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Equivariant coarse Novikov conjecture (injectivity of the equivariant coarse assembly map)

Establish rational injectivity of the equivariant coarse assembly map μ_X^Γ for every proper metric space X with bounded geometry and every countable discrete group Γ acting properly and isometrically on X, namely show that the map μ_X^Γ: lim_{d→∞} K_*(C^*_L(P_d(X))^Γ) → lim_{d→∞} K_*(C^*(P_d(X))^Γ) ≅ K_*(C^*(X)^Γ) is injective after tensoring with ℚ.

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Background

The paper recalls the equivariant localization-algebra approach and formulates the Novikov-type problem in the coarse, equivariant setting. In this formulation, the injectivity of the assembly map from the topological side (built from Rips complexes and localization algebras) to the analytic side (the K-theory of equivariant Roe algebras) is the coarse analogue of the classical Novikov conjecture.

The authors subsequently prove rational injectivity under specific geometric hypotheses (equivariant coarse embeddings into admissible Hilbert-Hadamard spaces), but the conjecture in full generality remains open and motivates much of the work discussed.

References

Then the equivariant coarse Novikov conjecture states as follows:

Let X be a proper metric space with bounded geometry, Γ a countable discrete group which acts properly and isometrically on X. Then the equivariant coarse assembly map induced by the evaluation map is rational injective, i.e., $$\mu_X\Gamma:\lim_{d\to\infty}K_(C^L(P_d(X)){\Gamma})\to\lim{d\to\infty}K_(C^(P_d(X){\Gamma}))\cong K_(C^(X){\Gamma})$$ is a (rational) injection.

— Hilbert-Hadamard spaces and the equivariant coarse Novikov conjecture (2411.18538 - Guo et al., 27 Nov 2024) in Section 3, “The equivariant coarse Novikov conjecture”