A criterion for slope 1 homological stability

TL;DR

This paper establishes a criterion linking diagonal vanishing lines in $ ext{E}_1$-homology of certain $ ext{E}_2$-algebras to slope 1 homological stability, extending previous results to an integral setting.

Contribution

It introduces a new criterion for slope 1 homological stability based on vanishing lines in $ ext{E}_1$-homology for $ ext{E}_2$-algebras, generalizing prior work.

Findings

Diagonal vanishing lines imply slope 1 stability.

Results apply to a class of $ ext{E}_2$-algebras.

Extends stability results to integral coefficients.

Abstract

We show that for nice enough $\mathbb{N}$-graded $\mathbb{E}_2$-algebras, a diagonal vanishing line in $\mathbb{E}_1$-homology of gives rise to slope $1$ homological stability. This is an integral version of a result by Kupers-Miller-Patzt.