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Borel asymptotic dimension of the Roller boundary of finite dimensional CAT(0) cube complexes

Koichi Oyakawa

TL;DR

The paper establishes that for any countable finite-dimensional CAT(0) cube complex $X$ of dimension $D$, the Borel asymptotic dimension of the Roller boundary median graph is at most $D$ and its connected-component relation is smooth. It achieves this by developing a Borel extension of Wright's construction: gluing cubes to a Borel median graph to form a standard Borel extended space $\widetilde{X}$, embedding into a larger complex $C(G)$, and proving Borel measurability for controlled colorings, the interpolation of quotients, and Wright projections. A key innovation is the componentwise, Borel-consistent implementation of Wright's interpolation and projection across all $G$-components, yielding a global Lipschitz cobornologous map into a quotient, and hence a finite Borel asymptotic dimension bound. This framework also shows that the Sageev-Roller duality extends to a Borel setting, enabling a smooth quotient while preserving median-graph structure. Consequently, the Roller boundary of a countable finite-dimensional CAT(0) cube complex exhibits a relatively tame Borel complexity, contrasting with more intricate Borel-tree examples, and yielding a concrete dimension bound tied to the underlying CAT(0) dimension.

Abstract

We prove that for any countable finite dimensional CAT(0) cube complex, the Borel median graph on its Roller compactification has the Borel asymptotic dimension bounded from above by its dimension.

Borel asymptotic dimension of the Roller boundary of finite dimensional CAT(0) cube complexes

TL;DR

The paper establishes that for any countable finite-dimensional CAT(0) cube complex of dimension , the Borel asymptotic dimension of the Roller boundary median graph is at most and its connected-component relation is smooth. It achieves this by developing a Borel extension of Wright's construction: gluing cubes to a Borel median graph to form a standard Borel extended space , embedding into a larger complex , and proving Borel measurability for controlled colorings, the interpolation of quotients, and Wright projections. A key innovation is the componentwise, Borel-consistent implementation of Wright's interpolation and projection across all -components, yielding a global Lipschitz cobornologous map into a quotient, and hence a finite Borel asymptotic dimension bound. This framework also shows that the Sageev-Roller duality extends to a Borel setting, enabling a smooth quotient while preserving median-graph structure. Consequently, the Roller boundary of a countable finite-dimensional CAT(0) cube complex exhibits a relatively tame Borel complexity, contrasting with more intricate Borel-tree examples, and yielding a concrete dimension bound tied to the underlying CAT(0) dimension.

Abstract

We prove that for any countable finite dimensional CAT(0) cube complex, the Borel median graph on its Roller compactification has the Borel asymptotic dimension bounded from above by its dimension.
Paper Structure (16 sections, 24 theorems, 91 equations, 3 figures)

This paper contains 16 sections, 24 theorems, 91 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. CAT(0) cube complex
  4. Wright's construction
  5. Controlled colorings
  6. Interpolation of a quotient map
  7. The Wright's projection theorem
  8. Descriptive set theory
  9. Gluing cubes to a Borel median graph
  10. Borel version of the Sageev-Roller duality
  11. Glueing cubes to a Borel median graph (smooth case)
  12. Borel measurability of Wright's construction
  13. Controlled colorings
  14. Interpolation of a quotient map
  15. The projection theorem
  16. ...and 1 more sections

Key Result

Theorem 1.1

For any countable CAT(0) cube complex $X$ of dimension $n \in \mathbb{N}\cup\{0\}$, the Borel median graph on the Roller boundary $\mathfrak{R}(X)$ of $X$ has Borel asymptotic dimension at most $n$ and its Borel equivalence relation consisting of connected components is smooth.

Figures (3)

  • Figure 1: Triangulation $T$ of $[0,1]^2$
  • Figure 2: Triangulation $T_1$ of $[0,1]^2$
  • Figure 3: The shaded region is the intersection of $\mathcal{S}_1$ and a 2-simplex in $\mathcal{T}$

Theorems & Definitions (122)

  • Theorem 1.1
  • Proposition 1.2
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • ...and 112 more