The Strong Novikov conjecture for finitely presented groups

Let Γ\Gamma be a finitely presented group, and let Pd(Γ)P_d(\Gamma) be its Rips complex for d>0d>0. Write CL(Pd(Γ))ΓC^*_L(P_d(\Gamma))^\Gamma for the equivariant localization algebra and Cr(Γ)C^*_r(\Gamma) for the reduced group CC^*-algebra. The evaluation map is

ev ⁣:limdCL(Pd(Γ)Γ)Cr(Γ).ev\colon \lim_{d\to\infty} C^*_L(P_d(\Gamma)^\Gamma)\to C^*_r(\Gamma).

Strong Novikov conjecture. The induced map

ev ⁣:limdK(CL(Pd(Γ))Γ)K(Cr(Γ))ev_*\colon \lim_{d\to\infty}K_*( C^*_L(P_d(\Gamma))^\Gamma)\to K_*( C^*_r(\Gamma))

is injective. This is the group-theoretic counterpart of the coarse Novikov conjecture, expressing injectivity of the assembly map for finitely presented groups. The source does not state whether it is resolved.

Sources & referencesView supporting material

Primary source

Jinmin Wang and Bo Zhu, “Sharp bottom spectrum and scalar curvature rigidity”, arXiv:2408.08245 (2026).

Additional references

4 papers in this index state this conjecture (2007–2024). The statement above is taken from the most recent of them; the others are arXiv:1604.00464, arXiv:1603.01829, arXiv:0705.2578.

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