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

TL;DR

This paper introduces a new A-by-FCE coarse fibration structure for metric spaces and proves the maximal coarse Baum-Connes conjecture for spaces with this structure, including certain expanders and box spaces.

Contribution

It generalizes the A-by-CE structure to A-by-FCE and establishes the conjecture for these broader classes of spaces.

Findings

Maximal coarse Baum-Connes conjecture holds for spaces with A-by-FCE structure.

Relative expanders and certain box spaces admit A-by-FCE structure.

The conjecture holds even when spaces do not admit fibred coarse embedding into Hilbert space.

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.