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Decomposition of Borel graphs and cohomology

Hiroki Ishikura

TL;DR

The work establishes a cohomological criterion for decomposing locally finite Borel graphs, paralleling Dunwoody's accessibility for groups. By introducing the $R_G$-cohomology and in particular $H^1(G,R_G)$, the authors show that finite generation of this cohomology forces a structural decomposition $G' o Tigast H$ with $T$ acyclic and $H$ uniformly at most one-ended, via injective Borel quasi-isometries. They prove an equivalent formulation: $ ext{cd}_R(G)\le 1$ iff $G$ is Lipschitz equivalent to a Borel acyclic graph, yielding a new proof of results on trees-like behavior in Borel graphs and enabling a quasi-isometric rigidity phenomenon for acyclicity. The methods hinge on structure trees built from Borel treesets, and on a detailed analysis of $H^1(G,R_G)$ through cutsets, W,G, and Z_G, linking cohomology to end-structure and global graph geometry with potential impacts on descriptive set theory and geometric group theory.

Abstract

We give a cohomological criterion for certain decomposition of Borel graphs, which is an analog of Dunwoody's work on accessibility of groups. As an application, we prove that a Borel graph $(X,G)$ with uniformly bounded degrees of cohomological dimension one is Lipschitz equivalent to a Borel acyclic graph on $X$. This gives a new proof of a result of Chen-Poulin-Tao-Tserunyan on Borel graphs with components quasi-isometric to trees.

Decomposition of Borel graphs and cohomology

TL;DR

The work establishes a cohomological criterion for decomposing locally finite Borel graphs, paralleling Dunwoody's accessibility for groups. By introducing the -cohomology and in particular , the authors show that finite generation of this cohomology forces a structural decomposition with acyclic and uniformly at most one-ended, via injective Borel quasi-isometries. They prove an equivalent formulation: iff is Lipschitz equivalent to a Borel acyclic graph, yielding a new proof of results on trees-like behavior in Borel graphs and enabling a quasi-isometric rigidity phenomenon for acyclicity. The methods hinge on structure trees built from Borel treesets, and on a detailed analysis of through cutsets, W,G, and Z_G, linking cohomology to end-structure and global graph geometry with potential impacts on descriptive set theory and geometric group theory.

Abstract

We give a cohomological criterion for certain decomposition of Borel graphs, which is an analog of Dunwoody's work on accessibility of groups. As an application, we prove that a Borel graph with uniformly bounded degrees of cohomological dimension one is Lipschitz equivalent to a Borel acyclic graph on . This gives a new proof of a result of Chen-Poulin-Tao-Tserunyan on Borel graphs with components quasi-isometric to trees.

Paper Structure

This paper contains 18 sections, 43 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Main theorem
  3. Borel graphs of cohomological dimension one
  4. Preliminaries
  5. Terminology of graph theory
  6. Basic homological algebra
  7. Structure trees
  8. Construction of trees
  9. Borel cutsets of Borel graphs
  10. Structure trees for Borel graphs
  11. Cohomology of graphs
  12. Computaion of cohomology
  13. Cohomological dimension
  14. Structure of H1(G,RG)
  15. Cutsets and H1(G,RG)
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $R$ be a non-zero commutative ring. Then a finitely generated group $\Gamma$ is accessible if and only if the cohomology group $\textup{H}^1(\Gamma,R\Gamma)$ is finitely generated as a right $R\Gamma$-module.

Theorems & Definitions (113)

  • Definition 1.1
  • Theorem 1.2: Dun1
  • Theorem 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.8
  • Example 1.9
  • Definition 1.10
  • Remark 1.11
  • ...and 103 more