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Linear and quadratic ranges in representation stability. (English) Zbl 1392.15030

Summary: We prove two general results concerning spectral sequences of FI-modules. These results can be used to significantly improve stable ranges in a large portion of the stability theorems for FI-modules currently in the literature. We work this out in detail for the cohomology of configuration spaces where we prove a linear stable range and the homology of congruence subgroups of general linear groups where we prove a quadratic stable range. Previously, the best stable ranges known in these examples were exponential. Up to an additive constant, our work on congruence subgroups verifies a conjecture of A. Djament [Méthodes fonctorielles pour l’étude de l’homologie stable des groupes. Nantes: Université de Nantes (Habil.) (2017)].

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)
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