Geometric Banach property (T) is defined via Banach representations of Roe algebras, shown to be coarsely invariant, equivalent to Kazhdan projections in Banach-Roe algebras, and linked to Banach property (T) of limit groups and box spaces.
The maximal coarse Baum-Connes conjecture for spaces that admit an A-by-FCE coarse fibration structure
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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.
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Geometric Banach property (T) for metric spaces via Banach representations of Roe algebras
Geometric Banach property (T) is defined via Banach representations of Roe algebras, shown to be coarsely invariant, equivalent to Kazhdan projections in Banach-Roe algebras, and linked to Banach property (T) of limit groups and box spaces.