Generalized opposite-category Farrell–Jones conjecture for td-groups

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Let GG be a td-group, let B\mathcal{B} be a Hecke category with GG-support, and let P(B,G)\mathcal{P}(\mathcal{B},G) be the set of primes belonging to P(G)\mathcal{P}(G) but not to P(B)\mathcal{P}(\mathcal{B}). Let Or⁡Cop(G)\operatorname{Or}_{{\mathcal{C}\hspace{-1pt}\mathrm{op}}}(G) denote the relevant orbit category, and consider the Cop{\mathcal{C}\hspace{-1pt}\mathrm{op}}-assembly map

hocolim⁡G/U∈Or⁡Cop(G)K(B[G/U])⟶K(B[G/G])≃K(B).\operatorname*{hocolim}_{G/U\in\operatorname{Or}_{{\mathcal{C}\hspace{-1pt}\mathrm{op}}}(G)}\mathbf{K}(\mathcal{B}[G/U])\longrightarrow \mathbf{K}(\mathcal{B}[G/G])\simeq\mathbf{K}(\mathcal{B}).

Generalized opposite-category Farrell–Jones conjecture. The td-group GG satisfies this conjecture if, for every Hecke category with GG-support, the displayed Cop{\mathcal{C}\hspace{-1pt}\mathrm{op}}-assembly map is a P(B,G)\mathcal{P}(\mathcal{B},G)-equivalence. The statement is presented as a version of the Farrell–Jones conjecture for Hecke categories and the excerpt gives no resolution status.

References

Primary source

Wolfgang Lueck, “Relative assembly maps and the K-theory of Hecke algebras in prime characteristic”, arXiv:2402.05278 (2024).

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