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The maximal coarse Baum-Connes conjecture for spaces that admit an A-by-FCE coarse fibration structure

Liang Guo, Qin Wang, Chen Zhang

TL;DR

This work introduces the A-by-FCE coarse fibration structure as a broad generalization of A-by-CE and proves that the maximal coarse Baum-Connes conjecture holds for discrete spaces of bounded geometry admitting this structure. The authors develop maximal twisted Roe algebras at infinity and localization algebras, and construct Bott and Dirac-type asymptotic morphisms that realize a geometric Bott periodicity in finite dimensions. Key applications include relative expanders from Arzhantseva–Tessera and amenable-by-Haagerup box spaces, which admit A-by-FCE structures and hence satisfy the maximal coarse Baum-Connes conjecture even when they do not fibred-embed into Hilbert space. The strategy combines cutting-and-pasting reductions to coarse disjoint unions, analysis of ideals over coherent systems, Mayer–Vietoris arguments, and localization techniques to reduce twisted problems to untwisted ones, culminating in a Dirac–Bott framework that yields the main isomorphism. The results extend prior A-by-CE surjectivity/injectivity in related settings and provide tools for studying higher indices on broad classes of non-embeddable spaces.

Abstract

In this paper, we introduce a concept of A-by-FCE coarse fibration structure for metric spaces, which serves as a generalization of the A-by-CE structure for a sequence of group extensions proposed by Deng, Wang, and Yu. We prove that the maximal coarse Baum-Connes conjecture holds for metric spaces with bounded geometry that admit an A-by-FCE coarse fibration structure. As an application, the relative expanders constructed by Arzhantseva and Tessera, as well as the box spaces derived from an ``amenable-by-Haagerup'' group extension, admit the A-by-FCE coarse fibration structure. Consequently, the maximal coarse Baum-Connes conjecture holds for these spaces, which may not admit an FCE structure, i.e. fibred coarse embedding into Hilbert space.

The maximal coarse Baum-Connes conjecture for spaces that admit an A-by-FCE coarse fibration structure

TL;DR

This work introduces the A-by-FCE coarse fibration structure as a broad generalization of A-by-CE and proves that the maximal coarse Baum-Connes conjecture holds for discrete spaces of bounded geometry admitting this structure. The authors develop maximal twisted Roe algebras at infinity and localization algebras, and construct Bott and Dirac-type asymptotic morphisms that realize a geometric Bott periodicity in finite dimensions. Key applications include relative expanders from Arzhantseva–Tessera and amenable-by-Haagerup box spaces, which admit A-by-FCE structures and hence satisfy the maximal coarse Baum-Connes conjecture even when they do not fibred-embed into Hilbert space. The strategy combines cutting-and-pasting reductions to coarse disjoint unions, analysis of ideals over coherent systems, Mayer–Vietoris arguments, and localization techniques to reduce twisted problems to untwisted ones, culminating in a Dirac–Bott framework that yields the main isomorphism. The results extend prior A-by-CE surjectivity/injectivity in related settings and provide tools for studying higher indices on broad classes of non-embeddable spaces.

Abstract

In this paper, we introduce a concept of A-by-FCE coarse fibration structure for metric spaces, which serves as a generalization of the A-by-CE structure for a sequence of group extensions proposed by Deng, Wang, and Yu. We prove that the maximal coarse Baum-Connes conjecture holds for metric spaces with bounded geometry that admit an A-by-FCE coarse fibration structure. As an application, the relative expanders constructed by Arzhantseva and Tessera, as well as the box spaces derived from an ``amenable-by-Haagerup'' group extension, admit the A-by-FCE coarse fibration structure. Consequently, the maximal coarse Baum-Connes conjecture holds for these spaces, which may not admit an FCE structure, i.e. fibred coarse embedding into Hilbert space.
Paper Structure (15 sections, 16 theorems, 224 equations, 1 figure)

This paper contains 15 sections, 16 theorems, 224 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A-by-FCE coarse fibration structure
  3. The maximal coarse Baum-Connes conjecture
  4. Reduction to the coarse disjoint union of metric spaces
  5. Maximal twisted Roe algebras at infinity
  6. Preliminary
  7. Maximal twisted Roe algebras at infinity
  8. Constructions of the Bott maps $\beta$ and $\beta_{L}$
  9. Reduction to cases with non-twisted coefficients
  10. Twisted algebras supported on certain coherent systems
  11. $K$-theory of ideals of the maximal twisted Roe algebras at infinity
  12. Gluing together the pieces
  13. Proof of the main theorem
  14. Constructions of the Dirac maps $\alpha$ and $\alpha_{L}$
  15. A geometric analogue of the Bott periodicity in finite dimensions

Key Result

Theorem 1.1

Let $X$ be a discrete metric space with bounded geometry. If $X$ admits an A-by-FCE coarse fibration structure, then the maximal coarse Baum-Connes conjecture holds for $X$.

Figures (1)

  • Figure 1: Diagram of the rotation process.

Theorems & Definitions (76)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2: CWY13
  • Definition 2.3
  • Definition 2.4: A-by-FCE coarse fibration structure
  • Example 2.5
  • Example 2.6
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 66 more