems, 47 equations, 4 figures)

This paper contains 24 sections, 37 theorems, 47 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Overview of nonamenability in the pmp setting
  3. Nonamenability in the quasi-pmp setting and our main result
  4. Application to percolation theory: Free $\w$-Maximal Spanning Forest
  5. Future directions
  6. Preliminaries
  7. Graphs
  8. End spaces
  9. Weighted graphs and ends
  10. Relative weight functions (cocycles) on graphs
  11. Borel and quasi-pmp graphs and equivalence relations
  12. $\w$-maximal subforests
  13. For a connected graph with enough trifurcation vertices
  14. For general connected graphs
  15. For Borel and quasi-pmp graphs
  16. ...and 9 more sections

Key Result

theorem 1

The connectedness relation $\#E_G$ of an ergodic pmp graph $G$ whose a.e. component has $\ge 3$ ends is of cost $>1$, hence $\#E_G$ is nonamenable. In fact, there is an ergodic subforest $F \subseteq \#E_G$ of cost $>1$ witnessing the nonamenability of $\#E_G$.

Figures (4)

  • Figure 1: A weighted tree with set of nonvanishing ends $F_\sigma$ but not $G_\delta$
  • Figure 2: Windmill graph described in \ref{['ex:windmill']}
  • Figure 3: This is an illustration of the proof of \ref{['thm:qpmp-ends-isol']}, where the large dots represent the maximal disjoint Borel family $F$ of $\w$-trifurcations.
  • Figure 4: The maps between the free pmp action $\Gamma \acts (X,\mu)$, its quotient $(X_o,\mu_o)$ by the stabilizer $\Gamma_o$ on which the Borel graph $G$ is defined, and the space of graphs on $V$ on which $\Gamma$ acts by shift.

Theorems & Definitions (107)

  • theorem 1: Gaboriau--Ghys; 2000
  • theorem 2: Tserunyan--Tucker-Drob; 2025
  • theorem 3: see \ref{['thm:mf-qpmp']}
  • remark 1
  • remark 2
  • remark 3
  • proposition 1
  • theorem 4
  • definition 1
  • definition 2
  • ...and 97 more