The Radon--Nikodym topography of acyclic measured graphs

TL;DR

This paper explores the geometric and measure-theoretic properties of acyclic measure-preserving graphs using Radon--Nikodym cocycles, extending classical dichotomies and characterizing smoothness and amenability through ends analysis.

Contribution

It introduces a topographic framework for acyclic measure-class-preserving graphs, extending the Adams dichotomy and characterizing smoothness and amenability via ends.

Findings

The number of nonvanishing ends governs amenability and smoothness.

Acyclic mcp graphs are amenable iff components have at most two nonvanishing ends.

Essentially smooth graphs have no nonvanishing ends in almost every component.

Abstract

We study locally countable acyclic measure-class-preserving (mcp) Borel graphs by analyzing their "topography" -- the interaction between the geometry and the associated Radon--Nikodym cocycle. We identify three notions of topographic significance for ends in such graphs and show that the number of nonvanishing ends governs both amenability and smoothness. More precisely, we extend the Adams dichotomy from the pmp to the mcp setting, replacing the number of ends with the number of nonvanishing ends: an acyclic mcp graph is amenable if and only if a.e. component has at most two nonvanishing ends, while it is nowhere amenable exactly when a.e. component has a nonempty perfect (closed) set of nonvanishing ends. We also characterize smoothness: an acyclic mcp graph is essentially smooth if and only if a.e. component has no nonvanishing ends. Furthermore, we show that the notion of nonvanishing ends depends only on the measure class and not on the specific measure. At the heart of our analysis lies the study of acyclic countable-to-one Borel functions. Our critical result is that, outside of the essentially two-ended setting, all back ends in a.e. orbit are vanishing and admit cocycle-finite geodesics. We also show that the number of barytropic ends controls the essential number of ends for such functions. This leads to a surprising topographic characterization of when such functions are essentially one-ended. Our proofs utilize mass transport, end selection, and the notion of the Radon--Nikodym core for acyclic mcp graphs, a new concept that serves as a guiding framework for our topographic analysis.