Assembly maps with coefficients in topological algebras and the integral K-theoretic Novikov conjecture

We prove that any countable discrete and torsion free subgroup of a general linear group over an arbitrary field or a similar subgroup of an almost connected Lie group satisfies the integral algebraic K-theoretic (split) Novikov conjecture over \cpt and \S, where \cpt denotes the C^-algebra of compact operators and \S denotes the algebra of Schatten class operators. We introduce assembly maps with finite coefficients and under an additional hypothesis, we prove that such a group also satisfies the algebraic K-theoretic Novikov conjecture over \bar{\mathbb{Q}} and \mathbb{C} with finite coefficients. For all torsion free Gromov hyperbolic groups G, we demonstrate that the canonical algebra homomorphism \cpt[G]\map C^_r(G)\hat{\otimes}\cpt induces an isomorphism between their algebraic K-theory groups.

💡 Research Summary

The paper establishes new integral algebraic K‑theoretic Novikov conjecture results for groups with coefficients in topological algebras, specifically the C∗‑algebra of compact operators K and the algebra of Schatten class operators S. The author proves two reduction principles. The first (Theorem 0.1) shows that if a countable discrete torsion‑free group satisfies the split injectivity part of the topological K‑theoretic Novikov conjecture with complex coefficients (i.e., the Baum–Connes assembly map μ_BC is split injective), then the same group satisfies the integral algebraic K‑theoretic Novikov conjecture with coefficients in K and S. Moreover, rational injectivity of μ_BC implies rational injectivity of the algebraic assembly map μ_L^K, while μ_L^S is already known to be rationally injective for all groups.

The second reduction (Theorem 0.2) introduces assembly maps with finite coefficients ℤ/n. Assuming the group satisfies the split injective Baum–Connes conjecture and that the ℤ/n‑homology of its classifying space B G is concentrated in even degrees, the author proves that the algebraic K‑theoretic Novikov conjecture holds over ℚ and ℂ with finite coefficients. The key technical step is to ensure that the natural map from connective to non‑connective K‑homology with finite coefficients is injective; concentration in even degrees guarantees this, though more general conditions may suffice.

To connect the topological and algebraic assembly maps, the paper uses the Davis–Lück framework and shows that for H‑unital, K‑regular ℚ‑algebras the Davis–Lück assembly map coincides with the Loday assembly map. This relies on excision in algebraic K‑theory for H‑unital algebras and the vanishing of Nil‑terms for torsion‑free groups. The author also constructs K‑ünneth type spectral sequences via symmetric spectra and S‑algebra models, providing a computational tool for the domains of the assembly maps.

A significant application is proved for torsion‑free Gromov‑hyperbolic groups: the canonical algebra homomorphism K