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

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, that is, fibered coarse embedding into Hilbert space.