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October 12, 2025Open Access

The twisted coarse Baum--Connes conjecture and relative hyperbolic groups

Key Points

The twisted coarse Baum-Connes conjecture shows permanence properties across coarse equivalences and subspaces.
We formulate the conjecture relative to stable coarse algebras, indicating the importance of bounded geometry.
The application of this theory reveals that a group $G$ meets the conjecture if all its relative subgroups do.
This work contributes to the understanding of higher index theory in the context of relative hyperbolicity.

Abstract

In this paper, we introduce a notion of stable coarse algebras for metric spaces with bounded geometry, and formulate the twisted coarse Baum--Connes conjecture with respect to stable coarse algebras. We prove permanence properties of this conjecture under coarse equivalences, unions and subspaces. As an application, we study higher index theory for a group G that is hyperbolic relative to a finite family of subgroups \H₁, H₂, , HN\. We prove that G satisfies the twisted coarse Baum--Connes conjecture with respect to any stable coarse algebra if and only if each subgroup Hᵢ does.

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Cite This Study

Deng et al. (Mon,) studied this question.

synapsesocial.com/papers/68ec1be02b8fa9b2b78ad2bd https://doi.org/https://doi.org/10.48550/arxiv.2509.06876

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