Representation stability and outer automorphism groups

TL;DR

The paper constructs the abelian framework $\mathcal{A}\mathcal{U}$ to encode families of outer-automorphism representations indexed by finite groups, and develops a comprehensive homological toolkit (generators, adjunctions, monoidal structure) to analyze finiteness, stability, and decomposition phenomena. It proves new locally noetherianity results for natural subfamilies (e.g., cyclic $p$-groups, finite abelian $p$-groups) and establishes central and representation stability in this generalized setting, extending classical FI/VI results. The work also links to rational global homotopy theory, showing that compact rational global spectra yield invariants with central stability, thus providing a bridge between representation theory of families of groups and global homotopy-theoretic objects. Collectively, the results give a robust structural theory for families of Out($G$)-representations, with precise conditions under which stability, finiteness, and decomposition properties hold and with explicit descriptions of projective/injective objects and their behavior under subcategory changes.

Abstract

In this paper we study families of representations of the outer automorphism groups indexed on a collection of finite groups $\mathcal{U}$. We encode this large amount of data into a convenient abelian category $\mathcal{A}\mathcal{U}$ which generalizes the category of VI-modules appearing in the representation theory of the finite general linear groups. Inspired by work of Church--Ellenberg--Farb, we investigate for which choices of $\mathcal{U}$ the abelian category is locally noetherian and deduce analogues of central stability and representation stability results in this setting. Finally, we show that some invariants coming from rational global homotopy theory exhibit representation stability.

Theorems & Definitions (272)

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