The Radon--Nikodym topography of acyclic measured graphs
Anush Tserunyan, Robin Tucker-Drob
TL;DR
This work develops a Radon–Nikodym topography framework for locally countable acyclic measure-class-preserving Borel graphs, connecting geometric ends with the Radon–Nikodym cocycle $ ho$. It extends Adams’s dichotomy from the pmp setting to mcp by replacing the end-count with the number of nonvanishing ends, yielding smoothness when all ends vanish, amenability when only one or two nonvanishing ends occur, and nowhere amenability when nonvanishing ends form a nonempty perfect set; crucially, nonvanishing-end behavior is invariant under change of measure class. A central construct is the Radon–Nikodym core, which isolates the topographically essential part of a treeing, and its vertex-core variants, whose properties drive the mass-transport and pruning arguments used to classify essential end-structures. The paper characterizes essential one-endedness through several equivalent conditions and proves cocycle-finiteness of back-geodesics and cocycle-vanishing of back-ends outside the essentially two-ended regime, with representative examples and a robust theory of end spaces. These results enrich the structural understanding of amenable subrelations in the mcp setting and provide tools for analyzing end-based obstructions to nonamenability within treeable, measure-class-preserving dynamics.
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.