Representation stability for filtrations of Torelli groups

TL;DR

This paper proves that certain algebraic structures related to Torelli groups exhibit uniform representation stability, confirming conjectures and extending stability results to their filtrations.

Contribution

It establishes uniform representation stability for filtrations of Torelli groups and proves two conjectures of Church and Farb regarding their quotients.

Findings

Finitely generated rational VIC and SI modules are uniformly stable.

Quotients of the lower central series of Torelli groups are stable.

Johnson filtrations also exhibit stability.

Abstract

We show, finitely generated rational $\mathsf{VIC}_{\mathbb Q}$-modules and $\mathsf{SI}_{\mathbb Q}$-modules are uniformly representation stable and all their submodules are finitely generated. We use this to prove two conjectures of Church and Farb, which state that the quotients of the lower central series of the Torelli subgroups of $\mathrm{Aut}(F_n)$ and $\mathrm{Mod}(\Sigma_{g,1})$ are uniformly representation stable as sequences of representations of the general linear groups and the symplectic groups, respectively. Furthermore we prove an analogous statement for their Johnson filtrations.