The Strong Novikov conjecture for finitely presented groups

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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 C∗C^*-algebra. The evaluation map is

ev ⁣:lim⁡d→∞CL∗(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∗ ⁣:lim⁡d→∞K∗(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.

References

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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