# Linear and quadratic ranges in representation stability

## TL;DR

This paper establishes improved linear and quadratic stable ranges for the cohomology of configuration spaces and homology of congruence subgroups, advancing the understanding of representation stability in these areas.

## Contribution

It proves two general spectral sequence results for $ extbf{FI}$-modules that significantly improve stable range bounds in key stability theorems.

## Key findings

- Linear stable range for configuration space cohomology
- Quadratic stable range for homology of congruence subgroups
- Verification of a conjecture of Djament up to an additive constant

## Abstract

We prove two general results concerning spectral sequences of $\mathbf{FI}$-modules. These results can be used to significantly improve stable ranges in a large portion of the stability theorems for $\mathbf{FI}$-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 Djament.